Research ArticleThree-Dimensional Vehicle-to-Vehicle ChannelModeling with Multiple Moving Scatterers
Derong Du Xiaoping Zeng Xin Jian LijuanMiao and HaoboWang
College of Communication Engineering Chongqing University Chongqing 400044 China
Correspondence should be addressed to Xin Jian jianxin_zg163com
Received 30 November 2016 Revised 9 April 2017 Accepted 3 May 2017 Published 10 July 2017
Academic Editor Barbara M Masini
Copyright copy 2017 Derong Du et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Connected vehicles have received much attention in recent years due to their significant societal benefit and commercialvalue However a suitable channel model for vehicle-to-vehicle (V2V) communications is difficult to build due to the dynamiccommunication environment In this paper a three-dimensional (3D) geometrical propagation model that includes line-of-sight(LoS) single bounced (SB) and multiple bounced (MB) rays is proposed Each of multiple scatterers in the model is movingwith a random velocity in a random direction Based on the geometrical propagation model a generalized 3D reference modelfor narrowband multiple-input-multiple-output (MIMO) V2V multipath fading channels is developed The corresponding space-time correlation functions (ST-CFs) time correlation functions (T-CFs) and space correlation functions (S-CFs) are analyticallyinvestigated and numerically simulated in terms of various factors Several notable ST-CFs for V2V and fixed-to-mobile (F2M)communications become the special cases of ST-CFs of the proposed model by adjusting the corresponding channel parametersFinally the theoretical results of the space-Doppler power spectral density (SD-PSD) are compared with the available measureddataThe close agreements between the theoretical andmeasured SD-PSD curves confirm the utility and generality of the proposedmodel
1 Introduction
Connected vehicles have the potential to improve the safetyand efficiency of the automobile transportation [1] andthey are expected to be a pillar of a smart society and torevolutionize the way people move Different from the con-ventional fixed-to-mobile (F2M) cellular systems vehicle-to-vehicle (V2V) is a kind of themobile-to-mobile (M2M) com-munication which allows both the transmitter and receiverto be in motion [2] Suitable channel models and channelcharacterizations are absolutely essential for successful designof V2V systems where the quality of wireless links betweenvehicles can vary greatly and rapidly from one environmentto another as one or both ends move [3]
Many M2M channel models have been proposed invarious ways some of which are summarized in [4] andthese models have important reference values in modelingthe V2V channel The models in M2M communicationenvironment can be traced back from two-dimensional (2D)[5ndash7] and three-dimensional (3D) [8ndash11] fixed scattering
models to 2D [12ndash17] and 3D [18ndash20] moving scatteringmodels The M2M channel models with the assumption ofstationary scatterers have been proposed in [5ndash11] Howevermoving scatterers are unavoidable in M2M communicationsMoving foliage walking pedestrians and passing vehiclesare only a few examples of scatterers in motion which canbe observed in most of the real-world radio propagationenvironments [14 21] The impact of moving scatterers onchannel characteristics has been studied in [22ndash25] for 2DF2M cellular and fixed-to-fixed (F2F) communications withline-of-sight (LoS) and single bounced (SB) rays
Recently modeling of M2MV2V channels in the pres-ence of moving scatterers has been discussed in [12ndash20] Anonstationarymultiple-input-multiple-output (MIMO) V2Vchannel model based on the geometrical street model wasderived and the impact of fixed and moving clusters ofscatterers on the channel statistics was studied in [12 15] Asingle-input-single-output (SISO) V2V channel model wasderived assuming a typical propagation scenario in whichthe local scatterers without specific constraints of positions
HindawiMobile Information SystemsVolume 2017 Article ID 7231417 14 pageshttpsdoiorg10115520177231417
2 Mobile Information Systems
moved with random velocities in random directions in [1314] The impact of mobile and stationary scattering clusterson the Doppler spectrum was investigated for widebandV2V communication channels in an urban canyon oncomingenvironment in [16 17] All previously reported models in[12ndash17] are 2D propagation models in which the rays are LoSSB or double bounced (DB) However this 2D assumptiondoes not seem to be appropriate for many communicationscenarios for example urbanV2V communications in whichthe transmitter and receiver antenna arrays are often locatedin close proximity to or lower than the surrounding scatterers
A 3D geometrical propagation model that included bothstationary and moving scatterers around the transmitter andreceiver was proposed in [18] and this work was extendedto wideband channels in [19] The models in [18 19] tookinto account that the rays in V2V channels could be bothSB and DB however they imposed some constraints on theposition of the local scatterers and assumed that the trans-mitter receiver and scatterers were in motion with constantvelocities in 2D spaceTherefore themodels in [18 19] cannotfully capture the 3D spatial information A preliminaryinvestigation of the impact of multiple moving scatterers onthe Doppler spectrum in 3D V2V communication scenarioswas presented in [20] However the work of [20] did notinvestigate and take into account the space-time correlationfunctions (ST-CFs) scatterer velocity distributions angledistributions and so on
It is unavoidable in V2V communications that the raysfrom the transmitter to receiver are multiple bounced (MB)by moving scatterers especially in environments with high-density scatterers for example urban area However to thebest of the authorsrsquo knowledge the model and statisticalproperties of 3D V2V or M2M channels in the presence ofmultiple moving scatterers have been investigated rarely sofarThis paper strives to alleviate the current lack of analyticalstudies by investigating the model and statistical propertiesof a narrowband 3D MIMO V2V channel in which the localmultiple scatterers are moving with random velocities inrandom directionsThe proposed referencemodel constructsthe channel impulse response as a combination of LoS andMB components and SB is considered as a special case ofMB Different from the assumption of all non-LoS (NLoS)path gains having the same size in [13 14] and meanwhilecompared with models in [18 19] SB DB and other MB rayshavemore flexible and precise power weights in the proposedmodel From the reference model the corresponding ST-CFs time correlation functions (T-CFs) and space corre-lation functions (S-CFs) are analytically investigated andnumerically simulated in terms of various factors such as themaximum bounces scattering forms scatterer velocity dis-tributions and spacing between adjacent antenna elementsFinally the theoretical space-Doppler power spectral density(SD-PSD) results with SD-PSDs in [18] and measured datain [26 27] are compared The close agreements betweenthe analytically and empirically obtained SD-PSDs confirmthe utility and generality of the proposed model and showthe importance of including multiple moving scatterers inpropagation models The contributions and novelties of thispaper are summarized as follows
(i) We propose a generalized geometrical model and ageneralized referencemodel that include LoS SB andMB rays between the transmitter and receiver for3D narrowband MIMO V2V communications Theproposed model can be adapted to a wide varietyof scenarios for example F2F F2M and M2M withcertain bounces by adjusting model parameters
(ii) To the best of the authorsrsquo knowledge the impactof raysrsquo different maximum bounces caused by themultiple moving scatterers on the ST-CFs and T-CFsis deeply investigated for the first time in V2V orM2M communication environments
(iii) The ST-CFs obtained from the proposed referencemodel are relatively generalized and can be reduced toseveral existing ST-CFs and T-CFs for example thosein [13 14 23 28 29]
The remainder of the paper is organized as followsSection 2 describes the geometrical propagation model andpresents a 3D reference model for narrowband MIMO V2Vchannels in the presence of multiple moving scatterersSection 3 derives the channel ST-CFs T-CFs S-CFs and SD-PSDs for different parametric sets Numerical results andcomparison between the theoretical results and measureddata are provided in Section 4 Finally Section 5 providessome concluding remarks
2 3D Geometrical PropagationModel and Reference Model
21 Geometrical PropagationModel This paper considers theMIMO V2V communication links between transmitter 119879119883and receiver 119877119883 as shown in Figure 1 The radio propagationenvironment is characterized by 3Dmultiple moving scatter-ing with either LoS or NLoS conditions between 119879119883 and 119877119883TheMBwaves emitted from the 119902th antenna element of119879119883 atan angle of departure (AOD) reach the 119901th antenna elementof 119877119883 at an angle of arrival (AOA) after being multiplescattered by the local moving scatterers
For ease of reference in this geometrical propagationmodel the main parameters are summarized in Table 1 andthe main assumptions are summarized as follows
(i) The received signal power consists of LoS SB andMBscattering components with the corresponding powerweights
(ii) Both 119879119883 and 119877119883 are equipped with uniform lineararrays consisting of omnidirectional antenna ele-ments
(iii) 119879119883 and 119877119883 move with constant velocities in 3D spacedescribed by the fixed azimuth angles and elevationangles
(iv) Each of the local scatterers on the links between 119879119883and 119877119883 for the multiple bounces is in motion with arandom velocity in a random direction in 3D space
Mobile Information Systems 3
y
z
x
D
Multiple scattering
LoS
nth path
12
12
훽q
TX
훽T
훽R
훽p훼q
훼T
훼R
훼p
훽Tpq
훽ipq
훽Rpq
훼Tpq
훼ipq
훼Rpq
SiS1 Sm
i
R
T
pq
RX
Figure 1 Geometrical propagation model with LoS and MB rays for the 3D MIMO V2V communication scenario Empty circles representthe antenna elements and solid squares represent the moving scatterers
Table 1 Definition of the parameters used in the geometrical model
Parameters Definition Attributes119878119894 The 119894th moving scatterer for the multiple bounces Symbol119863 The distance between 119879119883 and 119877119883 Deterministic119902 119901 The antenna element identifier of 119879119883 and 119877119883 respectively Deterministic119876 119875 The number of antenna elements of 119879119883 and 119877119883 respectively Deterministic119889119879 119889119877 The spacing between two adjacent antenna elements of 119879119883 and 119877119883 respectively Deterministic120572119902 120573119902 The azimuth angle and elevation angle of 119879119883rsquos antenna array respectively Deterministic120572119901 120573119901 The azimuth angle and elevation angle of 119877119883rsquos antenna array respectively Deterministic120572119879119901119902 120573119879119901119902 The azimuth angle and elevation angle of AOD respectively Statistical120572119877119901119902 120573119877119901119902 The azimuth angle and elevation angle of AOA respectively StatisticalV119879 V119877 The velocities of 119879119883 and 119877119883 respectively DeterministicV119894119901119902 The velocity of 119878119894 Statistical120572119879 120573119879 The azimuth angle and elevation angle of V119879 respectively Deterministic120572119877 120573119877 The azimuth angle and elevation angle of V119877 respectively Deterministic120572119894119901119902 120573119894119901119902 The azimuth angle and elevation angle of V119894119901119902 respectively Statistical
(v) The geometrical propagation model does not imposespecific constraints on the position of the local mov-ing scatterers like [13 14] Owing to high path loss weneglect the energy contribution of remote scatterers
22 Reference Model We can observe from Figure 1 that thecomplex faded envelope of the link between the 119902th antennaelement of 119879119883 and the 119901th antenna element of 119877119883 can bewritten as a superposition of LoS and MB components thatisℎ119901119902 (119905) = ℎLoS119901119902 (119905) + ℎMB
where 120588119901119902 119891119901119902120588 120579119901119902120588 119891119902119901119902120588 and 119891119901119901119902120588 denote the path gainDoppler shift phase shift frequency shift for 119902th antennaelement of119879119883 and frequency shift for119901th antenna element of119877119883 of ℎLoS119901119902 (119905) respectively119898 denotes the amount of bounces119872 is themaximumof119898119901119898 and119873119898 denote the power weightand path number of ℎ119898119901119902(119905) respectively 119888119901119902119899119898 119891119901119902119899119898 120579119901119902119899119898 119891119902119901119902119899 and 119891119901119901119902119899 denote the path gain Doppler shift phaseshift frequency shift for 119902th antenna element of 119879119883 andfrequency shift for 119901th antenna element of 119877119883 of the 119899119898thpath in ℎ119898119901119902(119905) The LoS component ℎLoS119901119902 (119905) can be describedby a complex sinusoid and (2) is an extension of the (317)in [30] ℎ119898119901119902(119905) denotes the NLoS component with119898 bounced
4 Mobile Information Systems
(119898B) rays and it is an extension of the (312) in [30] Notethat MB component ℎMB
119901119902 (119905) of the channel impulse responseconsists of 119872 clusters of rays reflected 119898 isin 1 2 3 119872times from moving scatterers
The channel gains (120588119901119902 119888119901119902119899119898
221 Channel Gains The central limit theorem states thatℎ119898119901119902(119905) equals a complex valued Gaussian random processwith zero mean and variance 21205902119898 = Varℎ119898119901119902(119905) =lim119873
119898rarrinfinsum119873119898119899
119898
119864[1198882119901119902119899119898
] 119901119898 in (3) denotes the power weightof the 119898th clusters of rays and sum119872119898=1 119901119898 = 1 The channelgain of ℎ119901119902(119905) is normalized ie 2sum119872119898=1 1199011198981205902119898 + 1205882119901119902 = 1and the Rice factor can be denoted as119870 = 12058821199011199022sum119872119898=1 1199011198981205902119898These parameters have to be either set during simulations orestimated from measurements
222 Doppler Shifts and Frequency Shifts The frequencyshifts 119891119902119901119902120588 119891119901119901119902120588 119891119902119901119902119899 and 119891119901119901119902119899 depend on the differenceof the propagation distance (TPD) changes between ℎ119901119902(119905)and ℎ11(119905) On the other hand the Doppler shifts 119891119901119902120588 and119891119901119902119899
119898
depend on the geometrical relation between directionsof movement of 119879119883 119877119883 and multiple moving scatterers andthe directions of AOD and AOA For max(119889119879 119889119877) ≪ 119863 boththe AOD and AOA of LoS rays are approximately equal tozero Appendix A shows that119891119902119901119902120588119891119901119901119902120588119891119902119901119902119899119891119901119901119902119899 and119891119901119902120588are respectively
119891119902119901119902120588 = (119902 minus 1) 1198891198791198910119888 cos120572119902 cos120573119902 (5)
119891119901119901119902120588 = (119901 minus 1) 1198891198771198910119888 cos120572119901 cos120573119901 (6)
119891119902119901119902119899 = (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(7)
119891119901119901119902119899 = (119901 minus 1) 1198891198771198910119888 [cos (120572119901 minus 120572119877119901119902) cos120573119901 cos120573119877119901119902+ sin120573119901 sin120573119877119901119902]
where 119875AOD119894119901119902 and 119875AOA119894119901119902 (119894 = 1 2 3 119898) are119875AOD119894119901119902 = cos (120572119894119901119902 minus 120572119879119901119902) cos120573119894119901119902 cos120573119879119901119902
+ sin120573119894119901119902 sin120573119879119901119902119875AOA119894119901119902 = cos (120572119894119901119902 minus 120572119877119901119902) cos120573119894119901119902 cos120573119877119901119902
Note that if 120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 and 119898 = 1(10) equals (7) in [13] and (5) in [14] regardless of the plus orminus signsThe differences among these plus or minus signsare caused by the different forms of anglesrsquo expression
223 Phases The phase shift 120579119901119902120588 can be assumed to beconstant [30] The phase shift 120579119901119902119899
119898
consists of the phasechange caused by the interaction of the transmitted signalwith the scatterers and the phase change caused by theTPD between the first and the last scatterers Without lossof generality we can assume that the phases 120579119901119902119899
119898
(119898 =1 2 3 119872) are independent random variables Here it isassumed that they are uniformly distributed on the interval[0 2120587) and independent of any other random variable
Mobile Information Systems 5
3 Space-Time Correlation Function andSpace-Doppler Power Spectral Density
Using the referencemodel described in Section 2 we can nowderive the key temporal and spatial characteristics of MIMOV2V narrowband multipath fading channels with the localmultiple moving scatters
31 Space-Time Correlation Function The normalized ST-CFbetween two complex faded envelopes ℎ119901119902(119905) and ℎ119901119902(119905) isdefined as
where (∙)lowast denotes the complex conjugate operation 119864(∙) isthe statistical expectation operator 119901 119901 isin 1 2 3 119875 and119902 119902 isin 1 2 3 119876 The normalized T-CF can be obtainedif 119901 = 119901 and 119902 = 119902 in (15) The normalized space correlationfunction (S-CF) can be obtained by setting 120591 to zero in (15)
Since ℎ1119901119902(119905) ℎ2119901119902(119905) ℎ119872119901119902(119905) and ℎLoS119901119902 (119905) are indepen-dent of each other (15) can be simplified to
where 119877119898119901119902119901119902(119889119879 119889119877 120591) and 119877LoS119901119902119901119902(119889119879 119889119877 120591) denote the nor-malized ST-CFs of the119898B and LoS components respectivelyand they are defined as
311 ST-CF of LoS Component By substituting (2) into (18)the expression for the ST-CF of the LoS component can bewritten as
119877LoS119901119902119901119902 (119889119879 119889119877 120591)= 120588119901119902120588119901119902119864 exp 119895 [2120587119905 (119891119901119902120588 minus 119891119901119902120588) + 2120587119891119901119902120588120591+ 2120587 (119891119902119901119902120588 + 119891119901119901119902120588 minus 119891119902119901119902120588 minus 119891119901119901119902120588) + 120579119901119902120588minus 120579119901119902120588]
(19)
For max(119889119879 119889119877) ≪ 119863 we assume 120588119901119902 = 120588119901119902 and 120579119901119902120588 =120579119901119902120588 Then (19) can be written as
Δ119891120588 = 119891119902119901119902120588 + 119891119901119901119902120588 minus 119891119902119901119902120588 minus 119891119901119901119902120588 (21)
By substituting (5) and (6) into (21) Δ119891120588 can be written as
Δ119891120588 = 1198910119888 [(119902 minus 119902) 119889119879 cos120572119902 cos120573119902+ (119901 minus 119901) 119889119877 cos120572119901 cos120573119901]
(22)
312 ST-CF of 119898119861 Component By substituting (4) into (17)the expression for the ST-CF of the 119898B component can bewritten as
times 119864 exp 119895 [2120587 (119891119901119902119899119898
minus 119891119901119902119899119899
) 119905 + 2120587119891119901119902119899119898
120591 + 2120587 (119891119902119901119902119899 + 119891119901119901119902119899 minus 119891119902119901119902119899 minus 119891119901119901119902119899) + 120579119901119902119899119898 minus 120579119901119902119899119899] (23)
Under the assumption that 120579119901119902119899119898
and 120579119901119902119899119899
are uniformlydistributed on the interval [0 2120587) and independent of eachother 119864exp[119895(120579119901119902119899
119898
minus120579119901119902119899119899
)] equals 1 It is assumed that allthe path gains of the119898B component have the same size thatis
= 119891119902119901119902119899 + 119891119901119901119902119899 minus 119891119902119901119902119899 minus 119891119901119901119902119899 (26)
It is assumed that AOD and AOA of ℎ119898119901119902(119905) and ℎ119898119901119902(119905) areindependent and identically distributed (iid) By substitut-ing (7) and (8) into (26) Δ119891119899
Since the number of local scatterers in the referencemodel is infinite the parameters 120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902 120572119894119901119902and 120573119894119901119902 can be seen as continuous random variables withcorresponding probability density functions (PDFs) Thenthe ST-CF of the 119898B component (25) can be writtenas
Now the complete expression of (16) can be obtainedby substituting (20) and (28) into (16) 119901(120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902)describes the joint distribution of AOD and AOA and itcan be optionally used to present some propagation channelmodels As a result the ST-CF in (28) can provide a suitableplatform to study the statistical properties of some differentchannelmodels such as the random scatteringmodel [13 14]Jakes model [28] one-ring model [23] and two-ring model[29] as described in Section 313Therefore the ST-CF in (28)is a generalized and parametric expression However due tothe complex nature of 119901(120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902) it is assumed thatAOD and AOA are independent [13 14 32 33] and azimuthangles and elevation angles in (28) are also independent[18 19 32] The parameters in (28) such as the velocities ofthe multiple moving scatterers and random angles can becalculated as follows
(i) Scatterer Velocity Distributions The Gaussian Laplaceexponential and uniform distributions can be used todescribe the velocity of moving scatterers [13] In fact thescatterer velocity ]119894119901119902 is always positive or equal to zeroWe use the uniform distribution in (29) and half-Gaussiandistribution in (30) to describe the velocity of multiplemoving scatterers
119901 (]119894119901119902) = 1V119894max 0 le ]119894119901119902 le V119894max (29)
where V119894max is the maximum of ]119894119901119902
21205902119901119902119894]] ]119894119901119902 ge 0 (30)
where 120590119901119902119894 is the standard deviation of ]119894119901119902
(ii) Angle Distributions To characterize the statistical anglesin Table 1 we use the uniform distribution in (31) in theisotropic scattering environment and use the von Misesdistribution in (32) and the cosine distribution in (33) inthe nonisotropic scattering environment In addition theinterval of azimuth angles 120572119879119901119902 120572119877119901119902 and 120572119894119901119902 is (minus120587 120587] and theinterval of elevation angles 120573119879119901119902 120573119877119901119902 and 120573119894119901119902 is (minus1205872 1205872]119901 (120574) = 11205742 minus 1205741 1205741 le 120574 le 1205742 (31)
119901 (120572) = exp [119896 cos (120572 minus 120572)]21205871198680 (119896) 1205721 le 120572 le 1205721 + 2120587 (32)
where 1198680(∙) is the zeroth-order modified Bessel function ofthe first kind 120572 is the mean angle and 119896 controls the spreadof angles around the mean The von Mises distribution PDFwith 120572 = 0 is used to describe the azimuth angles
119901 (120573) = 1205874 10038161003816100381610038161205731198981003816100381610038161003816 cos(1205872 120573120573119898) minus 120573119898 le 120573 le 120573119898 (33)
where 120573119898 is the maximum of 120573 The cosine distribution PDFis used to describe the elevation angles
313 Special Cases of 119898119861rsquos ST-CF If 120573119879 = 120573119877 = 120573119879119901119902 =120573119877119901119902 = 120573119894119901119902 = 0 that is the scattering environment is 2D
Mobile Information Systems 7
some different special cases can be derived from the generalexpression of the119898Brsquos ST-CF in (28)
In the 2D scattering environment (28) can be written as(B2) in Appendix B If 119901 = 119901 119902 = 119902 and119898 = 1 (B2) equals(11) in [13] and (6) in [14] regardless of the plus or minussigns under the assumption that the angles 120572119879 and 120572119877 areindependent of each other in [13 14] In isotropic scatteringenvironments some other special cases with closed-formexpressions can be derived as follows
Appendix B shows that an approximate ST-CF of (B2)can be written as
119877119898119901119902119901119902 (119889119879 119889119877 120591) = 212059021198981198690 1198960 [V119879120591 + (119902 minus 119902) 119889119879]times 1198690 1198960 [V119877120591 + (119901 minus 119901) 119889119877] int sdot sdot sdot int 1198690 (21198960]1119901119902120591)times 1198690 (21198960]119898119901119902120591)119898minus1prod
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind and 1198960 = 21205871198910119888 is the wave number
Note that if 119898 approaches to the infinity (34) can bewritten as
lim119898rarrinfin
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 212059021198981198690 [(119902 minus 119902) 1198960119889119879] 1198690 [(119901 minus 119901) 1198960119889119877] 120591 = 00 120591 gt 0
(35)
In the NLoS communication environment with veryhigh-density scatterers (35) implies that the ST-CFapproaches to zero at the nonzero time difference 120591 and hasnothing to do with velocities of scatterers 119879119883 and 119877119883 Thelarge-scale antenna arrays are very suitable to be used in thisenvironment because the antenna element spacing can bereduced to a smaller value
If ]119894119901119902 is constant (34) can be presented as the followingclosed-form expression
The T-CF of the classical F2M scenario with fixed scat-terers is obtained if V119879 = V119894119901119902 = 0 (119894 = 1 2 3 119898)119902 = 119902 and 119901 = 119901 in (36) In this case the ST-CF in(36) is 212059021198981198690(1198960V119877120591) which is known as the Jakes model[28] If V119879 = 0119898 = 1 119902 = 119902 and 119901 = 119901 the ST-CFin (36) results in 212059021198981198690(1198960V1119901119902120591)1198690(1198960V119877120591) which equals the
T-CF of F2M single-ring channel model in the presence ofmoving scatterers reported in (11) of [23] If V119894119901119902 = 0 (119894 =1 2 3 119898) 119902 = 119902 and 119901 = 119901 the ST-CF in (36) resultsin 212059021198981198690(1198960V119879120591)1198690(1198960V119877120591) which equals the T-CFs of theclassical M2M two-ring channel model in the presence offixed scatterers reported in (46) of [29] If V119894119901119902 = 120591 = 0 (119894 =1 2 3 119898) 119902 = 119901 = 2 and 119902 = 119901 = 1 the ST-CF in (36)results in 212059021198981198690(1198960119889119879)1198690(1198960119889119877) which equals the S-CF in (46)of [29]
The velocities ofmoving scatterers such asmoving foliagewalking pedestrians and passing vehicles generally are ran-dom variables If ]119894119901119902 is described by the uniform distributionin (29) (34) can be presented as the following closed-formexpression
times 1198690 1198960 [V119877120591 + (119901 minus 119901) 119889119877] [21198690 (21198960V1max120591)+ 1205871198691 (21198960V1max120591)1198670 (21198960V1max120591) minus 1205871198690 (21198960V1max120591)times 1198671 (21198960V1max120591)] [21198690 (21198960V119898max120591)+ 1205871198691 (21198960V119898max120591)1198670 (21198960V119898max120591) minus 1205871198690 (21198960V119898max120591)sdot 1198671 (21198960V119898max120591)] times 119898minus1prod
119894=2
[21198690 (41198960V119894max120591)+ 1205871198691 (41198960V119894max120591) times 1198670 (41198960V119894max120591)minus 1205871198690 (41198960V119894max120591)1198671 (41198960V119894max120591)]
(37)
where 1198691(∙) denotes the first-order Bessel function of the firstkind 1198670(∙) denotes the zeroth-order Struve function and1198671(∙) denotes the first-order Struve function
If ]119894119901119902 is described by the half-Gaussian distribution in(30) (34) can be presented as the following closed-formexpression
32 Space-Doppler Power Spectral Density The SD-PSD canbe obtained by taking the Fourier transform of the ST-CF in(16)with respect to time difference 120591 From (16) it follows that
8 Mobile Information Systems
Table 2 Parameters used in the numerical simulation
where 120575(∙) is the Dirac delta functionSince the ST-CF of the 119898B component is the multiple
integral as (28) the closed-form expression of SD-PSDcannot be derived In Section 42 we show the SD-PSD of the3DV2Vchannel in the presence ofmultiplemoving scatterersby means of numerical integrations
4 Numerical Results and Validation
This section demonstrates the normalized correlation func-tions and validates the Doppler power spectral density (D-PSD) described in Section 3 through theMATLABnumericalsimulations Unless indicated otherwise the values of thenumerical simulation parameters are summarized in Table 2
41 Numerical Results In this section the numerical curvesof ST-CFs T-CFs and S-CFs influenced by some importantcontributory factors are presented As in the typical urbanenvironments the power weight of the119898th cluster of rays hasbeen set to 1199011 = 1199012 = 12 for119872 = 2 1199011 = 1199012 = 1199013 = 13 for119872 = 3 1199011 = 1199012 = 13 1199013 = 1199014 = 16 for119872 = 4 1199011 = 131199012 = 14 1199013 = 1199014 = 16 and 1199015 = 112 for119872 = 5411 ST-CFs and T-CFs for Different Scattering Scenarios andBounces Figures 2ndash5 demonstrate the ST-CFs and T-CFs forthe different maximum bounces 119872 in the 3D isotropic and
0 05 1 15 2 25 3 35 4066
067
068
069
07
071
072
휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 2 ST-CFs in (16) in the isotropic scattering scenario fordifferent maximum bounces
nonisotropic scattering scenarios The scatterer velocity isuniformly distributed with an average speed of 25ms thatmay be the velocity of passing vehicles The other parametersused to obtain curves in Figures 2ndash5 are summarized in(Table 2 Cols 2 and 3) As shown in Figures 2 and 4 thelarger119872 is the faster the ST-CFs and T-CFs decrease in theisotropic scattering scenario However the descent rates ofST-CFs and T-CFs increase slowly when119872 is larger than 3The similar conclusions can be obtained fromFigures 3 and 5however the ST-CF curves in Figure 3 have oscillationswhichmay be caused by the nonisotropic scattering However thisconclusion differs from [27] which showed that the triple-or higher-order bounced rays had statistical properties verysimilar to those of the double-bounced rays and could beapproximated as double-bounced rays The discrepancy maybe caused by the different communication environmentsconsidered by us and [27] Specifically channel-soundingexperimental campaign in [27] was conducted along surface
Mobile Information Systems 9
066
067
068
069
07
071
072
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 3 ST-CFs in (16) in the nonisotropic scattering scenario fordifferent maximum bounces
065
07
075
08
085
09
095
1
M = 1M = 2M = 3
M = 4M = 5
0 05 1 15 2 25 3 35 4휏 (ms)
|R11
11(휏)|
Figure 4 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropicscattering scenario for different maximum bounces
streets around the Georgia Tech campus and on the Interstatehighways in the Midtown Atlanta metropolitan area whichmay have lower dense scatterers than our communicationenvironments The ST-CF curves in Figure 3 converge rel-atively slower for the larger 119872 that means there are morelocal scatterers inmotionwith randomvelocities and randomdirections As can be observed in Figures 4 and 5 theT-CFs inboth figures are very similar that indicates that the scatteringforms have no significant effect on the T-CFs of the 3D V2Vchannel in the presence of multiple moving scatterers Fur-thermore in Figures 2ndash5 it can be observed that a tendencyto a limit value curve of ST-CFs and T-CFs can be inferred if
065
07
075
08
085
09
095
1
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
|R11
11(휏)|
Figure 5 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the nonisotropicscattering scenario for different maximum bounces
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
066
067
068
069
071
072
1122(d
T=
05휆d
R=
05휆휏
)||R
07
Figure 6 ST-CFs in (16) in the isotropic and nonisotropic scatteringscenarios with119872 = 5 for different scatterer velocity distributions
119872 approaches infinity and the similar inference also can beobtained in (35) for the 2D scattering environment
412 ST-CFs and T-CFs for Different Scattering Scenarios andScatterer Velocity Distributions Figures 6 and 7 demonstratethe ST-CFs and T-CFs for the different scatterer velocitydistributions in the 3D isotropic and nonisotropic scatteringscenarios We use two uniform distributions in the isotropicscattering environment and use the von Mises distributionand the cosine distribution in the nonisotropic scatteringenvironment to describe all the statistical azimuth andelevation angles in Table 1 respectively The average speedof the uniform and half-Gaussian distributions is 25ms
10 Mobile Information Systems
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
065
07
075
08
085
09
095
1
|R11
11(휏)|
Figure 7 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropic andnonisotropic scattering scenarios with119872 = 5 for different scatterervelocity distributions
and the standard deviation of the half-Gaussian distribu-tion is 31ms The other parameters used to obtain curvesin Figures 6 and 7 are summarized in (Table 2 Cols 2and 3) As shown in Figure 6 the ST-CF for the uniformdistributed velocity of scatterers in the isotropic scatteringenvironment decreases fastest and both the nonisotropicscattering and the half-Gaussian distribution can result inswift oscillations of the fading curvesThe comparison resultsindicate that the scatterer velocity distributions have moresignificant effect than the scattering forms on the ST-CFsof the 3D V2V channel in the presence of multiple movingscatterers Figure 7 indicates that not like scattering formsthe scatterer velocity distributions have significant effect onthe T-CFs Generally the realistic propagation environmentis nonisotropic scattering and the scatterer velocity canbe modeled by the half-Gaussian or uniform distributionTherefore the results of Figures 6 and 7 may be helpful inanalysis of the system performance and modeling a specificpropagation channel for V2V communications
413 ST-CFs and S-CFs for Different Scattering Scenariosand Spacing between Adjacent Antenna Elements Figures8 and 9 demonstrate the 3D ST-CFs and S-CFs for thedifferent spacing between adjacent antenna elements of 119879119883and 119877119883 in isotropic and nonisotropic scattering scenariosThe set of the scatterer velocity in Figure 8 is the sameas Figures 2ndash5 and the other parameters used to obtaincurves in Figures 8 and 9 are summarized in (Table 2 Col4) Figures 8 and 9 show that spacing between adjacentantenna elements has significant effect on the 3D ST-CFsand S-CFs Furthermore the 3D ST-CFs in isotropic andnonisotropic scattering scenarios decrease to the bottomwhen 119889119877 = 119889119879 tends to 12120582 and 07120582 respectively Inaddition the 3D S-CF in nonisotropic scattering scenarios isexplicitly less than that in isotropic scattering scenarios for
|R11
22(d
Td
R휏)|
dR = d
T (휆) 휏 (ms)
0025
05075 1
12515 0
12
34
1
095
09
085
08
075
07
065
IsotropicNonisotropic
Figure 8 3D ST-CFs in (16) in the isotropic and nonisotropicscattering scenarios with119872 = 5 120582 is the carrier wavelength
|R11
22(d
Td
R)|
dR (휆)
dT (휆)
1
095
09
085
08
075
07
065
0 025 05 0751 125 15
0025
05075 1
12515
IsotropicNonisotropic
Figure 9 3D S-CFs in (16) with 120591 = 0 in the isotropic andnonisotropic scattering scenarios 120582 is the carrier wavelength
most spacing between adjacent antenna elements as shown inFigure 9Therefore the large-scale antenna arrays employingthe spatial multiplexing are more suitable to be used in thenonisotropic scattering scenarios owing to the high antennaspace utilization The spacing between adjacent antennaelements of antenna arrays is suggested to be more than 120582in the V2V communications especially in the environmentswith low-density scatterers
42 Comparison with Measurements The parametric natureof the proposed channel model makes it adaptable to avariety of propagation environments D-PSD is one of themost important and unique channel characteristics for V2Vcommunication channels To illustrate the validity of theproposed model we compare the modeled SD-PSDs of the3D V2V channel with multiple moving scatterers with themeasured F2F D-PSD in [26] the theoretical V2V SD-PSDin [18] and the measured V2V SD-PSD in [27]
Mobile Information Systems 11
minus2 minus15 minus1 minus05 0 05 1 15 2Doppler frequency f (kHz)
Figure 10 Comparison between the modeled SD-PSDs with119872 = 5and the measured D-PSDs of Figure 4 in [26]
Figure 10 shows the comparison between the modeledSD-PSDs obtained from the Fourier transform of the ST-CF presented in (16) in the isotropic scattering scenario andthe measured D-PSDs of Figure 4 in [26] The outdoor mea-surement experiments in [26] have shown that at millimeterwavelengths fading caused by foliage movement and themotion of nearby vehicles is a significantly deleterious effectin systems with static subscribers Figure 4 in [26] showsthe D-PSD generated by foliage movements and the D-PSDcaused by passing vehicles both in a fixed wireless channelat a carrier frequency of 295GHz We have used the half-Gaussian distribution with an average speed of 04ms andthe standard deviation of 05ms to model the velocity ofmoving foliage scatterers while the uniformdistributionwithan average speed of 6ms has been used tomodel the velocityof passing vehicles scatterers The power weight of the 119898thcluster of rays scattered by the moving foliage or passingvehicles has been set to 1199011 = 916 1199012 = 516 1199013 = 116and 1199014 = 1199015 = 132 The other parameters used to obtainthe modeled SD-PSD curves in Figure 10 are summarizedin (Table 2 Col 5) As can be observed in Figure 10 40 dBfading bandwidths for the effect of the foliage and vehiclesmovements are 500Hz and 1 kHz approximately respectivelyFade depth due to foliage is less than the variation due topassing vehicles Close agreements between the modeled andmeasured results are shown in Figure 10These close matchescan be seen not only for relatively slow moving scatterers butalso for relatively fast moving scatterers as well
Figure 11 shows the comparison among the modeled SD-PSD obtained from the Fourier transform of the ST-CFpresented in (16) in the nonisotropic scattering scenario thetheoretical SD-PSD of Figure 8 in [18] and themeasured SD-PSD of Figure 11 in [27]The theoretical SD-PSD reproducedhere is based on the two-cylinder model with moving andstationary scatterers for 119889119877 = 119889119879 = 0 The channel mea-surements in [27] for the measured SD-PSD were collectedat 2435GHz in the urban street surface environment and thespacing between two adjacent antenna elements of119879119883 and119877119883was set to zero We have used the half-Gaussian distributionwith average speeds of 2394ms 0798ms and 04ms
Figure 11 Comparison among the modeled SD-PSD with119872 = 5the theoretical SD-PSD of Figure 8 in [18] and the measured SD-PSD of Figure 11 in [27]
and the standard deviation of 3ms 1ms and 05ms tomodel the velocity of the first the second and the remainingmoving scatterers on the communication links respectivelyThe power weight of the 119898th cluster of rays has been set to1199011 = 815 1199012 = 13 1199013 = 115 1199014 = 130 and 1199015 = 130for119872 = 5 The other parameters used to obtain the modeledSD-PSD curve in Figure 11 are summarized in (Table 2 Col6) We note that different from the measured SD-PSDs in[26] themeasured SD-PSD curve in [27] has oscillationsTheoscillating pattern in measurements for the case 119889119877 = 119889119879 = 0may appear because it is not a true SISO scenario whereonly one transmit and one receive antenna are active [27] Ascan be observed in Figure 11 the modeled SD-PSD matchesbetter withmeasured SD-PSD than the theoretical SD-PSD in[18] This may be because of the introduction of the multiplebounced rays into the propagation model or the constraintson the position of local scatterers in the two-cylinder modelproposed in [18]
Figure 12 demonstrates the CDFs of the relative devi-ations between modeledtheoretical SD-PSDs and corre-sponding measured values in Figures 10 and 11 The meanrelative deviation of the modeled SD-PSDs for foliage inFigure 10 modeled SD-PSDs for vehicles in Figure 10 mod-eled SD-PSDs in Figure 11 and theoretical SD-PSDs inFigure 11 is 256 216 39 and 832 respectively Asshown in Figure 12 more than 80 of the relative deviationsof modeled SD-PSDs in Figures 10 and 11 are less than5 and the relative deviations of theoretical SD-PSDs inFigure 11 are explicitly larger than those of modeled SD-PSDs In addition from Figures 10 and 11 we note that themodeled SD-PSD matches better with measured SD-PSDs atthe lower and higher Doppler frequencies than the middleDoppler frequencies This may be because 119879119883 and 119877119883 werenot strictly fixed and not all the scatterers were in motion inmeasurementsThe approximate calculation of the frequencyshift in (A4) and Doppler shift in (10) may be anotherfactor which causes the mismatch at the middle Dopplerfrequencies andmore investigations concerning the accuracyof the proposed model will be addressed in future works
12 Mobile Information Systems
0 5 10 15 20 25 30 350
01
02
03
04
05
06
07
08
09
1
Relative deviation ()
Relat
ive d
evia
tion
CDF
Modeled SD-PSD (foliage)Modeled SD-PSD (vehicles)
Modeled SD-PSDTheoretical SD-PSD
Figure 12 The relative deviation cumulative distribution functions(CDFs) of the modeled and theoretical SD-PSDs in Figures 10 and11
However as shown in Figures 10ndash12 the acceptable matchesconfirm the utility and generality of the proposed model andshow the need for including multiple moving scatterers inpropagation model
5 Conclusion
Without specific constraints on the position of the local mov-ing scatterers a 3D geometrical V2V propagation model thatincludes LoS single bounced and multiple bounced linksbetween the transmitter and receiver was proposed Basedon the geometrical propagation model a 3D reference modelfor narrowband MIMO V2V multipath fading channels wasdeveloped From the reference model the correspondingmathematical expressions and numerical results of ST-CFsT-CFs S-CFs and SD-PSDs were studied for different para-metric sets It has been shown that the maximum bouncesscattering forms scatterer velocity distributions and spacingbetween adjacent antenna elements have significant effect onthe ST-CFs and only the maximum bounces and scatterervelocity distributions have significant effect on the T-CFsFinally the modeled SD-PSD results were compared withthe measured data and the other literaturersquos theoreticalresult The close agreements between the analytically andempirically obtained channel statistics confirmed the utilityand generality of the proposed model
Appendix
A Derivation Process of Frequency Shifts
The derivation process of 119891119902119901119902119899 in (7) is presented in thissection and 119891119901119901119902120588 119891119902119901119902119899 119891119901119901119902119899 and 119891119901119902120588 can be calculatedsimilarly
AOD
Δl
TX qth element
Figure 13 Projection from the coordinate vector of 119902th antennaelement to the AOD direction
According to Figure 1 the coordinate vectors of 119902thantenna element and the AOD direction can be written asrespectively
minus 1) 119889119879 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902 + sin120573119902 sin120573119879119901119902] (A3)
Because we mainly consider the difference of the fre-quency shifts caused by the TPD changes to calculate ST-CFsthe TPD change of ℎ11(119905) is assumed to be zero Eventually119891119902119901119902119899 can be calculated as
119891119902119901119902119899 asymp Δ1198971198910119888= (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(A4)
B Derivation Process of Approximate ST-CFs
In the 2D scattering environment the simplified and approx-imate expressions of (28) can be derived in this section If120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 (27) and (10) can be writtenas
Δ119891119899119898
= 1198910119888 [(119902 minus 119902) 119889119879 cos (120572119902 minus 120572119879119901119902)+ (119901 minus 119901) 119889119877 cos (120572119901 minus 120572119877119901119902)]
Mobile Information Systems 13
119891119901119902119899119898
= 1198910119888 [V119879 cos (120572119879 minus 120572119879119901119902) + V119877 cos (120572119877 minus 120572119877119901119902)
+ ]1119901119902 cos (1205721119901119902 minus 120572119879119901119902) + 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119879119901119902)+ ]119898119901119902 cos (120572119898119901119902 minus 120572119879119901119902) + ]1119901119902 cos (1205721119901119902 minus 120572119877119901119902)+ 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119877119901119902) + ]119898119901119902 cos (120572119898119901119902 minus 120572119877119901119902)] (B1)
If 120572119879 = 120572119902 and 120572119877 = 120572119901 that is the antenna array and itsvelocity have the same directions (28) can be simplified as bysubstituting (B1) into (28)
where 1198960 = 21205871198910119888 is the wave numberIn isotropic scattering environment we use the uniform
distribution in (31) to describe all the random azimuth anglesIn this case (B2) can be written as119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879] cos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877] cos (120572119877 minus 120572119877119901119902)
times1198690 [21198960]1119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times1198690 [21198960]119898119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times119898minus1prod119894=2
1198690 [41198960]119894119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind
The AOD 120572119879119901119902 and AOA 120572119877119901119902 are uniformly and indepen-dently distributed so120572119879119901119902minus120572119877119901119902 can be equal to zero in average
As a consequence the term cos[(120572119879119901119902 minus120572119877119901119902)2] in (B3) can beapproximated by one [13 14] and an approximate ST-CF canbe written as from (B3)
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879]sdotcos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877]sdot cos (120572119877 minus 120572119877119901119902) 1198690 (21198960]1119901119902120591)times1198690 (21198960]119898119901119902120591)119898minus1prod
Eventually the approximate ST-CF in (34) can be derivedfrom (B4) according to the definition of 1198690(∙)Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was funded by the National Natural Science Foun-dation of China (no 91438104 no 61501065 no 61571069and no 61601067) the Fundamental Research Funds forthe Central Universities (no 106112016CDJX160001) and theChongqing Research Program of Basic Research and FrontierTechnology (no CSTC2016JCYJA0021)
References
[1] A Bazzi B M Masini A Zanella and I Thibault ldquoBeaconingfrom connected vehicles IEEE 80211p vs LTE-V2Vrdquo in Pro-ceedings of 2016 IEEE 27th Annual International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRC)pp 1ndash6 Valencia Spain September 2016
[2] VM Rodrigo-Penarrocha J Reig L Rubio and et al ldquoAnalysisof small- scale fading distributions in vehicle-to-vehicle com-municationsrdquoMobile Information Systems vol 2016 Article ID9584815 7 pages 2016
[3] A Bazzi B M Masini A Zanella and G Pasolini ldquoIEEE80211p for cellular offloading in vehicular sensor networksrdquoComputer Communications vol 60 pp 97ndash108 2015
[4] X Cheng MIMO channel modelling and simulation for cellu-lar and mobile-to-mobile communication systems[Phd thesis]Heriot-Watt University 2009
[5] N Avazov and M Patzold ldquoA geometric street scatteringchannel model for car-to-car communication systemsrdquo inProceedings of 4th Annual International Conference onAdvanced
14 Mobile Information Systems
Technologies for Communications ATC2011 pp 224ndash230 vnmAugust 2011
[6] X Zhao X Liang S Li and K Haneda ldquoMobile-to-MobileWideband MIMO Channel Realization by Using a Two-RingGeometry-Based Stochastic Scattering Modelrdquo Wireless Per-sonal Communications vol 84 no 4 pp 2445ndash2465 2015
[7] M Walter D Shutin and U-C Fiebig ldquoDelay-dependentdoppler probability density functions for vehicle-to-vehiclescatter channelsrdquo IEEE Transactions on Antennas and Propaga-tion vol 62 no 4 pp 2238ndash2249 2014
[8] P T Samarasinghe T A Lamahewa T D Abhayapala and RA Kennedy ldquo3D mobile-to-mobile wireless channel modelrdquo inProceedings of 2010 Australian Communications Theory Work-shop AusCTW 2010 pp 30ndash34 aus February 2010
[9] M Riaz S J Nawaz and N M Khan ldquo3D ellipsoidal model formobile-to-mobile radio propagation environmentsrdquo WirelessPersonal Communications vol 72 no 4 pp 2465ndash2479 2013
[10] J Chen and T G Pratt ldquoThree-dimensional geometry-based stochastic modeling and performance of 4times4 space-polarization mobile-to-mobile wideband MIMO channelsrdquo inProceedings of 2013 IEEE Global Communications ConferenceGLOBECOM 2013 pp 3936ndash3941 usa December 2013
[11] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA Novel 3DNonstationary Channel Model Based on the von Mises-FisherScattering DistributionrdquoMobile Information Systems vol 2016Article ID 2161460 2016
[12] A Chelli and M Patzold ldquoThe impact of fixed and movingscatterers on the statistics of MIMO vehicle-to-vehicle chan-nelsrdquo in Proceedings of VTC Spring 2009 - IEEE 69th VehicularTechnology Conference esp April 2009
[13] A Borhani and M Patzold ldquoModeling of vehicle-to-vehiclechannels in the presence of moving scatterersrdquo in Proceedingsof 2012 IEEE Vehicular Technology Conference (VTC Fall) pp1ndash5 Quebec City QC Canada September 2012
[14] A Borhani andM Patzold ldquoCorrelation and spectral propertiesof vehicle-to-vehicle channels in the presence ofmoving scatter-ersrdquo IEEE Transactions on Vehicular Technology vol 62 no 9pp 4228ndash4239 2013
[15] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the IEEE 74th Vehicular TechnologyConference (VTC Fall rsquo11) pp 1ndash6 IEEE San Francisco CalifUSA September 2011
[16] M D Soltani M Alimadadi Y Seyedi and H AmindavarldquoModeling of doppler spectrum in V2V urban canyon oncom-ing environmentrdquo in Proceedings of 2014 7th InternationalSymposium on Telecommunications IST 2014 pp 1155ndash1160 irnSeptember 2014
[17] M D Soltani M Alimadadi and A Mohammadi ldquoModelingof mobile scatterer clusters for doppler spectrum in widebandvehicle-to-vehicle communication channelsrdquo IEEE Communi-cations Letters vol 18 no 4 pp 628ndash631 2014
[18] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014
[19] A G Zajic ldquoModeling impact of moving scatterers on dopplerspectrum in wideband vehicle-to-vehicle channelsrdquo in Proceed-ings of Proc Eur Conf Antennas Propag p 1 Lisbon May 2015
[20] J Zhang X Yin and X Cheng ldquoTheoretical analysis andmeasurements Doppler spectra of vehicular communication
channelsrdquo in Proceedings of 2012 12th International Confer-ence on ITS Telecommunications ITST 2012 pp 98ndash102 twnNovember 2012
[21] M Picone S Busanelli M Amoretti F Zanichelli and GFerrari Advanced Technologies for Intelligent TransportationSystems Springer 2015
[22] V H Pham M H Taieb J Y Chouinard S Roy and H THuynh ldquoOn the double Doppler effect generated by scatterermotionrdquo REV Journal on Electronics and Communications vol1 pp 30ndash37 2011
[23] S Roy H T Huynh and P Fortier ldquoCompound Dopplerspread effects of subscriber motion and scatterer motionrdquo AEU- International Journal of Electronics and Communications vol57 no 4 pp 237ndash246 2003
[24] J B Andersen J O Nielsen G F Pedersen G Bauch and GDietl ldquoDoppler spectrum from moving scatterers in a randomenvironmentrdquo IEEE Transactions on Wireless Communicationsvol 8 no 6 pp 3270ndash3277 2009
[25] H S Rad S Gazor and P Shariatpanahi ldquoNon-fixed scatterersand their effects on MIMOmulticarrier fading communicationchannelsrdquo in Proceedings of 50th Annual IEEE Global Telecom-munications Conference GLOBECOM2007 pp 3765ndash3769 usaNovember 2007
[26] N Naz and D Falconer ldquoTemporal variations characterizationfor fixed wireless at 295 GHzrdquo in Proceedings of 2000 IEEE51st Vehicular Technology Conference Proceedings VTC2000-Springer pp 2178ndash2182 Tokyo Japan 2000
[27] A G Zajic G L Stuber T G Pratt and S T NguyenldquoWideband MIMO mobile-to-mobile channels Geometry-based statistical modeling with experimental verificationrdquo IEEETransactions on Vehicular Technology vol 58 no 2 pp 517ndash5342009
[28] W C Jakes Microwave Mobile Communications Wiley-IEEEPress Piscataway NJ USA 1994
[29] M Patzold B O Hogstad and N Youssef ldquoModeling analysisand simulation of MIMO mobile-to-mobile fading channelsrdquoIEEE Transactions onWireless Communications vol 7 no 2 pp510ndash520 2008
[30] M PatzoldMobile Radio Channels Wiley Chichester UK 2ndedition 2011
[31] R Wang and D Cox ldquoDouble mobility mitigates fading in adhoc wireless networksrdquo in Proceedings of IEEE Antennas andPropagation Society International Symposium pp 306ndash309 SanAntonio TX USA
[32] G Bakhshi K Shahtalebi and H S Rad ldquoA novel full-three-dimensional MIMOmobile-to-mobile channel referencemodelrdquo in Proceedings of 3rd International Conference on SignalProcessing and Communication Systems ICSPCSrsquo2009 usaSeptember 2009
[33] W Zhou X Wang X Wang and W Chen ldquoA modified two-erose- ringmodel formimomobile-to-mobile fading channelsrdquoin Proceedings of 7th International Conference on WirelessCommunications Networking and Mobile Computing WiCOM2011 5 1 pages Wuhan China September 2011
moved with random velocities in random directions in [1314] The impact of mobile and stationary scattering clusterson the Doppler spectrum was investigated for widebandV2V communication channels in an urban canyon oncomingenvironment in [16 17] All previously reported models in[12ndash17] are 2D propagation models in which the rays are LoSSB or double bounced (DB) However this 2D assumptiondoes not seem to be appropriate for many communicationscenarios for example urbanV2V communications in whichthe transmitter and receiver antenna arrays are often locatedin close proximity to or lower than the surrounding scatterers
A 3D geometrical propagation model that included bothstationary and moving scatterers around the transmitter andreceiver was proposed in [18] and this work was extendedto wideband channels in [19] The models in [18 19] tookinto account that the rays in V2V channels could be bothSB and DB however they imposed some constraints on theposition of the local scatterers and assumed that the trans-mitter receiver and scatterers were in motion with constantvelocities in 2D spaceTherefore themodels in [18 19] cannotfully capture the 3D spatial information A preliminaryinvestigation of the impact of multiple moving scatterers onthe Doppler spectrum in 3D V2V communication scenarioswas presented in [20] However the work of [20] did notinvestigate and take into account the space-time correlationfunctions (ST-CFs) scatterer velocity distributions angledistributions and so on
It is unavoidable in V2V communications that the raysfrom the transmitter to receiver are multiple bounced (MB)by moving scatterers especially in environments with high-density scatterers for example urban area However to thebest of the authorsrsquo knowledge the model and statisticalproperties of 3D V2V or M2M channels in the presence ofmultiple moving scatterers have been investigated rarely sofarThis paper strives to alleviate the current lack of analyticalstudies by investigating the model and statistical propertiesof a narrowband 3D MIMO V2V channel in which the localmultiple scatterers are moving with random velocities inrandom directionsThe proposed referencemodel constructsthe channel impulse response as a combination of LoS andMB components and SB is considered as a special case ofMB Different from the assumption of all non-LoS (NLoS)path gains having the same size in [13 14] and meanwhilecompared with models in [18 19] SB DB and other MB rayshavemore flexible and precise power weights in the proposedmodel From the reference model the corresponding ST-CFs time correlation functions (T-CFs) and space corre-lation functions (S-CFs) are analytically investigated andnumerically simulated in terms of various factors such as themaximum bounces scattering forms scatterer velocity dis-tributions and spacing between adjacent antenna elementsFinally the theoretical space-Doppler power spectral density(SD-PSD) results with SD-PSDs in [18] and measured datain [26 27] are compared The close agreements betweenthe analytically and empirically obtained SD-PSDs confirmthe utility and generality of the proposed model and showthe importance of including multiple moving scatterers inpropagation models The contributions and novelties of thispaper are summarized as follows
(i) We propose a generalized geometrical model and ageneralized referencemodel that include LoS SB andMB rays between the transmitter and receiver for3D narrowband MIMO V2V communications Theproposed model can be adapted to a wide varietyof scenarios for example F2F F2M and M2M withcertain bounces by adjusting model parameters
(ii) To the best of the authorsrsquo knowledge the impactof raysrsquo different maximum bounces caused by themultiple moving scatterers on the ST-CFs and T-CFsis deeply investigated for the first time in V2V orM2M communication environments
(iii) The ST-CFs obtained from the proposed referencemodel are relatively generalized and can be reduced toseveral existing ST-CFs and T-CFs for example thosein [13 14 23 28 29]
The remainder of the paper is organized as followsSection 2 describes the geometrical propagation model andpresents a 3D reference model for narrowband MIMO V2Vchannels in the presence of multiple moving scatterersSection 3 derives the channel ST-CFs T-CFs S-CFs and SD-PSDs for different parametric sets Numerical results andcomparison between the theoretical results and measureddata are provided in Section 4 Finally Section 5 providessome concluding remarks
2 3D Geometrical PropagationModel and Reference Model
21 Geometrical PropagationModel This paper considers theMIMO V2V communication links between transmitter 119879119883and receiver 119877119883 as shown in Figure 1 The radio propagationenvironment is characterized by 3Dmultiple moving scatter-ing with either LoS or NLoS conditions between 119879119883 and 119877119883TheMBwaves emitted from the 119902th antenna element of119879119883 atan angle of departure (AOD) reach the 119901th antenna elementof 119877119883 at an angle of arrival (AOA) after being multiplescattered by the local moving scatterers
For ease of reference in this geometrical propagationmodel the main parameters are summarized in Table 1 andthe main assumptions are summarized as follows
(i) The received signal power consists of LoS SB andMBscattering components with the corresponding powerweights
(ii) Both 119879119883 and 119877119883 are equipped with uniform lineararrays consisting of omnidirectional antenna ele-ments
(iii) 119879119883 and 119877119883 move with constant velocities in 3D spacedescribed by the fixed azimuth angles and elevationangles
(iv) Each of the local scatterers on the links between 119879119883and 119877119883 for the multiple bounces is in motion with arandom velocity in a random direction in 3D space
Mobile Information Systems 3
y
z
x
D
Multiple scattering
LoS
nth path
12
12
훽q
TX
훽T
훽R
훽p훼q
훼T
훼R
훼p
훽Tpq
훽ipq
훽Rpq
훼Tpq
훼ipq
훼Rpq
SiS1 Sm
i
R
T
pq
RX
Figure 1 Geometrical propagation model with LoS and MB rays for the 3D MIMO V2V communication scenario Empty circles representthe antenna elements and solid squares represent the moving scatterers
Table 1 Definition of the parameters used in the geometrical model
Parameters Definition Attributes119878119894 The 119894th moving scatterer for the multiple bounces Symbol119863 The distance between 119879119883 and 119877119883 Deterministic119902 119901 The antenna element identifier of 119879119883 and 119877119883 respectively Deterministic119876 119875 The number of antenna elements of 119879119883 and 119877119883 respectively Deterministic119889119879 119889119877 The spacing between two adjacent antenna elements of 119879119883 and 119877119883 respectively Deterministic120572119902 120573119902 The azimuth angle and elevation angle of 119879119883rsquos antenna array respectively Deterministic120572119901 120573119901 The azimuth angle and elevation angle of 119877119883rsquos antenna array respectively Deterministic120572119879119901119902 120573119879119901119902 The azimuth angle and elevation angle of AOD respectively Statistical120572119877119901119902 120573119877119901119902 The azimuth angle and elevation angle of AOA respectively StatisticalV119879 V119877 The velocities of 119879119883 and 119877119883 respectively DeterministicV119894119901119902 The velocity of 119878119894 Statistical120572119879 120573119879 The azimuth angle and elevation angle of V119879 respectively Deterministic120572119877 120573119877 The azimuth angle and elevation angle of V119877 respectively Deterministic120572119894119901119902 120573119894119901119902 The azimuth angle and elevation angle of V119894119901119902 respectively Statistical
(v) The geometrical propagation model does not imposespecific constraints on the position of the local mov-ing scatterers like [13 14] Owing to high path loss weneglect the energy contribution of remote scatterers
22 Reference Model We can observe from Figure 1 that thecomplex faded envelope of the link between the 119902th antennaelement of 119879119883 and the 119901th antenna element of 119877119883 can bewritten as a superposition of LoS and MB components thatisℎ119901119902 (119905) = ℎLoS119901119902 (119905) + ℎMB
where 120588119901119902 119891119901119902120588 120579119901119902120588 119891119902119901119902120588 and 119891119901119901119902120588 denote the path gainDoppler shift phase shift frequency shift for 119902th antennaelement of119879119883 and frequency shift for119901th antenna element of119877119883 of ℎLoS119901119902 (119905) respectively119898 denotes the amount of bounces119872 is themaximumof119898119901119898 and119873119898 denote the power weightand path number of ℎ119898119901119902(119905) respectively 119888119901119902119899119898 119891119901119902119899119898 120579119901119902119899119898 119891119902119901119902119899 and 119891119901119901119902119899 denote the path gain Doppler shift phaseshift frequency shift for 119902th antenna element of 119879119883 andfrequency shift for 119901th antenna element of 119877119883 of the 119899119898thpath in ℎ119898119901119902(119905) The LoS component ℎLoS119901119902 (119905) can be describedby a complex sinusoid and (2) is an extension of the (317)in [30] ℎ119898119901119902(119905) denotes the NLoS component with119898 bounced
4 Mobile Information Systems
(119898B) rays and it is an extension of the (312) in [30] Notethat MB component ℎMB
119901119902 (119905) of the channel impulse responseconsists of 119872 clusters of rays reflected 119898 isin 1 2 3 119872times from moving scatterers
The channel gains (120588119901119902 119888119901119902119899119898
221 Channel Gains The central limit theorem states thatℎ119898119901119902(119905) equals a complex valued Gaussian random processwith zero mean and variance 21205902119898 = Varℎ119898119901119902(119905) =lim119873
119898rarrinfinsum119873119898119899
119898
119864[1198882119901119902119899119898
] 119901119898 in (3) denotes the power weightof the 119898th clusters of rays and sum119872119898=1 119901119898 = 1 The channelgain of ℎ119901119902(119905) is normalized ie 2sum119872119898=1 1199011198981205902119898 + 1205882119901119902 = 1and the Rice factor can be denoted as119870 = 12058821199011199022sum119872119898=1 1199011198981205902119898These parameters have to be either set during simulations orestimated from measurements
222 Doppler Shifts and Frequency Shifts The frequencyshifts 119891119902119901119902120588 119891119901119901119902120588 119891119902119901119902119899 and 119891119901119901119902119899 depend on the differenceof the propagation distance (TPD) changes between ℎ119901119902(119905)and ℎ11(119905) On the other hand the Doppler shifts 119891119901119902120588 and119891119901119902119899
119898
depend on the geometrical relation between directionsof movement of 119879119883 119877119883 and multiple moving scatterers andthe directions of AOD and AOA For max(119889119879 119889119877) ≪ 119863 boththe AOD and AOA of LoS rays are approximately equal tozero Appendix A shows that119891119902119901119902120588119891119901119901119902120588119891119902119901119902119899119891119901119901119902119899 and119891119901119902120588are respectively
119891119902119901119902120588 = (119902 minus 1) 1198891198791198910119888 cos120572119902 cos120573119902 (5)
119891119901119901119902120588 = (119901 minus 1) 1198891198771198910119888 cos120572119901 cos120573119901 (6)
119891119902119901119902119899 = (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(7)
119891119901119901119902119899 = (119901 minus 1) 1198891198771198910119888 [cos (120572119901 minus 120572119877119901119902) cos120573119901 cos120573119877119901119902+ sin120573119901 sin120573119877119901119902]
where 119875AOD119894119901119902 and 119875AOA119894119901119902 (119894 = 1 2 3 119898) are119875AOD119894119901119902 = cos (120572119894119901119902 minus 120572119879119901119902) cos120573119894119901119902 cos120573119879119901119902
+ sin120573119894119901119902 sin120573119879119901119902119875AOA119894119901119902 = cos (120572119894119901119902 minus 120572119877119901119902) cos120573119894119901119902 cos120573119877119901119902
Note that if 120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 and 119898 = 1(10) equals (7) in [13] and (5) in [14] regardless of the plus orminus signsThe differences among these plus or minus signsare caused by the different forms of anglesrsquo expression
223 Phases The phase shift 120579119901119902120588 can be assumed to beconstant [30] The phase shift 120579119901119902119899
119898
consists of the phasechange caused by the interaction of the transmitted signalwith the scatterers and the phase change caused by theTPD between the first and the last scatterers Without lossof generality we can assume that the phases 120579119901119902119899
119898
(119898 =1 2 3 119872) are independent random variables Here it isassumed that they are uniformly distributed on the interval[0 2120587) and independent of any other random variable
Mobile Information Systems 5
3 Space-Time Correlation Function andSpace-Doppler Power Spectral Density
Using the referencemodel described in Section 2 we can nowderive the key temporal and spatial characteristics of MIMOV2V narrowband multipath fading channels with the localmultiple moving scatters
31 Space-Time Correlation Function The normalized ST-CFbetween two complex faded envelopes ℎ119901119902(119905) and ℎ119901119902(119905) isdefined as
where (∙)lowast denotes the complex conjugate operation 119864(∙) isthe statistical expectation operator 119901 119901 isin 1 2 3 119875 and119902 119902 isin 1 2 3 119876 The normalized T-CF can be obtainedif 119901 = 119901 and 119902 = 119902 in (15) The normalized space correlationfunction (S-CF) can be obtained by setting 120591 to zero in (15)
Since ℎ1119901119902(119905) ℎ2119901119902(119905) ℎ119872119901119902(119905) and ℎLoS119901119902 (119905) are indepen-dent of each other (15) can be simplified to
where 119877119898119901119902119901119902(119889119879 119889119877 120591) and 119877LoS119901119902119901119902(119889119879 119889119877 120591) denote the nor-malized ST-CFs of the119898B and LoS components respectivelyand they are defined as
311 ST-CF of LoS Component By substituting (2) into (18)the expression for the ST-CF of the LoS component can bewritten as
119877LoS119901119902119901119902 (119889119879 119889119877 120591)= 120588119901119902120588119901119902119864 exp 119895 [2120587119905 (119891119901119902120588 minus 119891119901119902120588) + 2120587119891119901119902120588120591+ 2120587 (119891119902119901119902120588 + 119891119901119901119902120588 minus 119891119902119901119902120588 minus 119891119901119901119902120588) + 120579119901119902120588minus 120579119901119902120588]
(19)
For max(119889119879 119889119877) ≪ 119863 we assume 120588119901119902 = 120588119901119902 and 120579119901119902120588 =120579119901119902120588 Then (19) can be written as
Δ119891120588 = 119891119902119901119902120588 + 119891119901119901119902120588 minus 119891119902119901119902120588 minus 119891119901119901119902120588 (21)
By substituting (5) and (6) into (21) Δ119891120588 can be written as
Δ119891120588 = 1198910119888 [(119902 minus 119902) 119889119879 cos120572119902 cos120573119902+ (119901 minus 119901) 119889119877 cos120572119901 cos120573119901]
(22)
312 ST-CF of 119898119861 Component By substituting (4) into (17)the expression for the ST-CF of the 119898B component can bewritten as
times 119864 exp 119895 [2120587 (119891119901119902119899119898
minus 119891119901119902119899119899
) 119905 + 2120587119891119901119902119899119898
120591 + 2120587 (119891119902119901119902119899 + 119891119901119901119902119899 minus 119891119902119901119902119899 minus 119891119901119901119902119899) + 120579119901119902119899119898 minus 120579119901119902119899119899] (23)
Under the assumption that 120579119901119902119899119898
and 120579119901119902119899119899
are uniformlydistributed on the interval [0 2120587) and independent of eachother 119864exp[119895(120579119901119902119899
119898
minus120579119901119902119899119899
)] equals 1 It is assumed that allthe path gains of the119898B component have the same size thatis
= 119891119902119901119902119899 + 119891119901119901119902119899 minus 119891119902119901119902119899 minus 119891119901119901119902119899 (26)
It is assumed that AOD and AOA of ℎ119898119901119902(119905) and ℎ119898119901119902(119905) areindependent and identically distributed (iid) By substitut-ing (7) and (8) into (26) Δ119891119899
Since the number of local scatterers in the referencemodel is infinite the parameters 120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902 120572119894119901119902and 120573119894119901119902 can be seen as continuous random variables withcorresponding probability density functions (PDFs) Thenthe ST-CF of the 119898B component (25) can be writtenas
Now the complete expression of (16) can be obtainedby substituting (20) and (28) into (16) 119901(120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902)describes the joint distribution of AOD and AOA and itcan be optionally used to present some propagation channelmodels As a result the ST-CF in (28) can provide a suitableplatform to study the statistical properties of some differentchannelmodels such as the random scatteringmodel [13 14]Jakes model [28] one-ring model [23] and two-ring model[29] as described in Section 313Therefore the ST-CF in (28)is a generalized and parametric expression However due tothe complex nature of 119901(120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902) it is assumed thatAOD and AOA are independent [13 14 32 33] and azimuthangles and elevation angles in (28) are also independent[18 19 32] The parameters in (28) such as the velocities ofthe multiple moving scatterers and random angles can becalculated as follows
(i) Scatterer Velocity Distributions The Gaussian Laplaceexponential and uniform distributions can be used todescribe the velocity of moving scatterers [13] In fact thescatterer velocity ]119894119901119902 is always positive or equal to zeroWe use the uniform distribution in (29) and half-Gaussiandistribution in (30) to describe the velocity of multiplemoving scatterers
119901 (]119894119901119902) = 1V119894max 0 le ]119894119901119902 le V119894max (29)
where V119894max is the maximum of ]119894119901119902
21205902119901119902119894]] ]119894119901119902 ge 0 (30)
where 120590119901119902119894 is the standard deviation of ]119894119901119902
(ii) Angle Distributions To characterize the statistical anglesin Table 1 we use the uniform distribution in (31) in theisotropic scattering environment and use the von Misesdistribution in (32) and the cosine distribution in (33) inthe nonisotropic scattering environment In addition theinterval of azimuth angles 120572119879119901119902 120572119877119901119902 and 120572119894119901119902 is (minus120587 120587] and theinterval of elevation angles 120573119879119901119902 120573119877119901119902 and 120573119894119901119902 is (minus1205872 1205872]119901 (120574) = 11205742 minus 1205741 1205741 le 120574 le 1205742 (31)
119901 (120572) = exp [119896 cos (120572 minus 120572)]21205871198680 (119896) 1205721 le 120572 le 1205721 + 2120587 (32)
where 1198680(∙) is the zeroth-order modified Bessel function ofthe first kind 120572 is the mean angle and 119896 controls the spreadof angles around the mean The von Mises distribution PDFwith 120572 = 0 is used to describe the azimuth angles
119901 (120573) = 1205874 10038161003816100381610038161205731198981003816100381610038161003816 cos(1205872 120573120573119898) minus 120573119898 le 120573 le 120573119898 (33)
where 120573119898 is the maximum of 120573 The cosine distribution PDFis used to describe the elevation angles
313 Special Cases of 119898119861rsquos ST-CF If 120573119879 = 120573119877 = 120573119879119901119902 =120573119877119901119902 = 120573119894119901119902 = 0 that is the scattering environment is 2D
Mobile Information Systems 7
some different special cases can be derived from the generalexpression of the119898Brsquos ST-CF in (28)
In the 2D scattering environment (28) can be written as(B2) in Appendix B If 119901 = 119901 119902 = 119902 and119898 = 1 (B2) equals(11) in [13] and (6) in [14] regardless of the plus or minussigns under the assumption that the angles 120572119879 and 120572119877 areindependent of each other in [13 14] In isotropic scatteringenvironments some other special cases with closed-formexpressions can be derived as follows
Appendix B shows that an approximate ST-CF of (B2)can be written as
119877119898119901119902119901119902 (119889119879 119889119877 120591) = 212059021198981198690 1198960 [V119879120591 + (119902 minus 119902) 119889119879]times 1198690 1198960 [V119877120591 + (119901 minus 119901) 119889119877] int sdot sdot sdot int 1198690 (21198960]1119901119902120591)times 1198690 (21198960]119898119901119902120591)119898minus1prod
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind and 1198960 = 21205871198910119888 is the wave number
Note that if 119898 approaches to the infinity (34) can bewritten as
lim119898rarrinfin
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 212059021198981198690 [(119902 minus 119902) 1198960119889119879] 1198690 [(119901 minus 119901) 1198960119889119877] 120591 = 00 120591 gt 0
(35)
In the NLoS communication environment with veryhigh-density scatterers (35) implies that the ST-CFapproaches to zero at the nonzero time difference 120591 and hasnothing to do with velocities of scatterers 119879119883 and 119877119883 Thelarge-scale antenna arrays are very suitable to be used in thisenvironment because the antenna element spacing can bereduced to a smaller value
If ]119894119901119902 is constant (34) can be presented as the followingclosed-form expression
The T-CF of the classical F2M scenario with fixed scat-terers is obtained if V119879 = V119894119901119902 = 0 (119894 = 1 2 3 119898)119902 = 119902 and 119901 = 119901 in (36) In this case the ST-CF in(36) is 212059021198981198690(1198960V119877120591) which is known as the Jakes model[28] If V119879 = 0119898 = 1 119902 = 119902 and 119901 = 119901 the ST-CFin (36) results in 212059021198981198690(1198960V1119901119902120591)1198690(1198960V119877120591) which equals the
T-CF of F2M single-ring channel model in the presence ofmoving scatterers reported in (11) of [23] If V119894119901119902 = 0 (119894 =1 2 3 119898) 119902 = 119902 and 119901 = 119901 the ST-CF in (36) resultsin 212059021198981198690(1198960V119879120591)1198690(1198960V119877120591) which equals the T-CFs of theclassical M2M two-ring channel model in the presence offixed scatterers reported in (46) of [29] If V119894119901119902 = 120591 = 0 (119894 =1 2 3 119898) 119902 = 119901 = 2 and 119902 = 119901 = 1 the ST-CF in (36)results in 212059021198981198690(1198960119889119879)1198690(1198960119889119877) which equals the S-CF in (46)of [29]
The velocities ofmoving scatterers such asmoving foliagewalking pedestrians and passing vehicles generally are ran-dom variables If ]119894119901119902 is described by the uniform distributionin (29) (34) can be presented as the following closed-formexpression
times 1198690 1198960 [V119877120591 + (119901 minus 119901) 119889119877] [21198690 (21198960V1max120591)+ 1205871198691 (21198960V1max120591)1198670 (21198960V1max120591) minus 1205871198690 (21198960V1max120591)times 1198671 (21198960V1max120591)] [21198690 (21198960V119898max120591)+ 1205871198691 (21198960V119898max120591)1198670 (21198960V119898max120591) minus 1205871198690 (21198960V119898max120591)sdot 1198671 (21198960V119898max120591)] times 119898minus1prod
119894=2
[21198690 (41198960V119894max120591)+ 1205871198691 (41198960V119894max120591) times 1198670 (41198960V119894max120591)minus 1205871198690 (41198960V119894max120591)1198671 (41198960V119894max120591)]
(37)
where 1198691(∙) denotes the first-order Bessel function of the firstkind 1198670(∙) denotes the zeroth-order Struve function and1198671(∙) denotes the first-order Struve function
If ]119894119901119902 is described by the half-Gaussian distribution in(30) (34) can be presented as the following closed-formexpression
32 Space-Doppler Power Spectral Density The SD-PSD canbe obtained by taking the Fourier transform of the ST-CF in(16)with respect to time difference 120591 From (16) it follows that
8 Mobile Information Systems
Table 2 Parameters used in the numerical simulation
where 120575(∙) is the Dirac delta functionSince the ST-CF of the 119898B component is the multiple
integral as (28) the closed-form expression of SD-PSDcannot be derived In Section 42 we show the SD-PSD of the3DV2Vchannel in the presence ofmultiplemoving scatterersby means of numerical integrations
4 Numerical Results and Validation
This section demonstrates the normalized correlation func-tions and validates the Doppler power spectral density (D-PSD) described in Section 3 through theMATLABnumericalsimulations Unless indicated otherwise the values of thenumerical simulation parameters are summarized in Table 2
41 Numerical Results In this section the numerical curvesof ST-CFs T-CFs and S-CFs influenced by some importantcontributory factors are presented As in the typical urbanenvironments the power weight of the119898th cluster of rays hasbeen set to 1199011 = 1199012 = 12 for119872 = 2 1199011 = 1199012 = 1199013 = 13 for119872 = 3 1199011 = 1199012 = 13 1199013 = 1199014 = 16 for119872 = 4 1199011 = 131199012 = 14 1199013 = 1199014 = 16 and 1199015 = 112 for119872 = 5411 ST-CFs and T-CFs for Different Scattering Scenarios andBounces Figures 2ndash5 demonstrate the ST-CFs and T-CFs forthe different maximum bounces 119872 in the 3D isotropic and
0 05 1 15 2 25 3 35 4066
067
068
069
07
071
072
휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 2 ST-CFs in (16) in the isotropic scattering scenario fordifferent maximum bounces
nonisotropic scattering scenarios The scatterer velocity isuniformly distributed with an average speed of 25ms thatmay be the velocity of passing vehicles The other parametersused to obtain curves in Figures 2ndash5 are summarized in(Table 2 Cols 2 and 3) As shown in Figures 2 and 4 thelarger119872 is the faster the ST-CFs and T-CFs decrease in theisotropic scattering scenario However the descent rates ofST-CFs and T-CFs increase slowly when119872 is larger than 3The similar conclusions can be obtained fromFigures 3 and 5however the ST-CF curves in Figure 3 have oscillationswhichmay be caused by the nonisotropic scattering However thisconclusion differs from [27] which showed that the triple-or higher-order bounced rays had statistical properties verysimilar to those of the double-bounced rays and could beapproximated as double-bounced rays The discrepancy maybe caused by the different communication environmentsconsidered by us and [27] Specifically channel-soundingexperimental campaign in [27] was conducted along surface
Mobile Information Systems 9
066
067
068
069
07
071
072
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 3 ST-CFs in (16) in the nonisotropic scattering scenario fordifferent maximum bounces
065
07
075
08
085
09
095
1
M = 1M = 2M = 3
M = 4M = 5
0 05 1 15 2 25 3 35 4휏 (ms)
|R11
11(휏)|
Figure 4 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropicscattering scenario for different maximum bounces
streets around the Georgia Tech campus and on the Interstatehighways in the Midtown Atlanta metropolitan area whichmay have lower dense scatterers than our communicationenvironments The ST-CF curves in Figure 3 converge rel-atively slower for the larger 119872 that means there are morelocal scatterers inmotionwith randomvelocities and randomdirections As can be observed in Figures 4 and 5 theT-CFs inboth figures are very similar that indicates that the scatteringforms have no significant effect on the T-CFs of the 3D V2Vchannel in the presence of multiple moving scatterers Fur-thermore in Figures 2ndash5 it can be observed that a tendencyto a limit value curve of ST-CFs and T-CFs can be inferred if
065
07
075
08
085
09
095
1
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
|R11
11(휏)|
Figure 5 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the nonisotropicscattering scenario for different maximum bounces
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
066
067
068
069
071
072
1122(d
T=
05휆d
R=
05휆휏
)||R
07
Figure 6 ST-CFs in (16) in the isotropic and nonisotropic scatteringscenarios with119872 = 5 for different scatterer velocity distributions
119872 approaches infinity and the similar inference also can beobtained in (35) for the 2D scattering environment
412 ST-CFs and T-CFs for Different Scattering Scenarios andScatterer Velocity Distributions Figures 6 and 7 demonstratethe ST-CFs and T-CFs for the different scatterer velocitydistributions in the 3D isotropic and nonisotropic scatteringscenarios We use two uniform distributions in the isotropicscattering environment and use the von Mises distributionand the cosine distribution in the nonisotropic scatteringenvironment to describe all the statistical azimuth andelevation angles in Table 1 respectively The average speedof the uniform and half-Gaussian distributions is 25ms
10 Mobile Information Systems
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
065
07
075
08
085
09
095
1
|R11
11(휏)|
Figure 7 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropic andnonisotropic scattering scenarios with119872 = 5 for different scatterervelocity distributions
and the standard deviation of the half-Gaussian distribu-tion is 31ms The other parameters used to obtain curvesin Figures 6 and 7 are summarized in (Table 2 Cols 2and 3) As shown in Figure 6 the ST-CF for the uniformdistributed velocity of scatterers in the isotropic scatteringenvironment decreases fastest and both the nonisotropicscattering and the half-Gaussian distribution can result inswift oscillations of the fading curvesThe comparison resultsindicate that the scatterer velocity distributions have moresignificant effect than the scattering forms on the ST-CFsof the 3D V2V channel in the presence of multiple movingscatterers Figure 7 indicates that not like scattering formsthe scatterer velocity distributions have significant effect onthe T-CFs Generally the realistic propagation environmentis nonisotropic scattering and the scatterer velocity canbe modeled by the half-Gaussian or uniform distributionTherefore the results of Figures 6 and 7 may be helpful inanalysis of the system performance and modeling a specificpropagation channel for V2V communications
413 ST-CFs and S-CFs for Different Scattering Scenariosand Spacing between Adjacent Antenna Elements Figures8 and 9 demonstrate the 3D ST-CFs and S-CFs for thedifferent spacing between adjacent antenna elements of 119879119883and 119877119883 in isotropic and nonisotropic scattering scenariosThe set of the scatterer velocity in Figure 8 is the sameas Figures 2ndash5 and the other parameters used to obtaincurves in Figures 8 and 9 are summarized in (Table 2 Col4) Figures 8 and 9 show that spacing between adjacentantenna elements has significant effect on the 3D ST-CFsand S-CFs Furthermore the 3D ST-CFs in isotropic andnonisotropic scattering scenarios decrease to the bottomwhen 119889119877 = 119889119879 tends to 12120582 and 07120582 respectively Inaddition the 3D S-CF in nonisotropic scattering scenarios isexplicitly less than that in isotropic scattering scenarios for
|R11
22(d
Td
R휏)|
dR = d
T (휆) 휏 (ms)
0025
05075 1
12515 0
12
34
1
095
09
085
08
075
07
065
IsotropicNonisotropic
Figure 8 3D ST-CFs in (16) in the isotropic and nonisotropicscattering scenarios with119872 = 5 120582 is the carrier wavelength
|R11
22(d
Td
R)|
dR (휆)
dT (휆)
1
095
09
085
08
075
07
065
0 025 05 0751 125 15
0025
05075 1
12515
IsotropicNonisotropic
Figure 9 3D S-CFs in (16) with 120591 = 0 in the isotropic andnonisotropic scattering scenarios 120582 is the carrier wavelength
most spacing between adjacent antenna elements as shown inFigure 9Therefore the large-scale antenna arrays employingthe spatial multiplexing are more suitable to be used in thenonisotropic scattering scenarios owing to the high antennaspace utilization The spacing between adjacent antennaelements of antenna arrays is suggested to be more than 120582in the V2V communications especially in the environmentswith low-density scatterers
42 Comparison with Measurements The parametric natureof the proposed channel model makes it adaptable to avariety of propagation environments D-PSD is one of themost important and unique channel characteristics for V2Vcommunication channels To illustrate the validity of theproposed model we compare the modeled SD-PSDs of the3D V2V channel with multiple moving scatterers with themeasured F2F D-PSD in [26] the theoretical V2V SD-PSDin [18] and the measured V2V SD-PSD in [27]
Mobile Information Systems 11
minus2 minus15 minus1 minus05 0 05 1 15 2Doppler frequency f (kHz)
Figure 10 Comparison between the modeled SD-PSDs with119872 = 5and the measured D-PSDs of Figure 4 in [26]
Figure 10 shows the comparison between the modeledSD-PSDs obtained from the Fourier transform of the ST-CF presented in (16) in the isotropic scattering scenario andthe measured D-PSDs of Figure 4 in [26] The outdoor mea-surement experiments in [26] have shown that at millimeterwavelengths fading caused by foliage movement and themotion of nearby vehicles is a significantly deleterious effectin systems with static subscribers Figure 4 in [26] showsthe D-PSD generated by foliage movements and the D-PSDcaused by passing vehicles both in a fixed wireless channelat a carrier frequency of 295GHz We have used the half-Gaussian distribution with an average speed of 04ms andthe standard deviation of 05ms to model the velocity ofmoving foliage scatterers while the uniformdistributionwithan average speed of 6ms has been used tomodel the velocityof passing vehicles scatterers The power weight of the 119898thcluster of rays scattered by the moving foliage or passingvehicles has been set to 1199011 = 916 1199012 = 516 1199013 = 116and 1199014 = 1199015 = 132 The other parameters used to obtainthe modeled SD-PSD curves in Figure 10 are summarizedin (Table 2 Col 5) As can be observed in Figure 10 40 dBfading bandwidths for the effect of the foliage and vehiclesmovements are 500Hz and 1 kHz approximately respectivelyFade depth due to foliage is less than the variation due topassing vehicles Close agreements between the modeled andmeasured results are shown in Figure 10These close matchescan be seen not only for relatively slow moving scatterers butalso for relatively fast moving scatterers as well
Figure 11 shows the comparison among the modeled SD-PSD obtained from the Fourier transform of the ST-CFpresented in (16) in the nonisotropic scattering scenario thetheoretical SD-PSD of Figure 8 in [18] and themeasured SD-PSD of Figure 11 in [27]The theoretical SD-PSD reproducedhere is based on the two-cylinder model with moving andstationary scatterers for 119889119877 = 119889119879 = 0 The channel mea-surements in [27] for the measured SD-PSD were collectedat 2435GHz in the urban street surface environment and thespacing between two adjacent antenna elements of119879119883 and119877119883was set to zero We have used the half-Gaussian distributionwith average speeds of 2394ms 0798ms and 04ms
Figure 11 Comparison among the modeled SD-PSD with119872 = 5the theoretical SD-PSD of Figure 8 in [18] and the measured SD-PSD of Figure 11 in [27]
and the standard deviation of 3ms 1ms and 05ms tomodel the velocity of the first the second and the remainingmoving scatterers on the communication links respectivelyThe power weight of the 119898th cluster of rays has been set to1199011 = 815 1199012 = 13 1199013 = 115 1199014 = 130 and 1199015 = 130for119872 = 5 The other parameters used to obtain the modeledSD-PSD curve in Figure 11 are summarized in (Table 2 Col6) We note that different from the measured SD-PSDs in[26] themeasured SD-PSD curve in [27] has oscillationsTheoscillating pattern in measurements for the case 119889119877 = 119889119879 = 0may appear because it is not a true SISO scenario whereonly one transmit and one receive antenna are active [27] Ascan be observed in Figure 11 the modeled SD-PSD matchesbetter withmeasured SD-PSD than the theoretical SD-PSD in[18] This may be because of the introduction of the multiplebounced rays into the propagation model or the constraintson the position of local scatterers in the two-cylinder modelproposed in [18]
Figure 12 demonstrates the CDFs of the relative devi-ations between modeledtheoretical SD-PSDs and corre-sponding measured values in Figures 10 and 11 The meanrelative deviation of the modeled SD-PSDs for foliage inFigure 10 modeled SD-PSDs for vehicles in Figure 10 mod-eled SD-PSDs in Figure 11 and theoretical SD-PSDs inFigure 11 is 256 216 39 and 832 respectively Asshown in Figure 12 more than 80 of the relative deviationsof modeled SD-PSDs in Figures 10 and 11 are less than5 and the relative deviations of theoretical SD-PSDs inFigure 11 are explicitly larger than those of modeled SD-PSDs In addition from Figures 10 and 11 we note that themodeled SD-PSD matches better with measured SD-PSDs atthe lower and higher Doppler frequencies than the middleDoppler frequencies This may be because 119879119883 and 119877119883 werenot strictly fixed and not all the scatterers were in motion inmeasurementsThe approximate calculation of the frequencyshift in (A4) and Doppler shift in (10) may be anotherfactor which causes the mismatch at the middle Dopplerfrequencies andmore investigations concerning the accuracyof the proposed model will be addressed in future works
12 Mobile Information Systems
0 5 10 15 20 25 30 350
01
02
03
04
05
06
07
08
09
1
Relative deviation ()
Relat
ive d
evia
tion
CDF
Modeled SD-PSD (foliage)Modeled SD-PSD (vehicles)
Modeled SD-PSDTheoretical SD-PSD
Figure 12 The relative deviation cumulative distribution functions(CDFs) of the modeled and theoretical SD-PSDs in Figures 10 and11
However as shown in Figures 10ndash12 the acceptable matchesconfirm the utility and generality of the proposed model andshow the need for including multiple moving scatterers inpropagation model
5 Conclusion
Without specific constraints on the position of the local mov-ing scatterers a 3D geometrical V2V propagation model thatincludes LoS single bounced and multiple bounced linksbetween the transmitter and receiver was proposed Basedon the geometrical propagation model a 3D reference modelfor narrowband MIMO V2V multipath fading channels wasdeveloped From the reference model the correspondingmathematical expressions and numerical results of ST-CFsT-CFs S-CFs and SD-PSDs were studied for different para-metric sets It has been shown that the maximum bouncesscattering forms scatterer velocity distributions and spacingbetween adjacent antenna elements have significant effect onthe ST-CFs and only the maximum bounces and scatterervelocity distributions have significant effect on the T-CFsFinally the modeled SD-PSD results were compared withthe measured data and the other literaturersquos theoreticalresult The close agreements between the analytically andempirically obtained channel statistics confirmed the utilityand generality of the proposed model
Appendix
A Derivation Process of Frequency Shifts
The derivation process of 119891119902119901119902119899 in (7) is presented in thissection and 119891119901119901119902120588 119891119902119901119902119899 119891119901119901119902119899 and 119891119901119902120588 can be calculatedsimilarly
AOD
Δl
TX qth element
Figure 13 Projection from the coordinate vector of 119902th antennaelement to the AOD direction
According to Figure 1 the coordinate vectors of 119902thantenna element and the AOD direction can be written asrespectively
minus 1) 119889119879 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902 + sin120573119902 sin120573119879119901119902] (A3)
Because we mainly consider the difference of the fre-quency shifts caused by the TPD changes to calculate ST-CFsthe TPD change of ℎ11(119905) is assumed to be zero Eventually119891119902119901119902119899 can be calculated as
119891119902119901119902119899 asymp Δ1198971198910119888= (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(A4)
B Derivation Process of Approximate ST-CFs
In the 2D scattering environment the simplified and approx-imate expressions of (28) can be derived in this section If120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 (27) and (10) can be writtenas
Δ119891119899119898
= 1198910119888 [(119902 minus 119902) 119889119879 cos (120572119902 minus 120572119879119901119902)+ (119901 minus 119901) 119889119877 cos (120572119901 minus 120572119877119901119902)]
Mobile Information Systems 13
119891119901119902119899119898
= 1198910119888 [V119879 cos (120572119879 minus 120572119879119901119902) + V119877 cos (120572119877 minus 120572119877119901119902)
+ ]1119901119902 cos (1205721119901119902 minus 120572119879119901119902) + 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119879119901119902)+ ]119898119901119902 cos (120572119898119901119902 minus 120572119879119901119902) + ]1119901119902 cos (1205721119901119902 minus 120572119877119901119902)+ 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119877119901119902) + ]119898119901119902 cos (120572119898119901119902 minus 120572119877119901119902)] (B1)
If 120572119879 = 120572119902 and 120572119877 = 120572119901 that is the antenna array and itsvelocity have the same directions (28) can be simplified as bysubstituting (B1) into (28)
where 1198960 = 21205871198910119888 is the wave numberIn isotropic scattering environment we use the uniform
distribution in (31) to describe all the random azimuth anglesIn this case (B2) can be written as119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879] cos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877] cos (120572119877 minus 120572119877119901119902)
times1198690 [21198960]1119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times1198690 [21198960]119898119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times119898minus1prod119894=2
1198690 [41198960]119894119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind
The AOD 120572119879119901119902 and AOA 120572119877119901119902 are uniformly and indepen-dently distributed so120572119879119901119902minus120572119877119901119902 can be equal to zero in average
As a consequence the term cos[(120572119879119901119902 minus120572119877119901119902)2] in (B3) can beapproximated by one [13 14] and an approximate ST-CF canbe written as from (B3)
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879]sdotcos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877]sdot cos (120572119877 minus 120572119877119901119902) 1198690 (21198960]1119901119902120591)times1198690 (21198960]119898119901119902120591)119898minus1prod
Eventually the approximate ST-CF in (34) can be derivedfrom (B4) according to the definition of 1198690(∙)Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was funded by the National Natural Science Foun-dation of China (no 91438104 no 61501065 no 61571069and no 61601067) the Fundamental Research Funds forthe Central Universities (no 106112016CDJX160001) and theChongqing Research Program of Basic Research and FrontierTechnology (no CSTC2016JCYJA0021)
References
[1] A Bazzi B M Masini A Zanella and I Thibault ldquoBeaconingfrom connected vehicles IEEE 80211p vs LTE-V2Vrdquo in Pro-ceedings of 2016 IEEE 27th Annual International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRC)pp 1ndash6 Valencia Spain September 2016
[2] VM Rodrigo-Penarrocha J Reig L Rubio and et al ldquoAnalysisof small- scale fading distributions in vehicle-to-vehicle com-municationsrdquoMobile Information Systems vol 2016 Article ID9584815 7 pages 2016
[3] A Bazzi B M Masini A Zanella and G Pasolini ldquoIEEE80211p for cellular offloading in vehicular sensor networksrdquoComputer Communications vol 60 pp 97ndash108 2015
[4] X Cheng MIMO channel modelling and simulation for cellu-lar and mobile-to-mobile communication systems[Phd thesis]Heriot-Watt University 2009
[5] N Avazov and M Patzold ldquoA geometric street scatteringchannel model for car-to-car communication systemsrdquo inProceedings of 4th Annual International Conference onAdvanced
14 Mobile Information Systems
Technologies for Communications ATC2011 pp 224ndash230 vnmAugust 2011
[6] X Zhao X Liang S Li and K Haneda ldquoMobile-to-MobileWideband MIMO Channel Realization by Using a Two-RingGeometry-Based Stochastic Scattering Modelrdquo Wireless Per-sonal Communications vol 84 no 4 pp 2445ndash2465 2015
[7] M Walter D Shutin and U-C Fiebig ldquoDelay-dependentdoppler probability density functions for vehicle-to-vehiclescatter channelsrdquo IEEE Transactions on Antennas and Propaga-tion vol 62 no 4 pp 2238ndash2249 2014
[8] P T Samarasinghe T A Lamahewa T D Abhayapala and RA Kennedy ldquo3D mobile-to-mobile wireless channel modelrdquo inProceedings of 2010 Australian Communications Theory Work-shop AusCTW 2010 pp 30ndash34 aus February 2010
[9] M Riaz S J Nawaz and N M Khan ldquo3D ellipsoidal model formobile-to-mobile radio propagation environmentsrdquo WirelessPersonal Communications vol 72 no 4 pp 2465ndash2479 2013
[10] J Chen and T G Pratt ldquoThree-dimensional geometry-based stochastic modeling and performance of 4times4 space-polarization mobile-to-mobile wideband MIMO channelsrdquo inProceedings of 2013 IEEE Global Communications ConferenceGLOBECOM 2013 pp 3936ndash3941 usa December 2013
[11] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA Novel 3DNonstationary Channel Model Based on the von Mises-FisherScattering DistributionrdquoMobile Information Systems vol 2016Article ID 2161460 2016
[12] A Chelli and M Patzold ldquoThe impact of fixed and movingscatterers on the statistics of MIMO vehicle-to-vehicle chan-nelsrdquo in Proceedings of VTC Spring 2009 - IEEE 69th VehicularTechnology Conference esp April 2009
[13] A Borhani and M Patzold ldquoModeling of vehicle-to-vehiclechannels in the presence of moving scatterersrdquo in Proceedingsof 2012 IEEE Vehicular Technology Conference (VTC Fall) pp1ndash5 Quebec City QC Canada September 2012
[14] A Borhani andM Patzold ldquoCorrelation and spectral propertiesof vehicle-to-vehicle channels in the presence ofmoving scatter-ersrdquo IEEE Transactions on Vehicular Technology vol 62 no 9pp 4228ndash4239 2013
[15] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the IEEE 74th Vehicular TechnologyConference (VTC Fall rsquo11) pp 1ndash6 IEEE San Francisco CalifUSA September 2011
[16] M D Soltani M Alimadadi Y Seyedi and H AmindavarldquoModeling of doppler spectrum in V2V urban canyon oncom-ing environmentrdquo in Proceedings of 2014 7th InternationalSymposium on Telecommunications IST 2014 pp 1155ndash1160 irnSeptember 2014
[17] M D Soltani M Alimadadi and A Mohammadi ldquoModelingof mobile scatterer clusters for doppler spectrum in widebandvehicle-to-vehicle communication channelsrdquo IEEE Communi-cations Letters vol 18 no 4 pp 628ndash631 2014
[18] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014
[19] A G Zajic ldquoModeling impact of moving scatterers on dopplerspectrum in wideband vehicle-to-vehicle channelsrdquo in Proceed-ings of Proc Eur Conf Antennas Propag p 1 Lisbon May 2015
[20] J Zhang X Yin and X Cheng ldquoTheoretical analysis andmeasurements Doppler spectra of vehicular communication
channelsrdquo in Proceedings of 2012 12th International Confer-ence on ITS Telecommunications ITST 2012 pp 98ndash102 twnNovember 2012
[21] M Picone S Busanelli M Amoretti F Zanichelli and GFerrari Advanced Technologies for Intelligent TransportationSystems Springer 2015
[22] V H Pham M H Taieb J Y Chouinard S Roy and H THuynh ldquoOn the double Doppler effect generated by scatterermotionrdquo REV Journal on Electronics and Communications vol1 pp 30ndash37 2011
[23] S Roy H T Huynh and P Fortier ldquoCompound Dopplerspread effects of subscriber motion and scatterer motionrdquo AEU- International Journal of Electronics and Communications vol57 no 4 pp 237ndash246 2003
[24] J B Andersen J O Nielsen G F Pedersen G Bauch and GDietl ldquoDoppler spectrum from moving scatterers in a randomenvironmentrdquo IEEE Transactions on Wireless Communicationsvol 8 no 6 pp 3270ndash3277 2009
[25] H S Rad S Gazor and P Shariatpanahi ldquoNon-fixed scatterersand their effects on MIMOmulticarrier fading communicationchannelsrdquo in Proceedings of 50th Annual IEEE Global Telecom-munications Conference GLOBECOM2007 pp 3765ndash3769 usaNovember 2007
[26] N Naz and D Falconer ldquoTemporal variations characterizationfor fixed wireless at 295 GHzrdquo in Proceedings of 2000 IEEE51st Vehicular Technology Conference Proceedings VTC2000-Springer pp 2178ndash2182 Tokyo Japan 2000
[27] A G Zajic G L Stuber T G Pratt and S T NguyenldquoWideband MIMO mobile-to-mobile channels Geometry-based statistical modeling with experimental verificationrdquo IEEETransactions on Vehicular Technology vol 58 no 2 pp 517ndash5342009
[28] W C Jakes Microwave Mobile Communications Wiley-IEEEPress Piscataway NJ USA 1994
[29] M Patzold B O Hogstad and N Youssef ldquoModeling analysisand simulation of MIMO mobile-to-mobile fading channelsrdquoIEEE Transactions onWireless Communications vol 7 no 2 pp510ndash520 2008
[30] M PatzoldMobile Radio Channels Wiley Chichester UK 2ndedition 2011
[31] R Wang and D Cox ldquoDouble mobility mitigates fading in adhoc wireless networksrdquo in Proceedings of IEEE Antennas andPropagation Society International Symposium pp 306ndash309 SanAntonio TX USA
[32] G Bakhshi K Shahtalebi and H S Rad ldquoA novel full-three-dimensional MIMOmobile-to-mobile channel referencemodelrdquo in Proceedings of 3rd International Conference on SignalProcessing and Communication Systems ICSPCSrsquo2009 usaSeptember 2009
[33] W Zhou X Wang X Wang and W Chen ldquoA modified two-erose- ringmodel formimomobile-to-mobile fading channelsrdquoin Proceedings of 7th International Conference on WirelessCommunications Networking and Mobile Computing WiCOM2011 5 1 pages Wuhan China September 2011
Figure 1 Geometrical propagation model with LoS and MB rays for the 3D MIMO V2V communication scenario Empty circles representthe antenna elements and solid squares represent the moving scatterers
Table 1 Definition of the parameters used in the geometrical model
Parameters Definition Attributes119878119894 The 119894th moving scatterer for the multiple bounces Symbol119863 The distance between 119879119883 and 119877119883 Deterministic119902 119901 The antenna element identifier of 119879119883 and 119877119883 respectively Deterministic119876 119875 The number of antenna elements of 119879119883 and 119877119883 respectively Deterministic119889119879 119889119877 The spacing between two adjacent antenna elements of 119879119883 and 119877119883 respectively Deterministic120572119902 120573119902 The azimuth angle and elevation angle of 119879119883rsquos antenna array respectively Deterministic120572119901 120573119901 The azimuth angle and elevation angle of 119877119883rsquos antenna array respectively Deterministic120572119879119901119902 120573119879119901119902 The azimuth angle and elevation angle of AOD respectively Statistical120572119877119901119902 120573119877119901119902 The azimuth angle and elevation angle of AOA respectively StatisticalV119879 V119877 The velocities of 119879119883 and 119877119883 respectively DeterministicV119894119901119902 The velocity of 119878119894 Statistical120572119879 120573119879 The azimuth angle and elevation angle of V119879 respectively Deterministic120572119877 120573119877 The azimuth angle and elevation angle of V119877 respectively Deterministic120572119894119901119902 120573119894119901119902 The azimuth angle and elevation angle of V119894119901119902 respectively Statistical
(v) The geometrical propagation model does not imposespecific constraints on the position of the local mov-ing scatterers like [13 14] Owing to high path loss weneglect the energy contribution of remote scatterers
22 Reference Model We can observe from Figure 1 that thecomplex faded envelope of the link between the 119902th antennaelement of 119879119883 and the 119901th antenna element of 119877119883 can bewritten as a superposition of LoS and MB components thatisℎ119901119902 (119905) = ℎLoS119901119902 (119905) + ℎMB
where 120588119901119902 119891119901119902120588 120579119901119902120588 119891119902119901119902120588 and 119891119901119901119902120588 denote the path gainDoppler shift phase shift frequency shift for 119902th antennaelement of119879119883 and frequency shift for119901th antenna element of119877119883 of ℎLoS119901119902 (119905) respectively119898 denotes the amount of bounces119872 is themaximumof119898119901119898 and119873119898 denote the power weightand path number of ℎ119898119901119902(119905) respectively 119888119901119902119899119898 119891119901119902119899119898 120579119901119902119899119898 119891119902119901119902119899 and 119891119901119901119902119899 denote the path gain Doppler shift phaseshift frequency shift for 119902th antenna element of 119879119883 andfrequency shift for 119901th antenna element of 119877119883 of the 119899119898thpath in ℎ119898119901119902(119905) The LoS component ℎLoS119901119902 (119905) can be describedby a complex sinusoid and (2) is an extension of the (317)in [30] ℎ119898119901119902(119905) denotes the NLoS component with119898 bounced
4 Mobile Information Systems
(119898B) rays and it is an extension of the (312) in [30] Notethat MB component ℎMB
119901119902 (119905) of the channel impulse responseconsists of 119872 clusters of rays reflected 119898 isin 1 2 3 119872times from moving scatterers
The channel gains (120588119901119902 119888119901119902119899119898
221 Channel Gains The central limit theorem states thatℎ119898119901119902(119905) equals a complex valued Gaussian random processwith zero mean and variance 21205902119898 = Varℎ119898119901119902(119905) =lim119873
119898rarrinfinsum119873119898119899
119898
119864[1198882119901119902119899119898
] 119901119898 in (3) denotes the power weightof the 119898th clusters of rays and sum119872119898=1 119901119898 = 1 The channelgain of ℎ119901119902(119905) is normalized ie 2sum119872119898=1 1199011198981205902119898 + 1205882119901119902 = 1and the Rice factor can be denoted as119870 = 12058821199011199022sum119872119898=1 1199011198981205902119898These parameters have to be either set during simulations orestimated from measurements
222 Doppler Shifts and Frequency Shifts The frequencyshifts 119891119902119901119902120588 119891119901119901119902120588 119891119902119901119902119899 and 119891119901119901119902119899 depend on the differenceof the propagation distance (TPD) changes between ℎ119901119902(119905)and ℎ11(119905) On the other hand the Doppler shifts 119891119901119902120588 and119891119901119902119899
119898
depend on the geometrical relation between directionsof movement of 119879119883 119877119883 and multiple moving scatterers andthe directions of AOD and AOA For max(119889119879 119889119877) ≪ 119863 boththe AOD and AOA of LoS rays are approximately equal tozero Appendix A shows that119891119902119901119902120588119891119901119901119902120588119891119902119901119902119899119891119901119901119902119899 and119891119901119902120588are respectively
119891119902119901119902120588 = (119902 minus 1) 1198891198791198910119888 cos120572119902 cos120573119902 (5)
119891119901119901119902120588 = (119901 minus 1) 1198891198771198910119888 cos120572119901 cos120573119901 (6)
119891119902119901119902119899 = (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(7)
119891119901119901119902119899 = (119901 minus 1) 1198891198771198910119888 [cos (120572119901 minus 120572119877119901119902) cos120573119901 cos120573119877119901119902+ sin120573119901 sin120573119877119901119902]
where 119875AOD119894119901119902 and 119875AOA119894119901119902 (119894 = 1 2 3 119898) are119875AOD119894119901119902 = cos (120572119894119901119902 minus 120572119879119901119902) cos120573119894119901119902 cos120573119879119901119902
+ sin120573119894119901119902 sin120573119879119901119902119875AOA119894119901119902 = cos (120572119894119901119902 minus 120572119877119901119902) cos120573119894119901119902 cos120573119877119901119902
Note that if 120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 and 119898 = 1(10) equals (7) in [13] and (5) in [14] regardless of the plus orminus signsThe differences among these plus or minus signsare caused by the different forms of anglesrsquo expression
223 Phases The phase shift 120579119901119902120588 can be assumed to beconstant [30] The phase shift 120579119901119902119899
119898
consists of the phasechange caused by the interaction of the transmitted signalwith the scatterers and the phase change caused by theTPD between the first and the last scatterers Without lossof generality we can assume that the phases 120579119901119902119899
119898
(119898 =1 2 3 119872) are independent random variables Here it isassumed that they are uniformly distributed on the interval[0 2120587) and independent of any other random variable
Mobile Information Systems 5
3 Space-Time Correlation Function andSpace-Doppler Power Spectral Density
Using the referencemodel described in Section 2 we can nowderive the key temporal and spatial characteristics of MIMOV2V narrowband multipath fading channels with the localmultiple moving scatters
31 Space-Time Correlation Function The normalized ST-CFbetween two complex faded envelopes ℎ119901119902(119905) and ℎ119901119902(119905) isdefined as
where (∙)lowast denotes the complex conjugate operation 119864(∙) isthe statistical expectation operator 119901 119901 isin 1 2 3 119875 and119902 119902 isin 1 2 3 119876 The normalized T-CF can be obtainedif 119901 = 119901 and 119902 = 119902 in (15) The normalized space correlationfunction (S-CF) can be obtained by setting 120591 to zero in (15)
Since ℎ1119901119902(119905) ℎ2119901119902(119905) ℎ119872119901119902(119905) and ℎLoS119901119902 (119905) are indepen-dent of each other (15) can be simplified to
where 119877119898119901119902119901119902(119889119879 119889119877 120591) and 119877LoS119901119902119901119902(119889119879 119889119877 120591) denote the nor-malized ST-CFs of the119898B and LoS components respectivelyand they are defined as
311 ST-CF of LoS Component By substituting (2) into (18)the expression for the ST-CF of the LoS component can bewritten as
119877LoS119901119902119901119902 (119889119879 119889119877 120591)= 120588119901119902120588119901119902119864 exp 119895 [2120587119905 (119891119901119902120588 minus 119891119901119902120588) + 2120587119891119901119902120588120591+ 2120587 (119891119902119901119902120588 + 119891119901119901119902120588 minus 119891119902119901119902120588 minus 119891119901119901119902120588) + 120579119901119902120588minus 120579119901119902120588]
(19)
For max(119889119879 119889119877) ≪ 119863 we assume 120588119901119902 = 120588119901119902 and 120579119901119902120588 =120579119901119902120588 Then (19) can be written as
Δ119891120588 = 119891119902119901119902120588 + 119891119901119901119902120588 minus 119891119902119901119902120588 minus 119891119901119901119902120588 (21)
By substituting (5) and (6) into (21) Δ119891120588 can be written as
Δ119891120588 = 1198910119888 [(119902 minus 119902) 119889119879 cos120572119902 cos120573119902+ (119901 minus 119901) 119889119877 cos120572119901 cos120573119901]
(22)
312 ST-CF of 119898119861 Component By substituting (4) into (17)the expression for the ST-CF of the 119898B component can bewritten as
times 119864 exp 119895 [2120587 (119891119901119902119899119898
minus 119891119901119902119899119899
) 119905 + 2120587119891119901119902119899119898
120591 + 2120587 (119891119902119901119902119899 + 119891119901119901119902119899 minus 119891119902119901119902119899 minus 119891119901119901119902119899) + 120579119901119902119899119898 minus 120579119901119902119899119899] (23)
Under the assumption that 120579119901119902119899119898
and 120579119901119902119899119899
are uniformlydistributed on the interval [0 2120587) and independent of eachother 119864exp[119895(120579119901119902119899
119898
minus120579119901119902119899119899
)] equals 1 It is assumed that allthe path gains of the119898B component have the same size thatis
= 119891119902119901119902119899 + 119891119901119901119902119899 minus 119891119902119901119902119899 minus 119891119901119901119902119899 (26)
It is assumed that AOD and AOA of ℎ119898119901119902(119905) and ℎ119898119901119902(119905) areindependent and identically distributed (iid) By substitut-ing (7) and (8) into (26) Δ119891119899
Since the number of local scatterers in the referencemodel is infinite the parameters 120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902 120572119894119901119902and 120573119894119901119902 can be seen as continuous random variables withcorresponding probability density functions (PDFs) Thenthe ST-CF of the 119898B component (25) can be writtenas
Now the complete expression of (16) can be obtainedby substituting (20) and (28) into (16) 119901(120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902)describes the joint distribution of AOD and AOA and itcan be optionally used to present some propagation channelmodels As a result the ST-CF in (28) can provide a suitableplatform to study the statistical properties of some differentchannelmodels such as the random scatteringmodel [13 14]Jakes model [28] one-ring model [23] and two-ring model[29] as described in Section 313Therefore the ST-CF in (28)is a generalized and parametric expression However due tothe complex nature of 119901(120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902) it is assumed thatAOD and AOA are independent [13 14 32 33] and azimuthangles and elevation angles in (28) are also independent[18 19 32] The parameters in (28) such as the velocities ofthe multiple moving scatterers and random angles can becalculated as follows
(i) Scatterer Velocity Distributions The Gaussian Laplaceexponential and uniform distributions can be used todescribe the velocity of moving scatterers [13] In fact thescatterer velocity ]119894119901119902 is always positive or equal to zeroWe use the uniform distribution in (29) and half-Gaussiandistribution in (30) to describe the velocity of multiplemoving scatterers
119901 (]119894119901119902) = 1V119894max 0 le ]119894119901119902 le V119894max (29)
where V119894max is the maximum of ]119894119901119902
21205902119901119902119894]] ]119894119901119902 ge 0 (30)
where 120590119901119902119894 is the standard deviation of ]119894119901119902
(ii) Angle Distributions To characterize the statistical anglesin Table 1 we use the uniform distribution in (31) in theisotropic scattering environment and use the von Misesdistribution in (32) and the cosine distribution in (33) inthe nonisotropic scattering environment In addition theinterval of azimuth angles 120572119879119901119902 120572119877119901119902 and 120572119894119901119902 is (minus120587 120587] and theinterval of elevation angles 120573119879119901119902 120573119877119901119902 and 120573119894119901119902 is (minus1205872 1205872]119901 (120574) = 11205742 minus 1205741 1205741 le 120574 le 1205742 (31)
119901 (120572) = exp [119896 cos (120572 minus 120572)]21205871198680 (119896) 1205721 le 120572 le 1205721 + 2120587 (32)
where 1198680(∙) is the zeroth-order modified Bessel function ofthe first kind 120572 is the mean angle and 119896 controls the spreadof angles around the mean The von Mises distribution PDFwith 120572 = 0 is used to describe the azimuth angles
119901 (120573) = 1205874 10038161003816100381610038161205731198981003816100381610038161003816 cos(1205872 120573120573119898) minus 120573119898 le 120573 le 120573119898 (33)
where 120573119898 is the maximum of 120573 The cosine distribution PDFis used to describe the elevation angles
313 Special Cases of 119898119861rsquos ST-CF If 120573119879 = 120573119877 = 120573119879119901119902 =120573119877119901119902 = 120573119894119901119902 = 0 that is the scattering environment is 2D
Mobile Information Systems 7
some different special cases can be derived from the generalexpression of the119898Brsquos ST-CF in (28)
In the 2D scattering environment (28) can be written as(B2) in Appendix B If 119901 = 119901 119902 = 119902 and119898 = 1 (B2) equals(11) in [13] and (6) in [14] regardless of the plus or minussigns under the assumption that the angles 120572119879 and 120572119877 areindependent of each other in [13 14] In isotropic scatteringenvironments some other special cases with closed-formexpressions can be derived as follows
Appendix B shows that an approximate ST-CF of (B2)can be written as
119877119898119901119902119901119902 (119889119879 119889119877 120591) = 212059021198981198690 1198960 [V119879120591 + (119902 minus 119902) 119889119879]times 1198690 1198960 [V119877120591 + (119901 minus 119901) 119889119877] int sdot sdot sdot int 1198690 (21198960]1119901119902120591)times 1198690 (21198960]119898119901119902120591)119898minus1prod
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind and 1198960 = 21205871198910119888 is the wave number
Note that if 119898 approaches to the infinity (34) can bewritten as
lim119898rarrinfin
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 212059021198981198690 [(119902 minus 119902) 1198960119889119879] 1198690 [(119901 minus 119901) 1198960119889119877] 120591 = 00 120591 gt 0
(35)
In the NLoS communication environment with veryhigh-density scatterers (35) implies that the ST-CFapproaches to zero at the nonzero time difference 120591 and hasnothing to do with velocities of scatterers 119879119883 and 119877119883 Thelarge-scale antenna arrays are very suitable to be used in thisenvironment because the antenna element spacing can bereduced to a smaller value
If ]119894119901119902 is constant (34) can be presented as the followingclosed-form expression
The T-CF of the classical F2M scenario with fixed scat-terers is obtained if V119879 = V119894119901119902 = 0 (119894 = 1 2 3 119898)119902 = 119902 and 119901 = 119901 in (36) In this case the ST-CF in(36) is 212059021198981198690(1198960V119877120591) which is known as the Jakes model[28] If V119879 = 0119898 = 1 119902 = 119902 and 119901 = 119901 the ST-CFin (36) results in 212059021198981198690(1198960V1119901119902120591)1198690(1198960V119877120591) which equals the
T-CF of F2M single-ring channel model in the presence ofmoving scatterers reported in (11) of [23] If V119894119901119902 = 0 (119894 =1 2 3 119898) 119902 = 119902 and 119901 = 119901 the ST-CF in (36) resultsin 212059021198981198690(1198960V119879120591)1198690(1198960V119877120591) which equals the T-CFs of theclassical M2M two-ring channel model in the presence offixed scatterers reported in (46) of [29] If V119894119901119902 = 120591 = 0 (119894 =1 2 3 119898) 119902 = 119901 = 2 and 119902 = 119901 = 1 the ST-CF in (36)results in 212059021198981198690(1198960119889119879)1198690(1198960119889119877) which equals the S-CF in (46)of [29]
The velocities ofmoving scatterers such asmoving foliagewalking pedestrians and passing vehicles generally are ran-dom variables If ]119894119901119902 is described by the uniform distributionin (29) (34) can be presented as the following closed-formexpression
times 1198690 1198960 [V119877120591 + (119901 minus 119901) 119889119877] [21198690 (21198960V1max120591)+ 1205871198691 (21198960V1max120591)1198670 (21198960V1max120591) minus 1205871198690 (21198960V1max120591)times 1198671 (21198960V1max120591)] [21198690 (21198960V119898max120591)+ 1205871198691 (21198960V119898max120591)1198670 (21198960V119898max120591) minus 1205871198690 (21198960V119898max120591)sdot 1198671 (21198960V119898max120591)] times 119898minus1prod
119894=2
[21198690 (41198960V119894max120591)+ 1205871198691 (41198960V119894max120591) times 1198670 (41198960V119894max120591)minus 1205871198690 (41198960V119894max120591)1198671 (41198960V119894max120591)]
(37)
where 1198691(∙) denotes the first-order Bessel function of the firstkind 1198670(∙) denotes the zeroth-order Struve function and1198671(∙) denotes the first-order Struve function
If ]119894119901119902 is described by the half-Gaussian distribution in(30) (34) can be presented as the following closed-formexpression
32 Space-Doppler Power Spectral Density The SD-PSD canbe obtained by taking the Fourier transform of the ST-CF in(16)with respect to time difference 120591 From (16) it follows that
8 Mobile Information Systems
Table 2 Parameters used in the numerical simulation
where 120575(∙) is the Dirac delta functionSince the ST-CF of the 119898B component is the multiple
integral as (28) the closed-form expression of SD-PSDcannot be derived In Section 42 we show the SD-PSD of the3DV2Vchannel in the presence ofmultiplemoving scatterersby means of numerical integrations
4 Numerical Results and Validation
This section demonstrates the normalized correlation func-tions and validates the Doppler power spectral density (D-PSD) described in Section 3 through theMATLABnumericalsimulations Unless indicated otherwise the values of thenumerical simulation parameters are summarized in Table 2
41 Numerical Results In this section the numerical curvesof ST-CFs T-CFs and S-CFs influenced by some importantcontributory factors are presented As in the typical urbanenvironments the power weight of the119898th cluster of rays hasbeen set to 1199011 = 1199012 = 12 for119872 = 2 1199011 = 1199012 = 1199013 = 13 for119872 = 3 1199011 = 1199012 = 13 1199013 = 1199014 = 16 for119872 = 4 1199011 = 131199012 = 14 1199013 = 1199014 = 16 and 1199015 = 112 for119872 = 5411 ST-CFs and T-CFs for Different Scattering Scenarios andBounces Figures 2ndash5 demonstrate the ST-CFs and T-CFs forthe different maximum bounces 119872 in the 3D isotropic and
0 05 1 15 2 25 3 35 4066
067
068
069
07
071
072
휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 2 ST-CFs in (16) in the isotropic scattering scenario fordifferent maximum bounces
nonisotropic scattering scenarios The scatterer velocity isuniformly distributed with an average speed of 25ms thatmay be the velocity of passing vehicles The other parametersused to obtain curves in Figures 2ndash5 are summarized in(Table 2 Cols 2 and 3) As shown in Figures 2 and 4 thelarger119872 is the faster the ST-CFs and T-CFs decrease in theisotropic scattering scenario However the descent rates ofST-CFs and T-CFs increase slowly when119872 is larger than 3The similar conclusions can be obtained fromFigures 3 and 5however the ST-CF curves in Figure 3 have oscillationswhichmay be caused by the nonisotropic scattering However thisconclusion differs from [27] which showed that the triple-or higher-order bounced rays had statistical properties verysimilar to those of the double-bounced rays and could beapproximated as double-bounced rays The discrepancy maybe caused by the different communication environmentsconsidered by us and [27] Specifically channel-soundingexperimental campaign in [27] was conducted along surface
Mobile Information Systems 9
066
067
068
069
07
071
072
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 3 ST-CFs in (16) in the nonisotropic scattering scenario fordifferent maximum bounces
065
07
075
08
085
09
095
1
M = 1M = 2M = 3
M = 4M = 5
0 05 1 15 2 25 3 35 4휏 (ms)
|R11
11(휏)|
Figure 4 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropicscattering scenario for different maximum bounces
streets around the Georgia Tech campus and on the Interstatehighways in the Midtown Atlanta metropolitan area whichmay have lower dense scatterers than our communicationenvironments The ST-CF curves in Figure 3 converge rel-atively slower for the larger 119872 that means there are morelocal scatterers inmotionwith randomvelocities and randomdirections As can be observed in Figures 4 and 5 theT-CFs inboth figures are very similar that indicates that the scatteringforms have no significant effect on the T-CFs of the 3D V2Vchannel in the presence of multiple moving scatterers Fur-thermore in Figures 2ndash5 it can be observed that a tendencyto a limit value curve of ST-CFs and T-CFs can be inferred if
065
07
075
08
085
09
095
1
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
|R11
11(휏)|
Figure 5 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the nonisotropicscattering scenario for different maximum bounces
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
066
067
068
069
071
072
1122(d
T=
05휆d
R=
05휆휏
)||R
07
Figure 6 ST-CFs in (16) in the isotropic and nonisotropic scatteringscenarios with119872 = 5 for different scatterer velocity distributions
119872 approaches infinity and the similar inference also can beobtained in (35) for the 2D scattering environment
412 ST-CFs and T-CFs for Different Scattering Scenarios andScatterer Velocity Distributions Figures 6 and 7 demonstratethe ST-CFs and T-CFs for the different scatterer velocitydistributions in the 3D isotropic and nonisotropic scatteringscenarios We use two uniform distributions in the isotropicscattering environment and use the von Mises distributionand the cosine distribution in the nonisotropic scatteringenvironment to describe all the statistical azimuth andelevation angles in Table 1 respectively The average speedof the uniform and half-Gaussian distributions is 25ms
10 Mobile Information Systems
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
065
07
075
08
085
09
095
1
|R11
11(휏)|
Figure 7 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropic andnonisotropic scattering scenarios with119872 = 5 for different scatterervelocity distributions
and the standard deviation of the half-Gaussian distribu-tion is 31ms The other parameters used to obtain curvesin Figures 6 and 7 are summarized in (Table 2 Cols 2and 3) As shown in Figure 6 the ST-CF for the uniformdistributed velocity of scatterers in the isotropic scatteringenvironment decreases fastest and both the nonisotropicscattering and the half-Gaussian distribution can result inswift oscillations of the fading curvesThe comparison resultsindicate that the scatterer velocity distributions have moresignificant effect than the scattering forms on the ST-CFsof the 3D V2V channel in the presence of multiple movingscatterers Figure 7 indicates that not like scattering formsthe scatterer velocity distributions have significant effect onthe T-CFs Generally the realistic propagation environmentis nonisotropic scattering and the scatterer velocity canbe modeled by the half-Gaussian or uniform distributionTherefore the results of Figures 6 and 7 may be helpful inanalysis of the system performance and modeling a specificpropagation channel for V2V communications
413 ST-CFs and S-CFs for Different Scattering Scenariosand Spacing between Adjacent Antenna Elements Figures8 and 9 demonstrate the 3D ST-CFs and S-CFs for thedifferent spacing between adjacent antenna elements of 119879119883and 119877119883 in isotropic and nonisotropic scattering scenariosThe set of the scatterer velocity in Figure 8 is the sameas Figures 2ndash5 and the other parameters used to obtaincurves in Figures 8 and 9 are summarized in (Table 2 Col4) Figures 8 and 9 show that spacing between adjacentantenna elements has significant effect on the 3D ST-CFsand S-CFs Furthermore the 3D ST-CFs in isotropic andnonisotropic scattering scenarios decrease to the bottomwhen 119889119877 = 119889119879 tends to 12120582 and 07120582 respectively Inaddition the 3D S-CF in nonisotropic scattering scenarios isexplicitly less than that in isotropic scattering scenarios for
|R11
22(d
Td
R휏)|
dR = d
T (휆) 휏 (ms)
0025
05075 1
12515 0
12
34
1
095
09
085
08
075
07
065
IsotropicNonisotropic
Figure 8 3D ST-CFs in (16) in the isotropic and nonisotropicscattering scenarios with119872 = 5 120582 is the carrier wavelength
|R11
22(d
Td
R)|
dR (휆)
dT (휆)
1
095
09
085
08
075
07
065
0 025 05 0751 125 15
0025
05075 1
12515
IsotropicNonisotropic
Figure 9 3D S-CFs in (16) with 120591 = 0 in the isotropic andnonisotropic scattering scenarios 120582 is the carrier wavelength
most spacing between adjacent antenna elements as shown inFigure 9Therefore the large-scale antenna arrays employingthe spatial multiplexing are more suitable to be used in thenonisotropic scattering scenarios owing to the high antennaspace utilization The spacing between adjacent antennaelements of antenna arrays is suggested to be more than 120582in the V2V communications especially in the environmentswith low-density scatterers
42 Comparison with Measurements The parametric natureof the proposed channel model makes it adaptable to avariety of propagation environments D-PSD is one of themost important and unique channel characteristics for V2Vcommunication channels To illustrate the validity of theproposed model we compare the modeled SD-PSDs of the3D V2V channel with multiple moving scatterers with themeasured F2F D-PSD in [26] the theoretical V2V SD-PSDin [18] and the measured V2V SD-PSD in [27]
Mobile Information Systems 11
minus2 minus15 minus1 minus05 0 05 1 15 2Doppler frequency f (kHz)
Figure 10 Comparison between the modeled SD-PSDs with119872 = 5and the measured D-PSDs of Figure 4 in [26]
Figure 10 shows the comparison between the modeledSD-PSDs obtained from the Fourier transform of the ST-CF presented in (16) in the isotropic scattering scenario andthe measured D-PSDs of Figure 4 in [26] The outdoor mea-surement experiments in [26] have shown that at millimeterwavelengths fading caused by foliage movement and themotion of nearby vehicles is a significantly deleterious effectin systems with static subscribers Figure 4 in [26] showsthe D-PSD generated by foliage movements and the D-PSDcaused by passing vehicles both in a fixed wireless channelat a carrier frequency of 295GHz We have used the half-Gaussian distribution with an average speed of 04ms andthe standard deviation of 05ms to model the velocity ofmoving foliage scatterers while the uniformdistributionwithan average speed of 6ms has been used tomodel the velocityof passing vehicles scatterers The power weight of the 119898thcluster of rays scattered by the moving foliage or passingvehicles has been set to 1199011 = 916 1199012 = 516 1199013 = 116and 1199014 = 1199015 = 132 The other parameters used to obtainthe modeled SD-PSD curves in Figure 10 are summarizedin (Table 2 Col 5) As can be observed in Figure 10 40 dBfading bandwidths for the effect of the foliage and vehiclesmovements are 500Hz and 1 kHz approximately respectivelyFade depth due to foliage is less than the variation due topassing vehicles Close agreements between the modeled andmeasured results are shown in Figure 10These close matchescan be seen not only for relatively slow moving scatterers butalso for relatively fast moving scatterers as well
Figure 11 shows the comparison among the modeled SD-PSD obtained from the Fourier transform of the ST-CFpresented in (16) in the nonisotropic scattering scenario thetheoretical SD-PSD of Figure 8 in [18] and themeasured SD-PSD of Figure 11 in [27]The theoretical SD-PSD reproducedhere is based on the two-cylinder model with moving andstationary scatterers for 119889119877 = 119889119879 = 0 The channel mea-surements in [27] for the measured SD-PSD were collectedat 2435GHz in the urban street surface environment and thespacing between two adjacent antenna elements of119879119883 and119877119883was set to zero We have used the half-Gaussian distributionwith average speeds of 2394ms 0798ms and 04ms
Figure 11 Comparison among the modeled SD-PSD with119872 = 5the theoretical SD-PSD of Figure 8 in [18] and the measured SD-PSD of Figure 11 in [27]
and the standard deviation of 3ms 1ms and 05ms tomodel the velocity of the first the second and the remainingmoving scatterers on the communication links respectivelyThe power weight of the 119898th cluster of rays has been set to1199011 = 815 1199012 = 13 1199013 = 115 1199014 = 130 and 1199015 = 130for119872 = 5 The other parameters used to obtain the modeledSD-PSD curve in Figure 11 are summarized in (Table 2 Col6) We note that different from the measured SD-PSDs in[26] themeasured SD-PSD curve in [27] has oscillationsTheoscillating pattern in measurements for the case 119889119877 = 119889119879 = 0may appear because it is not a true SISO scenario whereonly one transmit and one receive antenna are active [27] Ascan be observed in Figure 11 the modeled SD-PSD matchesbetter withmeasured SD-PSD than the theoretical SD-PSD in[18] This may be because of the introduction of the multiplebounced rays into the propagation model or the constraintson the position of local scatterers in the two-cylinder modelproposed in [18]
Figure 12 demonstrates the CDFs of the relative devi-ations between modeledtheoretical SD-PSDs and corre-sponding measured values in Figures 10 and 11 The meanrelative deviation of the modeled SD-PSDs for foliage inFigure 10 modeled SD-PSDs for vehicles in Figure 10 mod-eled SD-PSDs in Figure 11 and theoretical SD-PSDs inFigure 11 is 256 216 39 and 832 respectively Asshown in Figure 12 more than 80 of the relative deviationsof modeled SD-PSDs in Figures 10 and 11 are less than5 and the relative deviations of theoretical SD-PSDs inFigure 11 are explicitly larger than those of modeled SD-PSDs In addition from Figures 10 and 11 we note that themodeled SD-PSD matches better with measured SD-PSDs atthe lower and higher Doppler frequencies than the middleDoppler frequencies This may be because 119879119883 and 119877119883 werenot strictly fixed and not all the scatterers were in motion inmeasurementsThe approximate calculation of the frequencyshift in (A4) and Doppler shift in (10) may be anotherfactor which causes the mismatch at the middle Dopplerfrequencies andmore investigations concerning the accuracyof the proposed model will be addressed in future works
12 Mobile Information Systems
0 5 10 15 20 25 30 350
01
02
03
04
05
06
07
08
09
1
Relative deviation ()
Relat
ive d
evia
tion
CDF
Modeled SD-PSD (foliage)Modeled SD-PSD (vehicles)
Modeled SD-PSDTheoretical SD-PSD
Figure 12 The relative deviation cumulative distribution functions(CDFs) of the modeled and theoretical SD-PSDs in Figures 10 and11
However as shown in Figures 10ndash12 the acceptable matchesconfirm the utility and generality of the proposed model andshow the need for including multiple moving scatterers inpropagation model
5 Conclusion
Without specific constraints on the position of the local mov-ing scatterers a 3D geometrical V2V propagation model thatincludes LoS single bounced and multiple bounced linksbetween the transmitter and receiver was proposed Basedon the geometrical propagation model a 3D reference modelfor narrowband MIMO V2V multipath fading channels wasdeveloped From the reference model the correspondingmathematical expressions and numerical results of ST-CFsT-CFs S-CFs and SD-PSDs were studied for different para-metric sets It has been shown that the maximum bouncesscattering forms scatterer velocity distributions and spacingbetween adjacent antenna elements have significant effect onthe ST-CFs and only the maximum bounces and scatterervelocity distributions have significant effect on the T-CFsFinally the modeled SD-PSD results were compared withthe measured data and the other literaturersquos theoreticalresult The close agreements between the analytically andempirically obtained channel statistics confirmed the utilityand generality of the proposed model
Appendix
A Derivation Process of Frequency Shifts
The derivation process of 119891119902119901119902119899 in (7) is presented in thissection and 119891119901119901119902120588 119891119902119901119902119899 119891119901119901119902119899 and 119891119901119902120588 can be calculatedsimilarly
AOD
Δl
TX qth element
Figure 13 Projection from the coordinate vector of 119902th antennaelement to the AOD direction
According to Figure 1 the coordinate vectors of 119902thantenna element and the AOD direction can be written asrespectively
minus 1) 119889119879 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902 + sin120573119902 sin120573119879119901119902] (A3)
Because we mainly consider the difference of the fre-quency shifts caused by the TPD changes to calculate ST-CFsthe TPD change of ℎ11(119905) is assumed to be zero Eventually119891119902119901119902119899 can be calculated as
119891119902119901119902119899 asymp Δ1198971198910119888= (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(A4)
B Derivation Process of Approximate ST-CFs
In the 2D scattering environment the simplified and approx-imate expressions of (28) can be derived in this section If120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 (27) and (10) can be writtenas
Δ119891119899119898
= 1198910119888 [(119902 minus 119902) 119889119879 cos (120572119902 minus 120572119879119901119902)+ (119901 minus 119901) 119889119877 cos (120572119901 minus 120572119877119901119902)]
Mobile Information Systems 13
119891119901119902119899119898
= 1198910119888 [V119879 cos (120572119879 minus 120572119879119901119902) + V119877 cos (120572119877 minus 120572119877119901119902)
+ ]1119901119902 cos (1205721119901119902 minus 120572119879119901119902) + 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119879119901119902)+ ]119898119901119902 cos (120572119898119901119902 minus 120572119879119901119902) + ]1119901119902 cos (1205721119901119902 minus 120572119877119901119902)+ 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119877119901119902) + ]119898119901119902 cos (120572119898119901119902 minus 120572119877119901119902)] (B1)
If 120572119879 = 120572119902 and 120572119877 = 120572119901 that is the antenna array and itsvelocity have the same directions (28) can be simplified as bysubstituting (B1) into (28)
where 1198960 = 21205871198910119888 is the wave numberIn isotropic scattering environment we use the uniform
distribution in (31) to describe all the random azimuth anglesIn this case (B2) can be written as119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879] cos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877] cos (120572119877 minus 120572119877119901119902)
times1198690 [21198960]1119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times1198690 [21198960]119898119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times119898minus1prod119894=2
1198690 [41198960]119894119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind
The AOD 120572119879119901119902 and AOA 120572119877119901119902 are uniformly and indepen-dently distributed so120572119879119901119902minus120572119877119901119902 can be equal to zero in average
As a consequence the term cos[(120572119879119901119902 minus120572119877119901119902)2] in (B3) can beapproximated by one [13 14] and an approximate ST-CF canbe written as from (B3)
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879]sdotcos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877]sdot cos (120572119877 minus 120572119877119901119902) 1198690 (21198960]1119901119902120591)times1198690 (21198960]119898119901119902120591)119898minus1prod
Eventually the approximate ST-CF in (34) can be derivedfrom (B4) according to the definition of 1198690(∙)Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was funded by the National Natural Science Foun-dation of China (no 91438104 no 61501065 no 61571069and no 61601067) the Fundamental Research Funds forthe Central Universities (no 106112016CDJX160001) and theChongqing Research Program of Basic Research and FrontierTechnology (no CSTC2016JCYJA0021)
References
[1] A Bazzi B M Masini A Zanella and I Thibault ldquoBeaconingfrom connected vehicles IEEE 80211p vs LTE-V2Vrdquo in Pro-ceedings of 2016 IEEE 27th Annual International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRC)pp 1ndash6 Valencia Spain September 2016
[2] VM Rodrigo-Penarrocha J Reig L Rubio and et al ldquoAnalysisof small- scale fading distributions in vehicle-to-vehicle com-municationsrdquoMobile Information Systems vol 2016 Article ID9584815 7 pages 2016
[3] A Bazzi B M Masini A Zanella and G Pasolini ldquoIEEE80211p for cellular offloading in vehicular sensor networksrdquoComputer Communications vol 60 pp 97ndash108 2015
[4] X Cheng MIMO channel modelling and simulation for cellu-lar and mobile-to-mobile communication systems[Phd thesis]Heriot-Watt University 2009
[5] N Avazov and M Patzold ldquoA geometric street scatteringchannel model for car-to-car communication systemsrdquo inProceedings of 4th Annual International Conference onAdvanced
14 Mobile Information Systems
Technologies for Communications ATC2011 pp 224ndash230 vnmAugust 2011
[6] X Zhao X Liang S Li and K Haneda ldquoMobile-to-MobileWideband MIMO Channel Realization by Using a Two-RingGeometry-Based Stochastic Scattering Modelrdquo Wireless Per-sonal Communications vol 84 no 4 pp 2445ndash2465 2015
[7] M Walter D Shutin and U-C Fiebig ldquoDelay-dependentdoppler probability density functions for vehicle-to-vehiclescatter channelsrdquo IEEE Transactions on Antennas and Propaga-tion vol 62 no 4 pp 2238ndash2249 2014
[8] P T Samarasinghe T A Lamahewa T D Abhayapala and RA Kennedy ldquo3D mobile-to-mobile wireless channel modelrdquo inProceedings of 2010 Australian Communications Theory Work-shop AusCTW 2010 pp 30ndash34 aus February 2010
[9] M Riaz S J Nawaz and N M Khan ldquo3D ellipsoidal model formobile-to-mobile radio propagation environmentsrdquo WirelessPersonal Communications vol 72 no 4 pp 2465ndash2479 2013
[10] J Chen and T G Pratt ldquoThree-dimensional geometry-based stochastic modeling and performance of 4times4 space-polarization mobile-to-mobile wideband MIMO channelsrdquo inProceedings of 2013 IEEE Global Communications ConferenceGLOBECOM 2013 pp 3936ndash3941 usa December 2013
[11] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA Novel 3DNonstationary Channel Model Based on the von Mises-FisherScattering DistributionrdquoMobile Information Systems vol 2016Article ID 2161460 2016
[12] A Chelli and M Patzold ldquoThe impact of fixed and movingscatterers on the statistics of MIMO vehicle-to-vehicle chan-nelsrdquo in Proceedings of VTC Spring 2009 - IEEE 69th VehicularTechnology Conference esp April 2009
[13] A Borhani and M Patzold ldquoModeling of vehicle-to-vehiclechannels in the presence of moving scatterersrdquo in Proceedingsof 2012 IEEE Vehicular Technology Conference (VTC Fall) pp1ndash5 Quebec City QC Canada September 2012
[14] A Borhani andM Patzold ldquoCorrelation and spectral propertiesof vehicle-to-vehicle channels in the presence ofmoving scatter-ersrdquo IEEE Transactions on Vehicular Technology vol 62 no 9pp 4228ndash4239 2013
[15] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the IEEE 74th Vehicular TechnologyConference (VTC Fall rsquo11) pp 1ndash6 IEEE San Francisco CalifUSA September 2011
[16] M D Soltani M Alimadadi Y Seyedi and H AmindavarldquoModeling of doppler spectrum in V2V urban canyon oncom-ing environmentrdquo in Proceedings of 2014 7th InternationalSymposium on Telecommunications IST 2014 pp 1155ndash1160 irnSeptember 2014
[17] M D Soltani M Alimadadi and A Mohammadi ldquoModelingof mobile scatterer clusters for doppler spectrum in widebandvehicle-to-vehicle communication channelsrdquo IEEE Communi-cations Letters vol 18 no 4 pp 628ndash631 2014
[18] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014
[19] A G Zajic ldquoModeling impact of moving scatterers on dopplerspectrum in wideband vehicle-to-vehicle channelsrdquo in Proceed-ings of Proc Eur Conf Antennas Propag p 1 Lisbon May 2015
[20] J Zhang X Yin and X Cheng ldquoTheoretical analysis andmeasurements Doppler spectra of vehicular communication
channelsrdquo in Proceedings of 2012 12th International Confer-ence on ITS Telecommunications ITST 2012 pp 98ndash102 twnNovember 2012
[21] M Picone S Busanelli M Amoretti F Zanichelli and GFerrari Advanced Technologies for Intelligent TransportationSystems Springer 2015
[22] V H Pham M H Taieb J Y Chouinard S Roy and H THuynh ldquoOn the double Doppler effect generated by scatterermotionrdquo REV Journal on Electronics and Communications vol1 pp 30ndash37 2011
[23] S Roy H T Huynh and P Fortier ldquoCompound Dopplerspread effects of subscriber motion and scatterer motionrdquo AEU- International Journal of Electronics and Communications vol57 no 4 pp 237ndash246 2003
[24] J B Andersen J O Nielsen G F Pedersen G Bauch and GDietl ldquoDoppler spectrum from moving scatterers in a randomenvironmentrdquo IEEE Transactions on Wireless Communicationsvol 8 no 6 pp 3270ndash3277 2009
[25] H S Rad S Gazor and P Shariatpanahi ldquoNon-fixed scatterersand their effects on MIMOmulticarrier fading communicationchannelsrdquo in Proceedings of 50th Annual IEEE Global Telecom-munications Conference GLOBECOM2007 pp 3765ndash3769 usaNovember 2007
[26] N Naz and D Falconer ldquoTemporal variations characterizationfor fixed wireless at 295 GHzrdquo in Proceedings of 2000 IEEE51st Vehicular Technology Conference Proceedings VTC2000-Springer pp 2178ndash2182 Tokyo Japan 2000
[27] A G Zajic G L Stuber T G Pratt and S T NguyenldquoWideband MIMO mobile-to-mobile channels Geometry-based statistical modeling with experimental verificationrdquo IEEETransactions on Vehicular Technology vol 58 no 2 pp 517ndash5342009
[28] W C Jakes Microwave Mobile Communications Wiley-IEEEPress Piscataway NJ USA 1994
[29] M Patzold B O Hogstad and N Youssef ldquoModeling analysisand simulation of MIMO mobile-to-mobile fading channelsrdquoIEEE Transactions onWireless Communications vol 7 no 2 pp510ndash520 2008
[30] M PatzoldMobile Radio Channels Wiley Chichester UK 2ndedition 2011
[31] R Wang and D Cox ldquoDouble mobility mitigates fading in adhoc wireless networksrdquo in Proceedings of IEEE Antennas andPropagation Society International Symposium pp 306ndash309 SanAntonio TX USA
[32] G Bakhshi K Shahtalebi and H S Rad ldquoA novel full-three-dimensional MIMOmobile-to-mobile channel referencemodelrdquo in Proceedings of 3rd International Conference on SignalProcessing and Communication Systems ICSPCSrsquo2009 usaSeptember 2009
[33] W Zhou X Wang X Wang and W Chen ldquoA modified two-erose- ringmodel formimomobile-to-mobile fading channelsrdquoin Proceedings of 7th International Conference on WirelessCommunications Networking and Mobile Computing WiCOM2011 5 1 pages Wuhan China September 2011
(119898B) rays and it is an extension of the (312) in [30] Notethat MB component ℎMB
119901119902 (119905) of the channel impulse responseconsists of 119872 clusters of rays reflected 119898 isin 1 2 3 119872times from moving scatterers
The channel gains (120588119901119902 119888119901119902119899119898
221 Channel Gains The central limit theorem states thatℎ119898119901119902(119905) equals a complex valued Gaussian random processwith zero mean and variance 21205902119898 = Varℎ119898119901119902(119905) =lim119873
119898rarrinfinsum119873119898119899
119898
119864[1198882119901119902119899119898
] 119901119898 in (3) denotes the power weightof the 119898th clusters of rays and sum119872119898=1 119901119898 = 1 The channelgain of ℎ119901119902(119905) is normalized ie 2sum119872119898=1 1199011198981205902119898 + 1205882119901119902 = 1and the Rice factor can be denoted as119870 = 12058821199011199022sum119872119898=1 1199011198981205902119898These parameters have to be either set during simulations orestimated from measurements
222 Doppler Shifts and Frequency Shifts The frequencyshifts 119891119902119901119902120588 119891119901119901119902120588 119891119902119901119902119899 and 119891119901119901119902119899 depend on the differenceof the propagation distance (TPD) changes between ℎ119901119902(119905)and ℎ11(119905) On the other hand the Doppler shifts 119891119901119902120588 and119891119901119902119899
119898
depend on the geometrical relation between directionsof movement of 119879119883 119877119883 and multiple moving scatterers andthe directions of AOD and AOA For max(119889119879 119889119877) ≪ 119863 boththe AOD and AOA of LoS rays are approximately equal tozero Appendix A shows that119891119902119901119902120588119891119901119901119902120588119891119902119901119902119899119891119901119901119902119899 and119891119901119902120588are respectively
119891119902119901119902120588 = (119902 minus 1) 1198891198791198910119888 cos120572119902 cos120573119902 (5)
119891119901119901119902120588 = (119901 minus 1) 1198891198771198910119888 cos120572119901 cos120573119901 (6)
119891119902119901119902119899 = (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(7)
119891119901119901119902119899 = (119901 minus 1) 1198891198771198910119888 [cos (120572119901 minus 120572119877119901119902) cos120573119901 cos120573119877119901119902+ sin120573119901 sin120573119877119901119902]
where 119875AOD119894119901119902 and 119875AOA119894119901119902 (119894 = 1 2 3 119898) are119875AOD119894119901119902 = cos (120572119894119901119902 minus 120572119879119901119902) cos120573119894119901119902 cos120573119879119901119902
+ sin120573119894119901119902 sin120573119879119901119902119875AOA119894119901119902 = cos (120572119894119901119902 minus 120572119877119901119902) cos120573119894119901119902 cos120573119877119901119902
Note that if 120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 and 119898 = 1(10) equals (7) in [13] and (5) in [14] regardless of the plus orminus signsThe differences among these plus or minus signsare caused by the different forms of anglesrsquo expression
223 Phases The phase shift 120579119901119902120588 can be assumed to beconstant [30] The phase shift 120579119901119902119899
119898
consists of the phasechange caused by the interaction of the transmitted signalwith the scatterers and the phase change caused by theTPD between the first and the last scatterers Without lossof generality we can assume that the phases 120579119901119902119899
119898
(119898 =1 2 3 119872) are independent random variables Here it isassumed that they are uniformly distributed on the interval[0 2120587) and independent of any other random variable
Mobile Information Systems 5
3 Space-Time Correlation Function andSpace-Doppler Power Spectral Density
Using the referencemodel described in Section 2 we can nowderive the key temporal and spatial characteristics of MIMOV2V narrowband multipath fading channels with the localmultiple moving scatters
31 Space-Time Correlation Function The normalized ST-CFbetween two complex faded envelopes ℎ119901119902(119905) and ℎ119901119902(119905) isdefined as
where (∙)lowast denotes the complex conjugate operation 119864(∙) isthe statistical expectation operator 119901 119901 isin 1 2 3 119875 and119902 119902 isin 1 2 3 119876 The normalized T-CF can be obtainedif 119901 = 119901 and 119902 = 119902 in (15) The normalized space correlationfunction (S-CF) can be obtained by setting 120591 to zero in (15)
Since ℎ1119901119902(119905) ℎ2119901119902(119905) ℎ119872119901119902(119905) and ℎLoS119901119902 (119905) are indepen-dent of each other (15) can be simplified to
where 119877119898119901119902119901119902(119889119879 119889119877 120591) and 119877LoS119901119902119901119902(119889119879 119889119877 120591) denote the nor-malized ST-CFs of the119898B and LoS components respectivelyand they are defined as
311 ST-CF of LoS Component By substituting (2) into (18)the expression for the ST-CF of the LoS component can bewritten as
119877LoS119901119902119901119902 (119889119879 119889119877 120591)= 120588119901119902120588119901119902119864 exp 119895 [2120587119905 (119891119901119902120588 minus 119891119901119902120588) + 2120587119891119901119902120588120591+ 2120587 (119891119902119901119902120588 + 119891119901119901119902120588 minus 119891119902119901119902120588 minus 119891119901119901119902120588) + 120579119901119902120588minus 120579119901119902120588]
(19)
For max(119889119879 119889119877) ≪ 119863 we assume 120588119901119902 = 120588119901119902 and 120579119901119902120588 =120579119901119902120588 Then (19) can be written as
Δ119891120588 = 119891119902119901119902120588 + 119891119901119901119902120588 minus 119891119902119901119902120588 minus 119891119901119901119902120588 (21)
By substituting (5) and (6) into (21) Δ119891120588 can be written as
Δ119891120588 = 1198910119888 [(119902 minus 119902) 119889119879 cos120572119902 cos120573119902+ (119901 minus 119901) 119889119877 cos120572119901 cos120573119901]
(22)
312 ST-CF of 119898119861 Component By substituting (4) into (17)the expression for the ST-CF of the 119898B component can bewritten as
times 119864 exp 119895 [2120587 (119891119901119902119899119898
minus 119891119901119902119899119899
) 119905 + 2120587119891119901119902119899119898
120591 + 2120587 (119891119902119901119902119899 + 119891119901119901119902119899 minus 119891119902119901119902119899 minus 119891119901119901119902119899) + 120579119901119902119899119898 minus 120579119901119902119899119899] (23)
Under the assumption that 120579119901119902119899119898
and 120579119901119902119899119899
are uniformlydistributed on the interval [0 2120587) and independent of eachother 119864exp[119895(120579119901119902119899
119898
minus120579119901119902119899119899
)] equals 1 It is assumed that allthe path gains of the119898B component have the same size thatis
= 119891119902119901119902119899 + 119891119901119901119902119899 minus 119891119902119901119902119899 minus 119891119901119901119902119899 (26)
It is assumed that AOD and AOA of ℎ119898119901119902(119905) and ℎ119898119901119902(119905) areindependent and identically distributed (iid) By substitut-ing (7) and (8) into (26) Δ119891119899
Since the number of local scatterers in the referencemodel is infinite the parameters 120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902 120572119894119901119902and 120573119894119901119902 can be seen as continuous random variables withcorresponding probability density functions (PDFs) Thenthe ST-CF of the 119898B component (25) can be writtenas
Now the complete expression of (16) can be obtainedby substituting (20) and (28) into (16) 119901(120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902)describes the joint distribution of AOD and AOA and itcan be optionally used to present some propagation channelmodels As a result the ST-CF in (28) can provide a suitableplatform to study the statistical properties of some differentchannelmodels such as the random scatteringmodel [13 14]Jakes model [28] one-ring model [23] and two-ring model[29] as described in Section 313Therefore the ST-CF in (28)is a generalized and parametric expression However due tothe complex nature of 119901(120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902) it is assumed thatAOD and AOA are independent [13 14 32 33] and azimuthangles and elevation angles in (28) are also independent[18 19 32] The parameters in (28) such as the velocities ofthe multiple moving scatterers and random angles can becalculated as follows
(i) Scatterer Velocity Distributions The Gaussian Laplaceexponential and uniform distributions can be used todescribe the velocity of moving scatterers [13] In fact thescatterer velocity ]119894119901119902 is always positive or equal to zeroWe use the uniform distribution in (29) and half-Gaussiandistribution in (30) to describe the velocity of multiplemoving scatterers
119901 (]119894119901119902) = 1V119894max 0 le ]119894119901119902 le V119894max (29)
where V119894max is the maximum of ]119894119901119902
21205902119901119902119894]] ]119894119901119902 ge 0 (30)
where 120590119901119902119894 is the standard deviation of ]119894119901119902
(ii) Angle Distributions To characterize the statistical anglesin Table 1 we use the uniform distribution in (31) in theisotropic scattering environment and use the von Misesdistribution in (32) and the cosine distribution in (33) inthe nonisotropic scattering environment In addition theinterval of azimuth angles 120572119879119901119902 120572119877119901119902 and 120572119894119901119902 is (minus120587 120587] and theinterval of elevation angles 120573119879119901119902 120573119877119901119902 and 120573119894119901119902 is (minus1205872 1205872]119901 (120574) = 11205742 minus 1205741 1205741 le 120574 le 1205742 (31)
119901 (120572) = exp [119896 cos (120572 minus 120572)]21205871198680 (119896) 1205721 le 120572 le 1205721 + 2120587 (32)
where 1198680(∙) is the zeroth-order modified Bessel function ofthe first kind 120572 is the mean angle and 119896 controls the spreadof angles around the mean The von Mises distribution PDFwith 120572 = 0 is used to describe the azimuth angles
119901 (120573) = 1205874 10038161003816100381610038161205731198981003816100381610038161003816 cos(1205872 120573120573119898) minus 120573119898 le 120573 le 120573119898 (33)
where 120573119898 is the maximum of 120573 The cosine distribution PDFis used to describe the elevation angles
313 Special Cases of 119898119861rsquos ST-CF If 120573119879 = 120573119877 = 120573119879119901119902 =120573119877119901119902 = 120573119894119901119902 = 0 that is the scattering environment is 2D
Mobile Information Systems 7
some different special cases can be derived from the generalexpression of the119898Brsquos ST-CF in (28)
In the 2D scattering environment (28) can be written as(B2) in Appendix B If 119901 = 119901 119902 = 119902 and119898 = 1 (B2) equals(11) in [13] and (6) in [14] regardless of the plus or minussigns under the assumption that the angles 120572119879 and 120572119877 areindependent of each other in [13 14] In isotropic scatteringenvironments some other special cases with closed-formexpressions can be derived as follows
Appendix B shows that an approximate ST-CF of (B2)can be written as
119877119898119901119902119901119902 (119889119879 119889119877 120591) = 212059021198981198690 1198960 [V119879120591 + (119902 minus 119902) 119889119879]times 1198690 1198960 [V119877120591 + (119901 minus 119901) 119889119877] int sdot sdot sdot int 1198690 (21198960]1119901119902120591)times 1198690 (21198960]119898119901119902120591)119898minus1prod
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind and 1198960 = 21205871198910119888 is the wave number
Note that if 119898 approaches to the infinity (34) can bewritten as
lim119898rarrinfin
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 212059021198981198690 [(119902 minus 119902) 1198960119889119879] 1198690 [(119901 minus 119901) 1198960119889119877] 120591 = 00 120591 gt 0
(35)
In the NLoS communication environment with veryhigh-density scatterers (35) implies that the ST-CFapproaches to zero at the nonzero time difference 120591 and hasnothing to do with velocities of scatterers 119879119883 and 119877119883 Thelarge-scale antenna arrays are very suitable to be used in thisenvironment because the antenna element spacing can bereduced to a smaller value
If ]119894119901119902 is constant (34) can be presented as the followingclosed-form expression
The T-CF of the classical F2M scenario with fixed scat-terers is obtained if V119879 = V119894119901119902 = 0 (119894 = 1 2 3 119898)119902 = 119902 and 119901 = 119901 in (36) In this case the ST-CF in(36) is 212059021198981198690(1198960V119877120591) which is known as the Jakes model[28] If V119879 = 0119898 = 1 119902 = 119902 and 119901 = 119901 the ST-CFin (36) results in 212059021198981198690(1198960V1119901119902120591)1198690(1198960V119877120591) which equals the
T-CF of F2M single-ring channel model in the presence ofmoving scatterers reported in (11) of [23] If V119894119901119902 = 0 (119894 =1 2 3 119898) 119902 = 119902 and 119901 = 119901 the ST-CF in (36) resultsin 212059021198981198690(1198960V119879120591)1198690(1198960V119877120591) which equals the T-CFs of theclassical M2M two-ring channel model in the presence offixed scatterers reported in (46) of [29] If V119894119901119902 = 120591 = 0 (119894 =1 2 3 119898) 119902 = 119901 = 2 and 119902 = 119901 = 1 the ST-CF in (36)results in 212059021198981198690(1198960119889119879)1198690(1198960119889119877) which equals the S-CF in (46)of [29]
The velocities ofmoving scatterers such asmoving foliagewalking pedestrians and passing vehicles generally are ran-dom variables If ]119894119901119902 is described by the uniform distributionin (29) (34) can be presented as the following closed-formexpression
times 1198690 1198960 [V119877120591 + (119901 minus 119901) 119889119877] [21198690 (21198960V1max120591)+ 1205871198691 (21198960V1max120591)1198670 (21198960V1max120591) minus 1205871198690 (21198960V1max120591)times 1198671 (21198960V1max120591)] [21198690 (21198960V119898max120591)+ 1205871198691 (21198960V119898max120591)1198670 (21198960V119898max120591) minus 1205871198690 (21198960V119898max120591)sdot 1198671 (21198960V119898max120591)] times 119898minus1prod
119894=2
[21198690 (41198960V119894max120591)+ 1205871198691 (41198960V119894max120591) times 1198670 (41198960V119894max120591)minus 1205871198690 (41198960V119894max120591)1198671 (41198960V119894max120591)]
(37)
where 1198691(∙) denotes the first-order Bessel function of the firstkind 1198670(∙) denotes the zeroth-order Struve function and1198671(∙) denotes the first-order Struve function
If ]119894119901119902 is described by the half-Gaussian distribution in(30) (34) can be presented as the following closed-formexpression
32 Space-Doppler Power Spectral Density The SD-PSD canbe obtained by taking the Fourier transform of the ST-CF in(16)with respect to time difference 120591 From (16) it follows that
8 Mobile Information Systems
Table 2 Parameters used in the numerical simulation
where 120575(∙) is the Dirac delta functionSince the ST-CF of the 119898B component is the multiple
integral as (28) the closed-form expression of SD-PSDcannot be derived In Section 42 we show the SD-PSD of the3DV2Vchannel in the presence ofmultiplemoving scatterersby means of numerical integrations
4 Numerical Results and Validation
This section demonstrates the normalized correlation func-tions and validates the Doppler power spectral density (D-PSD) described in Section 3 through theMATLABnumericalsimulations Unless indicated otherwise the values of thenumerical simulation parameters are summarized in Table 2
41 Numerical Results In this section the numerical curvesof ST-CFs T-CFs and S-CFs influenced by some importantcontributory factors are presented As in the typical urbanenvironments the power weight of the119898th cluster of rays hasbeen set to 1199011 = 1199012 = 12 for119872 = 2 1199011 = 1199012 = 1199013 = 13 for119872 = 3 1199011 = 1199012 = 13 1199013 = 1199014 = 16 for119872 = 4 1199011 = 131199012 = 14 1199013 = 1199014 = 16 and 1199015 = 112 for119872 = 5411 ST-CFs and T-CFs for Different Scattering Scenarios andBounces Figures 2ndash5 demonstrate the ST-CFs and T-CFs forthe different maximum bounces 119872 in the 3D isotropic and
0 05 1 15 2 25 3 35 4066
067
068
069
07
071
072
휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 2 ST-CFs in (16) in the isotropic scattering scenario fordifferent maximum bounces
nonisotropic scattering scenarios The scatterer velocity isuniformly distributed with an average speed of 25ms thatmay be the velocity of passing vehicles The other parametersused to obtain curves in Figures 2ndash5 are summarized in(Table 2 Cols 2 and 3) As shown in Figures 2 and 4 thelarger119872 is the faster the ST-CFs and T-CFs decrease in theisotropic scattering scenario However the descent rates ofST-CFs and T-CFs increase slowly when119872 is larger than 3The similar conclusions can be obtained fromFigures 3 and 5however the ST-CF curves in Figure 3 have oscillationswhichmay be caused by the nonisotropic scattering However thisconclusion differs from [27] which showed that the triple-or higher-order bounced rays had statistical properties verysimilar to those of the double-bounced rays and could beapproximated as double-bounced rays The discrepancy maybe caused by the different communication environmentsconsidered by us and [27] Specifically channel-soundingexperimental campaign in [27] was conducted along surface
Mobile Information Systems 9
066
067
068
069
07
071
072
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 3 ST-CFs in (16) in the nonisotropic scattering scenario fordifferent maximum bounces
065
07
075
08
085
09
095
1
M = 1M = 2M = 3
M = 4M = 5
0 05 1 15 2 25 3 35 4휏 (ms)
|R11
11(휏)|
Figure 4 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropicscattering scenario for different maximum bounces
streets around the Georgia Tech campus and on the Interstatehighways in the Midtown Atlanta metropolitan area whichmay have lower dense scatterers than our communicationenvironments The ST-CF curves in Figure 3 converge rel-atively slower for the larger 119872 that means there are morelocal scatterers inmotionwith randomvelocities and randomdirections As can be observed in Figures 4 and 5 theT-CFs inboth figures are very similar that indicates that the scatteringforms have no significant effect on the T-CFs of the 3D V2Vchannel in the presence of multiple moving scatterers Fur-thermore in Figures 2ndash5 it can be observed that a tendencyto a limit value curve of ST-CFs and T-CFs can be inferred if
065
07
075
08
085
09
095
1
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
|R11
11(휏)|
Figure 5 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the nonisotropicscattering scenario for different maximum bounces
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
066
067
068
069
071
072
1122(d
T=
05휆d
R=
05휆휏
)||R
07
Figure 6 ST-CFs in (16) in the isotropic and nonisotropic scatteringscenarios with119872 = 5 for different scatterer velocity distributions
119872 approaches infinity and the similar inference also can beobtained in (35) for the 2D scattering environment
412 ST-CFs and T-CFs for Different Scattering Scenarios andScatterer Velocity Distributions Figures 6 and 7 demonstratethe ST-CFs and T-CFs for the different scatterer velocitydistributions in the 3D isotropic and nonisotropic scatteringscenarios We use two uniform distributions in the isotropicscattering environment and use the von Mises distributionand the cosine distribution in the nonisotropic scatteringenvironment to describe all the statistical azimuth andelevation angles in Table 1 respectively The average speedof the uniform and half-Gaussian distributions is 25ms
10 Mobile Information Systems
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
065
07
075
08
085
09
095
1
|R11
11(휏)|
Figure 7 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropic andnonisotropic scattering scenarios with119872 = 5 for different scatterervelocity distributions
and the standard deviation of the half-Gaussian distribu-tion is 31ms The other parameters used to obtain curvesin Figures 6 and 7 are summarized in (Table 2 Cols 2and 3) As shown in Figure 6 the ST-CF for the uniformdistributed velocity of scatterers in the isotropic scatteringenvironment decreases fastest and both the nonisotropicscattering and the half-Gaussian distribution can result inswift oscillations of the fading curvesThe comparison resultsindicate that the scatterer velocity distributions have moresignificant effect than the scattering forms on the ST-CFsof the 3D V2V channel in the presence of multiple movingscatterers Figure 7 indicates that not like scattering formsthe scatterer velocity distributions have significant effect onthe T-CFs Generally the realistic propagation environmentis nonisotropic scattering and the scatterer velocity canbe modeled by the half-Gaussian or uniform distributionTherefore the results of Figures 6 and 7 may be helpful inanalysis of the system performance and modeling a specificpropagation channel for V2V communications
413 ST-CFs and S-CFs for Different Scattering Scenariosand Spacing between Adjacent Antenna Elements Figures8 and 9 demonstrate the 3D ST-CFs and S-CFs for thedifferent spacing between adjacent antenna elements of 119879119883and 119877119883 in isotropic and nonisotropic scattering scenariosThe set of the scatterer velocity in Figure 8 is the sameas Figures 2ndash5 and the other parameters used to obtaincurves in Figures 8 and 9 are summarized in (Table 2 Col4) Figures 8 and 9 show that spacing between adjacentantenna elements has significant effect on the 3D ST-CFsand S-CFs Furthermore the 3D ST-CFs in isotropic andnonisotropic scattering scenarios decrease to the bottomwhen 119889119877 = 119889119879 tends to 12120582 and 07120582 respectively Inaddition the 3D S-CF in nonisotropic scattering scenarios isexplicitly less than that in isotropic scattering scenarios for
|R11
22(d
Td
R휏)|
dR = d
T (휆) 휏 (ms)
0025
05075 1
12515 0
12
34
1
095
09
085
08
075
07
065
IsotropicNonisotropic
Figure 8 3D ST-CFs in (16) in the isotropic and nonisotropicscattering scenarios with119872 = 5 120582 is the carrier wavelength
|R11
22(d
Td
R)|
dR (휆)
dT (휆)
1
095
09
085
08
075
07
065
0 025 05 0751 125 15
0025
05075 1
12515
IsotropicNonisotropic
Figure 9 3D S-CFs in (16) with 120591 = 0 in the isotropic andnonisotropic scattering scenarios 120582 is the carrier wavelength
most spacing between adjacent antenna elements as shown inFigure 9Therefore the large-scale antenna arrays employingthe spatial multiplexing are more suitable to be used in thenonisotropic scattering scenarios owing to the high antennaspace utilization The spacing between adjacent antennaelements of antenna arrays is suggested to be more than 120582in the V2V communications especially in the environmentswith low-density scatterers
42 Comparison with Measurements The parametric natureof the proposed channel model makes it adaptable to avariety of propagation environments D-PSD is one of themost important and unique channel characteristics for V2Vcommunication channels To illustrate the validity of theproposed model we compare the modeled SD-PSDs of the3D V2V channel with multiple moving scatterers with themeasured F2F D-PSD in [26] the theoretical V2V SD-PSDin [18] and the measured V2V SD-PSD in [27]
Mobile Information Systems 11
minus2 minus15 minus1 minus05 0 05 1 15 2Doppler frequency f (kHz)
Figure 10 Comparison between the modeled SD-PSDs with119872 = 5and the measured D-PSDs of Figure 4 in [26]
Figure 10 shows the comparison between the modeledSD-PSDs obtained from the Fourier transform of the ST-CF presented in (16) in the isotropic scattering scenario andthe measured D-PSDs of Figure 4 in [26] The outdoor mea-surement experiments in [26] have shown that at millimeterwavelengths fading caused by foliage movement and themotion of nearby vehicles is a significantly deleterious effectin systems with static subscribers Figure 4 in [26] showsthe D-PSD generated by foliage movements and the D-PSDcaused by passing vehicles both in a fixed wireless channelat a carrier frequency of 295GHz We have used the half-Gaussian distribution with an average speed of 04ms andthe standard deviation of 05ms to model the velocity ofmoving foliage scatterers while the uniformdistributionwithan average speed of 6ms has been used tomodel the velocityof passing vehicles scatterers The power weight of the 119898thcluster of rays scattered by the moving foliage or passingvehicles has been set to 1199011 = 916 1199012 = 516 1199013 = 116and 1199014 = 1199015 = 132 The other parameters used to obtainthe modeled SD-PSD curves in Figure 10 are summarizedin (Table 2 Col 5) As can be observed in Figure 10 40 dBfading bandwidths for the effect of the foliage and vehiclesmovements are 500Hz and 1 kHz approximately respectivelyFade depth due to foliage is less than the variation due topassing vehicles Close agreements between the modeled andmeasured results are shown in Figure 10These close matchescan be seen not only for relatively slow moving scatterers butalso for relatively fast moving scatterers as well
Figure 11 shows the comparison among the modeled SD-PSD obtained from the Fourier transform of the ST-CFpresented in (16) in the nonisotropic scattering scenario thetheoretical SD-PSD of Figure 8 in [18] and themeasured SD-PSD of Figure 11 in [27]The theoretical SD-PSD reproducedhere is based on the two-cylinder model with moving andstationary scatterers for 119889119877 = 119889119879 = 0 The channel mea-surements in [27] for the measured SD-PSD were collectedat 2435GHz in the urban street surface environment and thespacing between two adjacent antenna elements of119879119883 and119877119883was set to zero We have used the half-Gaussian distributionwith average speeds of 2394ms 0798ms and 04ms
Figure 11 Comparison among the modeled SD-PSD with119872 = 5the theoretical SD-PSD of Figure 8 in [18] and the measured SD-PSD of Figure 11 in [27]
and the standard deviation of 3ms 1ms and 05ms tomodel the velocity of the first the second and the remainingmoving scatterers on the communication links respectivelyThe power weight of the 119898th cluster of rays has been set to1199011 = 815 1199012 = 13 1199013 = 115 1199014 = 130 and 1199015 = 130for119872 = 5 The other parameters used to obtain the modeledSD-PSD curve in Figure 11 are summarized in (Table 2 Col6) We note that different from the measured SD-PSDs in[26] themeasured SD-PSD curve in [27] has oscillationsTheoscillating pattern in measurements for the case 119889119877 = 119889119879 = 0may appear because it is not a true SISO scenario whereonly one transmit and one receive antenna are active [27] Ascan be observed in Figure 11 the modeled SD-PSD matchesbetter withmeasured SD-PSD than the theoretical SD-PSD in[18] This may be because of the introduction of the multiplebounced rays into the propagation model or the constraintson the position of local scatterers in the two-cylinder modelproposed in [18]
Figure 12 demonstrates the CDFs of the relative devi-ations between modeledtheoretical SD-PSDs and corre-sponding measured values in Figures 10 and 11 The meanrelative deviation of the modeled SD-PSDs for foliage inFigure 10 modeled SD-PSDs for vehicles in Figure 10 mod-eled SD-PSDs in Figure 11 and theoretical SD-PSDs inFigure 11 is 256 216 39 and 832 respectively Asshown in Figure 12 more than 80 of the relative deviationsof modeled SD-PSDs in Figures 10 and 11 are less than5 and the relative deviations of theoretical SD-PSDs inFigure 11 are explicitly larger than those of modeled SD-PSDs In addition from Figures 10 and 11 we note that themodeled SD-PSD matches better with measured SD-PSDs atthe lower and higher Doppler frequencies than the middleDoppler frequencies This may be because 119879119883 and 119877119883 werenot strictly fixed and not all the scatterers were in motion inmeasurementsThe approximate calculation of the frequencyshift in (A4) and Doppler shift in (10) may be anotherfactor which causes the mismatch at the middle Dopplerfrequencies andmore investigations concerning the accuracyof the proposed model will be addressed in future works
12 Mobile Information Systems
0 5 10 15 20 25 30 350
01
02
03
04
05
06
07
08
09
1
Relative deviation ()
Relat
ive d
evia
tion
CDF
Modeled SD-PSD (foliage)Modeled SD-PSD (vehicles)
Modeled SD-PSDTheoretical SD-PSD
Figure 12 The relative deviation cumulative distribution functions(CDFs) of the modeled and theoretical SD-PSDs in Figures 10 and11
However as shown in Figures 10ndash12 the acceptable matchesconfirm the utility and generality of the proposed model andshow the need for including multiple moving scatterers inpropagation model
5 Conclusion
Without specific constraints on the position of the local mov-ing scatterers a 3D geometrical V2V propagation model thatincludes LoS single bounced and multiple bounced linksbetween the transmitter and receiver was proposed Basedon the geometrical propagation model a 3D reference modelfor narrowband MIMO V2V multipath fading channels wasdeveloped From the reference model the correspondingmathematical expressions and numerical results of ST-CFsT-CFs S-CFs and SD-PSDs were studied for different para-metric sets It has been shown that the maximum bouncesscattering forms scatterer velocity distributions and spacingbetween adjacent antenna elements have significant effect onthe ST-CFs and only the maximum bounces and scatterervelocity distributions have significant effect on the T-CFsFinally the modeled SD-PSD results were compared withthe measured data and the other literaturersquos theoreticalresult The close agreements between the analytically andempirically obtained channel statistics confirmed the utilityand generality of the proposed model
Appendix
A Derivation Process of Frequency Shifts
The derivation process of 119891119902119901119902119899 in (7) is presented in thissection and 119891119901119901119902120588 119891119902119901119902119899 119891119901119901119902119899 and 119891119901119902120588 can be calculatedsimilarly
AOD
Δl
TX qth element
Figure 13 Projection from the coordinate vector of 119902th antennaelement to the AOD direction
According to Figure 1 the coordinate vectors of 119902thantenna element and the AOD direction can be written asrespectively
minus 1) 119889119879 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902 + sin120573119902 sin120573119879119901119902] (A3)
Because we mainly consider the difference of the fre-quency shifts caused by the TPD changes to calculate ST-CFsthe TPD change of ℎ11(119905) is assumed to be zero Eventually119891119902119901119902119899 can be calculated as
119891119902119901119902119899 asymp Δ1198971198910119888= (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(A4)
B Derivation Process of Approximate ST-CFs
In the 2D scattering environment the simplified and approx-imate expressions of (28) can be derived in this section If120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 (27) and (10) can be writtenas
Δ119891119899119898
= 1198910119888 [(119902 minus 119902) 119889119879 cos (120572119902 minus 120572119879119901119902)+ (119901 minus 119901) 119889119877 cos (120572119901 minus 120572119877119901119902)]
Mobile Information Systems 13
119891119901119902119899119898
= 1198910119888 [V119879 cos (120572119879 minus 120572119879119901119902) + V119877 cos (120572119877 minus 120572119877119901119902)
+ ]1119901119902 cos (1205721119901119902 minus 120572119879119901119902) + 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119879119901119902)+ ]119898119901119902 cos (120572119898119901119902 minus 120572119879119901119902) + ]1119901119902 cos (1205721119901119902 minus 120572119877119901119902)+ 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119877119901119902) + ]119898119901119902 cos (120572119898119901119902 minus 120572119877119901119902)] (B1)
If 120572119879 = 120572119902 and 120572119877 = 120572119901 that is the antenna array and itsvelocity have the same directions (28) can be simplified as bysubstituting (B1) into (28)
where 1198960 = 21205871198910119888 is the wave numberIn isotropic scattering environment we use the uniform
distribution in (31) to describe all the random azimuth anglesIn this case (B2) can be written as119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879] cos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877] cos (120572119877 minus 120572119877119901119902)
times1198690 [21198960]1119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times1198690 [21198960]119898119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times119898minus1prod119894=2
1198690 [41198960]119894119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind
The AOD 120572119879119901119902 and AOA 120572119877119901119902 are uniformly and indepen-dently distributed so120572119879119901119902minus120572119877119901119902 can be equal to zero in average
As a consequence the term cos[(120572119879119901119902 minus120572119877119901119902)2] in (B3) can beapproximated by one [13 14] and an approximate ST-CF canbe written as from (B3)
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879]sdotcos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877]sdot cos (120572119877 minus 120572119877119901119902) 1198690 (21198960]1119901119902120591)times1198690 (21198960]119898119901119902120591)119898minus1prod
Eventually the approximate ST-CF in (34) can be derivedfrom (B4) according to the definition of 1198690(∙)Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was funded by the National Natural Science Foun-dation of China (no 91438104 no 61501065 no 61571069and no 61601067) the Fundamental Research Funds forthe Central Universities (no 106112016CDJX160001) and theChongqing Research Program of Basic Research and FrontierTechnology (no CSTC2016JCYJA0021)
References
[1] A Bazzi B M Masini A Zanella and I Thibault ldquoBeaconingfrom connected vehicles IEEE 80211p vs LTE-V2Vrdquo in Pro-ceedings of 2016 IEEE 27th Annual International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRC)pp 1ndash6 Valencia Spain September 2016
[2] VM Rodrigo-Penarrocha J Reig L Rubio and et al ldquoAnalysisof small- scale fading distributions in vehicle-to-vehicle com-municationsrdquoMobile Information Systems vol 2016 Article ID9584815 7 pages 2016
[3] A Bazzi B M Masini A Zanella and G Pasolini ldquoIEEE80211p for cellular offloading in vehicular sensor networksrdquoComputer Communications vol 60 pp 97ndash108 2015
[4] X Cheng MIMO channel modelling and simulation for cellu-lar and mobile-to-mobile communication systems[Phd thesis]Heriot-Watt University 2009
[5] N Avazov and M Patzold ldquoA geometric street scatteringchannel model for car-to-car communication systemsrdquo inProceedings of 4th Annual International Conference onAdvanced
14 Mobile Information Systems
Technologies for Communications ATC2011 pp 224ndash230 vnmAugust 2011
[6] X Zhao X Liang S Li and K Haneda ldquoMobile-to-MobileWideband MIMO Channel Realization by Using a Two-RingGeometry-Based Stochastic Scattering Modelrdquo Wireless Per-sonal Communications vol 84 no 4 pp 2445ndash2465 2015
[7] M Walter D Shutin and U-C Fiebig ldquoDelay-dependentdoppler probability density functions for vehicle-to-vehiclescatter channelsrdquo IEEE Transactions on Antennas and Propaga-tion vol 62 no 4 pp 2238ndash2249 2014
[8] P T Samarasinghe T A Lamahewa T D Abhayapala and RA Kennedy ldquo3D mobile-to-mobile wireless channel modelrdquo inProceedings of 2010 Australian Communications Theory Work-shop AusCTW 2010 pp 30ndash34 aus February 2010
[9] M Riaz S J Nawaz and N M Khan ldquo3D ellipsoidal model formobile-to-mobile radio propagation environmentsrdquo WirelessPersonal Communications vol 72 no 4 pp 2465ndash2479 2013
[10] J Chen and T G Pratt ldquoThree-dimensional geometry-based stochastic modeling and performance of 4times4 space-polarization mobile-to-mobile wideband MIMO channelsrdquo inProceedings of 2013 IEEE Global Communications ConferenceGLOBECOM 2013 pp 3936ndash3941 usa December 2013
[11] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA Novel 3DNonstationary Channel Model Based on the von Mises-FisherScattering DistributionrdquoMobile Information Systems vol 2016Article ID 2161460 2016
[12] A Chelli and M Patzold ldquoThe impact of fixed and movingscatterers on the statistics of MIMO vehicle-to-vehicle chan-nelsrdquo in Proceedings of VTC Spring 2009 - IEEE 69th VehicularTechnology Conference esp April 2009
[13] A Borhani and M Patzold ldquoModeling of vehicle-to-vehiclechannels in the presence of moving scatterersrdquo in Proceedingsof 2012 IEEE Vehicular Technology Conference (VTC Fall) pp1ndash5 Quebec City QC Canada September 2012
[14] A Borhani andM Patzold ldquoCorrelation and spectral propertiesof vehicle-to-vehicle channels in the presence ofmoving scatter-ersrdquo IEEE Transactions on Vehicular Technology vol 62 no 9pp 4228ndash4239 2013
[15] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the IEEE 74th Vehicular TechnologyConference (VTC Fall rsquo11) pp 1ndash6 IEEE San Francisco CalifUSA September 2011
[16] M D Soltani M Alimadadi Y Seyedi and H AmindavarldquoModeling of doppler spectrum in V2V urban canyon oncom-ing environmentrdquo in Proceedings of 2014 7th InternationalSymposium on Telecommunications IST 2014 pp 1155ndash1160 irnSeptember 2014
[17] M D Soltani M Alimadadi and A Mohammadi ldquoModelingof mobile scatterer clusters for doppler spectrum in widebandvehicle-to-vehicle communication channelsrdquo IEEE Communi-cations Letters vol 18 no 4 pp 628ndash631 2014
[18] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014
[19] A G Zajic ldquoModeling impact of moving scatterers on dopplerspectrum in wideband vehicle-to-vehicle channelsrdquo in Proceed-ings of Proc Eur Conf Antennas Propag p 1 Lisbon May 2015
[20] J Zhang X Yin and X Cheng ldquoTheoretical analysis andmeasurements Doppler spectra of vehicular communication
channelsrdquo in Proceedings of 2012 12th International Confer-ence on ITS Telecommunications ITST 2012 pp 98ndash102 twnNovember 2012
[21] M Picone S Busanelli M Amoretti F Zanichelli and GFerrari Advanced Technologies for Intelligent TransportationSystems Springer 2015
[22] V H Pham M H Taieb J Y Chouinard S Roy and H THuynh ldquoOn the double Doppler effect generated by scatterermotionrdquo REV Journal on Electronics and Communications vol1 pp 30ndash37 2011
[23] S Roy H T Huynh and P Fortier ldquoCompound Dopplerspread effects of subscriber motion and scatterer motionrdquo AEU- International Journal of Electronics and Communications vol57 no 4 pp 237ndash246 2003
[24] J B Andersen J O Nielsen G F Pedersen G Bauch and GDietl ldquoDoppler spectrum from moving scatterers in a randomenvironmentrdquo IEEE Transactions on Wireless Communicationsvol 8 no 6 pp 3270ndash3277 2009
[25] H S Rad S Gazor and P Shariatpanahi ldquoNon-fixed scatterersand their effects on MIMOmulticarrier fading communicationchannelsrdquo in Proceedings of 50th Annual IEEE Global Telecom-munications Conference GLOBECOM2007 pp 3765ndash3769 usaNovember 2007
[26] N Naz and D Falconer ldquoTemporal variations characterizationfor fixed wireless at 295 GHzrdquo in Proceedings of 2000 IEEE51st Vehicular Technology Conference Proceedings VTC2000-Springer pp 2178ndash2182 Tokyo Japan 2000
[27] A G Zajic G L Stuber T G Pratt and S T NguyenldquoWideband MIMO mobile-to-mobile channels Geometry-based statistical modeling with experimental verificationrdquo IEEETransactions on Vehicular Technology vol 58 no 2 pp 517ndash5342009
[28] W C Jakes Microwave Mobile Communications Wiley-IEEEPress Piscataway NJ USA 1994
[29] M Patzold B O Hogstad and N Youssef ldquoModeling analysisand simulation of MIMO mobile-to-mobile fading channelsrdquoIEEE Transactions onWireless Communications vol 7 no 2 pp510ndash520 2008
[30] M PatzoldMobile Radio Channels Wiley Chichester UK 2ndedition 2011
[31] R Wang and D Cox ldquoDouble mobility mitigates fading in adhoc wireless networksrdquo in Proceedings of IEEE Antennas andPropagation Society International Symposium pp 306ndash309 SanAntonio TX USA
[32] G Bakhshi K Shahtalebi and H S Rad ldquoA novel full-three-dimensional MIMOmobile-to-mobile channel referencemodelrdquo in Proceedings of 3rd International Conference on SignalProcessing and Communication Systems ICSPCSrsquo2009 usaSeptember 2009
[33] W Zhou X Wang X Wang and W Chen ldquoA modified two-erose- ringmodel formimomobile-to-mobile fading channelsrdquoin Proceedings of 7th International Conference on WirelessCommunications Networking and Mobile Computing WiCOM2011 5 1 pages Wuhan China September 2011
3 Space-Time Correlation Function andSpace-Doppler Power Spectral Density
Using the referencemodel described in Section 2 we can nowderive the key temporal and spatial characteristics of MIMOV2V narrowband multipath fading channels with the localmultiple moving scatters
31 Space-Time Correlation Function The normalized ST-CFbetween two complex faded envelopes ℎ119901119902(119905) and ℎ119901119902(119905) isdefined as
where (∙)lowast denotes the complex conjugate operation 119864(∙) isthe statistical expectation operator 119901 119901 isin 1 2 3 119875 and119902 119902 isin 1 2 3 119876 The normalized T-CF can be obtainedif 119901 = 119901 and 119902 = 119902 in (15) The normalized space correlationfunction (S-CF) can be obtained by setting 120591 to zero in (15)
Since ℎ1119901119902(119905) ℎ2119901119902(119905) ℎ119872119901119902(119905) and ℎLoS119901119902 (119905) are indepen-dent of each other (15) can be simplified to
where 119877119898119901119902119901119902(119889119879 119889119877 120591) and 119877LoS119901119902119901119902(119889119879 119889119877 120591) denote the nor-malized ST-CFs of the119898B and LoS components respectivelyand they are defined as
311 ST-CF of LoS Component By substituting (2) into (18)the expression for the ST-CF of the LoS component can bewritten as
119877LoS119901119902119901119902 (119889119879 119889119877 120591)= 120588119901119902120588119901119902119864 exp 119895 [2120587119905 (119891119901119902120588 minus 119891119901119902120588) + 2120587119891119901119902120588120591+ 2120587 (119891119902119901119902120588 + 119891119901119901119902120588 minus 119891119902119901119902120588 minus 119891119901119901119902120588) + 120579119901119902120588minus 120579119901119902120588]
(19)
For max(119889119879 119889119877) ≪ 119863 we assume 120588119901119902 = 120588119901119902 and 120579119901119902120588 =120579119901119902120588 Then (19) can be written as
Δ119891120588 = 119891119902119901119902120588 + 119891119901119901119902120588 minus 119891119902119901119902120588 minus 119891119901119901119902120588 (21)
By substituting (5) and (6) into (21) Δ119891120588 can be written as
Δ119891120588 = 1198910119888 [(119902 minus 119902) 119889119879 cos120572119902 cos120573119902+ (119901 minus 119901) 119889119877 cos120572119901 cos120573119901]
(22)
312 ST-CF of 119898119861 Component By substituting (4) into (17)the expression for the ST-CF of the 119898B component can bewritten as
times 119864 exp 119895 [2120587 (119891119901119902119899119898
minus 119891119901119902119899119899
) 119905 + 2120587119891119901119902119899119898
120591 + 2120587 (119891119902119901119902119899 + 119891119901119901119902119899 minus 119891119902119901119902119899 minus 119891119901119901119902119899) + 120579119901119902119899119898 minus 120579119901119902119899119899] (23)
Under the assumption that 120579119901119902119899119898
and 120579119901119902119899119899
are uniformlydistributed on the interval [0 2120587) and independent of eachother 119864exp[119895(120579119901119902119899
119898
minus120579119901119902119899119899
)] equals 1 It is assumed that allthe path gains of the119898B component have the same size thatis
= 119891119902119901119902119899 + 119891119901119901119902119899 minus 119891119902119901119902119899 minus 119891119901119901119902119899 (26)
It is assumed that AOD and AOA of ℎ119898119901119902(119905) and ℎ119898119901119902(119905) areindependent and identically distributed (iid) By substitut-ing (7) and (8) into (26) Δ119891119899
Since the number of local scatterers in the referencemodel is infinite the parameters 120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902 120572119894119901119902and 120573119894119901119902 can be seen as continuous random variables withcorresponding probability density functions (PDFs) Thenthe ST-CF of the 119898B component (25) can be writtenas
Now the complete expression of (16) can be obtainedby substituting (20) and (28) into (16) 119901(120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902)describes the joint distribution of AOD and AOA and itcan be optionally used to present some propagation channelmodels As a result the ST-CF in (28) can provide a suitableplatform to study the statistical properties of some differentchannelmodels such as the random scatteringmodel [13 14]Jakes model [28] one-ring model [23] and two-ring model[29] as described in Section 313Therefore the ST-CF in (28)is a generalized and parametric expression However due tothe complex nature of 119901(120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902) it is assumed thatAOD and AOA are independent [13 14 32 33] and azimuthangles and elevation angles in (28) are also independent[18 19 32] The parameters in (28) such as the velocities ofthe multiple moving scatterers and random angles can becalculated as follows
(i) Scatterer Velocity Distributions The Gaussian Laplaceexponential and uniform distributions can be used todescribe the velocity of moving scatterers [13] In fact thescatterer velocity ]119894119901119902 is always positive or equal to zeroWe use the uniform distribution in (29) and half-Gaussiandistribution in (30) to describe the velocity of multiplemoving scatterers
119901 (]119894119901119902) = 1V119894max 0 le ]119894119901119902 le V119894max (29)
where V119894max is the maximum of ]119894119901119902
21205902119901119902119894]] ]119894119901119902 ge 0 (30)
where 120590119901119902119894 is the standard deviation of ]119894119901119902
(ii) Angle Distributions To characterize the statistical anglesin Table 1 we use the uniform distribution in (31) in theisotropic scattering environment and use the von Misesdistribution in (32) and the cosine distribution in (33) inthe nonisotropic scattering environment In addition theinterval of azimuth angles 120572119879119901119902 120572119877119901119902 and 120572119894119901119902 is (minus120587 120587] and theinterval of elevation angles 120573119879119901119902 120573119877119901119902 and 120573119894119901119902 is (minus1205872 1205872]119901 (120574) = 11205742 minus 1205741 1205741 le 120574 le 1205742 (31)
119901 (120572) = exp [119896 cos (120572 minus 120572)]21205871198680 (119896) 1205721 le 120572 le 1205721 + 2120587 (32)
where 1198680(∙) is the zeroth-order modified Bessel function ofthe first kind 120572 is the mean angle and 119896 controls the spreadof angles around the mean The von Mises distribution PDFwith 120572 = 0 is used to describe the azimuth angles
119901 (120573) = 1205874 10038161003816100381610038161205731198981003816100381610038161003816 cos(1205872 120573120573119898) minus 120573119898 le 120573 le 120573119898 (33)
where 120573119898 is the maximum of 120573 The cosine distribution PDFis used to describe the elevation angles
313 Special Cases of 119898119861rsquos ST-CF If 120573119879 = 120573119877 = 120573119879119901119902 =120573119877119901119902 = 120573119894119901119902 = 0 that is the scattering environment is 2D
Mobile Information Systems 7
some different special cases can be derived from the generalexpression of the119898Brsquos ST-CF in (28)
In the 2D scattering environment (28) can be written as(B2) in Appendix B If 119901 = 119901 119902 = 119902 and119898 = 1 (B2) equals(11) in [13] and (6) in [14] regardless of the plus or minussigns under the assumption that the angles 120572119879 and 120572119877 areindependent of each other in [13 14] In isotropic scatteringenvironments some other special cases with closed-formexpressions can be derived as follows
Appendix B shows that an approximate ST-CF of (B2)can be written as
119877119898119901119902119901119902 (119889119879 119889119877 120591) = 212059021198981198690 1198960 [V119879120591 + (119902 minus 119902) 119889119879]times 1198690 1198960 [V119877120591 + (119901 minus 119901) 119889119877] int sdot sdot sdot int 1198690 (21198960]1119901119902120591)times 1198690 (21198960]119898119901119902120591)119898minus1prod
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind and 1198960 = 21205871198910119888 is the wave number
Note that if 119898 approaches to the infinity (34) can bewritten as
lim119898rarrinfin
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 212059021198981198690 [(119902 minus 119902) 1198960119889119879] 1198690 [(119901 minus 119901) 1198960119889119877] 120591 = 00 120591 gt 0
(35)
In the NLoS communication environment with veryhigh-density scatterers (35) implies that the ST-CFapproaches to zero at the nonzero time difference 120591 and hasnothing to do with velocities of scatterers 119879119883 and 119877119883 Thelarge-scale antenna arrays are very suitable to be used in thisenvironment because the antenna element spacing can bereduced to a smaller value
If ]119894119901119902 is constant (34) can be presented as the followingclosed-form expression
The T-CF of the classical F2M scenario with fixed scat-terers is obtained if V119879 = V119894119901119902 = 0 (119894 = 1 2 3 119898)119902 = 119902 and 119901 = 119901 in (36) In this case the ST-CF in(36) is 212059021198981198690(1198960V119877120591) which is known as the Jakes model[28] If V119879 = 0119898 = 1 119902 = 119902 and 119901 = 119901 the ST-CFin (36) results in 212059021198981198690(1198960V1119901119902120591)1198690(1198960V119877120591) which equals the
T-CF of F2M single-ring channel model in the presence ofmoving scatterers reported in (11) of [23] If V119894119901119902 = 0 (119894 =1 2 3 119898) 119902 = 119902 and 119901 = 119901 the ST-CF in (36) resultsin 212059021198981198690(1198960V119879120591)1198690(1198960V119877120591) which equals the T-CFs of theclassical M2M two-ring channel model in the presence offixed scatterers reported in (46) of [29] If V119894119901119902 = 120591 = 0 (119894 =1 2 3 119898) 119902 = 119901 = 2 and 119902 = 119901 = 1 the ST-CF in (36)results in 212059021198981198690(1198960119889119879)1198690(1198960119889119877) which equals the S-CF in (46)of [29]
The velocities ofmoving scatterers such asmoving foliagewalking pedestrians and passing vehicles generally are ran-dom variables If ]119894119901119902 is described by the uniform distributionin (29) (34) can be presented as the following closed-formexpression
times 1198690 1198960 [V119877120591 + (119901 minus 119901) 119889119877] [21198690 (21198960V1max120591)+ 1205871198691 (21198960V1max120591)1198670 (21198960V1max120591) minus 1205871198690 (21198960V1max120591)times 1198671 (21198960V1max120591)] [21198690 (21198960V119898max120591)+ 1205871198691 (21198960V119898max120591)1198670 (21198960V119898max120591) minus 1205871198690 (21198960V119898max120591)sdot 1198671 (21198960V119898max120591)] times 119898minus1prod
119894=2
[21198690 (41198960V119894max120591)+ 1205871198691 (41198960V119894max120591) times 1198670 (41198960V119894max120591)minus 1205871198690 (41198960V119894max120591)1198671 (41198960V119894max120591)]
(37)
where 1198691(∙) denotes the first-order Bessel function of the firstkind 1198670(∙) denotes the zeroth-order Struve function and1198671(∙) denotes the first-order Struve function
If ]119894119901119902 is described by the half-Gaussian distribution in(30) (34) can be presented as the following closed-formexpression
32 Space-Doppler Power Spectral Density The SD-PSD canbe obtained by taking the Fourier transform of the ST-CF in(16)with respect to time difference 120591 From (16) it follows that
8 Mobile Information Systems
Table 2 Parameters used in the numerical simulation
where 120575(∙) is the Dirac delta functionSince the ST-CF of the 119898B component is the multiple
integral as (28) the closed-form expression of SD-PSDcannot be derived In Section 42 we show the SD-PSD of the3DV2Vchannel in the presence ofmultiplemoving scatterersby means of numerical integrations
4 Numerical Results and Validation
This section demonstrates the normalized correlation func-tions and validates the Doppler power spectral density (D-PSD) described in Section 3 through theMATLABnumericalsimulations Unless indicated otherwise the values of thenumerical simulation parameters are summarized in Table 2
41 Numerical Results In this section the numerical curvesof ST-CFs T-CFs and S-CFs influenced by some importantcontributory factors are presented As in the typical urbanenvironments the power weight of the119898th cluster of rays hasbeen set to 1199011 = 1199012 = 12 for119872 = 2 1199011 = 1199012 = 1199013 = 13 for119872 = 3 1199011 = 1199012 = 13 1199013 = 1199014 = 16 for119872 = 4 1199011 = 131199012 = 14 1199013 = 1199014 = 16 and 1199015 = 112 for119872 = 5411 ST-CFs and T-CFs for Different Scattering Scenarios andBounces Figures 2ndash5 demonstrate the ST-CFs and T-CFs forthe different maximum bounces 119872 in the 3D isotropic and
0 05 1 15 2 25 3 35 4066
067
068
069
07
071
072
휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 2 ST-CFs in (16) in the isotropic scattering scenario fordifferent maximum bounces
nonisotropic scattering scenarios The scatterer velocity isuniformly distributed with an average speed of 25ms thatmay be the velocity of passing vehicles The other parametersused to obtain curves in Figures 2ndash5 are summarized in(Table 2 Cols 2 and 3) As shown in Figures 2 and 4 thelarger119872 is the faster the ST-CFs and T-CFs decrease in theisotropic scattering scenario However the descent rates ofST-CFs and T-CFs increase slowly when119872 is larger than 3The similar conclusions can be obtained fromFigures 3 and 5however the ST-CF curves in Figure 3 have oscillationswhichmay be caused by the nonisotropic scattering However thisconclusion differs from [27] which showed that the triple-or higher-order bounced rays had statistical properties verysimilar to those of the double-bounced rays and could beapproximated as double-bounced rays The discrepancy maybe caused by the different communication environmentsconsidered by us and [27] Specifically channel-soundingexperimental campaign in [27] was conducted along surface
Mobile Information Systems 9
066
067
068
069
07
071
072
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 3 ST-CFs in (16) in the nonisotropic scattering scenario fordifferent maximum bounces
065
07
075
08
085
09
095
1
M = 1M = 2M = 3
M = 4M = 5
0 05 1 15 2 25 3 35 4휏 (ms)
|R11
11(휏)|
Figure 4 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropicscattering scenario for different maximum bounces
streets around the Georgia Tech campus and on the Interstatehighways in the Midtown Atlanta metropolitan area whichmay have lower dense scatterers than our communicationenvironments The ST-CF curves in Figure 3 converge rel-atively slower for the larger 119872 that means there are morelocal scatterers inmotionwith randomvelocities and randomdirections As can be observed in Figures 4 and 5 theT-CFs inboth figures are very similar that indicates that the scatteringforms have no significant effect on the T-CFs of the 3D V2Vchannel in the presence of multiple moving scatterers Fur-thermore in Figures 2ndash5 it can be observed that a tendencyto a limit value curve of ST-CFs and T-CFs can be inferred if
065
07
075
08
085
09
095
1
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
|R11
11(휏)|
Figure 5 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the nonisotropicscattering scenario for different maximum bounces
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
066
067
068
069
071
072
1122(d
T=
05휆d
R=
05휆휏
)||R
07
Figure 6 ST-CFs in (16) in the isotropic and nonisotropic scatteringscenarios with119872 = 5 for different scatterer velocity distributions
119872 approaches infinity and the similar inference also can beobtained in (35) for the 2D scattering environment
412 ST-CFs and T-CFs for Different Scattering Scenarios andScatterer Velocity Distributions Figures 6 and 7 demonstratethe ST-CFs and T-CFs for the different scatterer velocitydistributions in the 3D isotropic and nonisotropic scatteringscenarios We use two uniform distributions in the isotropicscattering environment and use the von Mises distributionand the cosine distribution in the nonisotropic scatteringenvironment to describe all the statistical azimuth andelevation angles in Table 1 respectively The average speedof the uniform and half-Gaussian distributions is 25ms
10 Mobile Information Systems
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
065
07
075
08
085
09
095
1
|R11
11(휏)|
Figure 7 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropic andnonisotropic scattering scenarios with119872 = 5 for different scatterervelocity distributions
and the standard deviation of the half-Gaussian distribu-tion is 31ms The other parameters used to obtain curvesin Figures 6 and 7 are summarized in (Table 2 Cols 2and 3) As shown in Figure 6 the ST-CF for the uniformdistributed velocity of scatterers in the isotropic scatteringenvironment decreases fastest and both the nonisotropicscattering and the half-Gaussian distribution can result inswift oscillations of the fading curvesThe comparison resultsindicate that the scatterer velocity distributions have moresignificant effect than the scattering forms on the ST-CFsof the 3D V2V channel in the presence of multiple movingscatterers Figure 7 indicates that not like scattering formsthe scatterer velocity distributions have significant effect onthe T-CFs Generally the realistic propagation environmentis nonisotropic scattering and the scatterer velocity canbe modeled by the half-Gaussian or uniform distributionTherefore the results of Figures 6 and 7 may be helpful inanalysis of the system performance and modeling a specificpropagation channel for V2V communications
413 ST-CFs and S-CFs for Different Scattering Scenariosand Spacing between Adjacent Antenna Elements Figures8 and 9 demonstrate the 3D ST-CFs and S-CFs for thedifferent spacing between adjacent antenna elements of 119879119883and 119877119883 in isotropic and nonisotropic scattering scenariosThe set of the scatterer velocity in Figure 8 is the sameas Figures 2ndash5 and the other parameters used to obtaincurves in Figures 8 and 9 are summarized in (Table 2 Col4) Figures 8 and 9 show that spacing between adjacentantenna elements has significant effect on the 3D ST-CFsand S-CFs Furthermore the 3D ST-CFs in isotropic andnonisotropic scattering scenarios decrease to the bottomwhen 119889119877 = 119889119879 tends to 12120582 and 07120582 respectively Inaddition the 3D S-CF in nonisotropic scattering scenarios isexplicitly less than that in isotropic scattering scenarios for
|R11
22(d
Td
R휏)|
dR = d
T (휆) 휏 (ms)
0025
05075 1
12515 0
12
34
1
095
09
085
08
075
07
065
IsotropicNonisotropic
Figure 8 3D ST-CFs in (16) in the isotropic and nonisotropicscattering scenarios with119872 = 5 120582 is the carrier wavelength
|R11
22(d
Td
R)|
dR (휆)
dT (휆)
1
095
09
085
08
075
07
065
0 025 05 0751 125 15
0025
05075 1
12515
IsotropicNonisotropic
Figure 9 3D S-CFs in (16) with 120591 = 0 in the isotropic andnonisotropic scattering scenarios 120582 is the carrier wavelength
most spacing between adjacent antenna elements as shown inFigure 9Therefore the large-scale antenna arrays employingthe spatial multiplexing are more suitable to be used in thenonisotropic scattering scenarios owing to the high antennaspace utilization The spacing between adjacent antennaelements of antenna arrays is suggested to be more than 120582in the V2V communications especially in the environmentswith low-density scatterers
42 Comparison with Measurements The parametric natureof the proposed channel model makes it adaptable to avariety of propagation environments D-PSD is one of themost important and unique channel characteristics for V2Vcommunication channels To illustrate the validity of theproposed model we compare the modeled SD-PSDs of the3D V2V channel with multiple moving scatterers with themeasured F2F D-PSD in [26] the theoretical V2V SD-PSDin [18] and the measured V2V SD-PSD in [27]
Mobile Information Systems 11
minus2 minus15 minus1 minus05 0 05 1 15 2Doppler frequency f (kHz)
Figure 10 Comparison between the modeled SD-PSDs with119872 = 5and the measured D-PSDs of Figure 4 in [26]
Figure 10 shows the comparison between the modeledSD-PSDs obtained from the Fourier transform of the ST-CF presented in (16) in the isotropic scattering scenario andthe measured D-PSDs of Figure 4 in [26] The outdoor mea-surement experiments in [26] have shown that at millimeterwavelengths fading caused by foliage movement and themotion of nearby vehicles is a significantly deleterious effectin systems with static subscribers Figure 4 in [26] showsthe D-PSD generated by foliage movements and the D-PSDcaused by passing vehicles both in a fixed wireless channelat a carrier frequency of 295GHz We have used the half-Gaussian distribution with an average speed of 04ms andthe standard deviation of 05ms to model the velocity ofmoving foliage scatterers while the uniformdistributionwithan average speed of 6ms has been used tomodel the velocityof passing vehicles scatterers The power weight of the 119898thcluster of rays scattered by the moving foliage or passingvehicles has been set to 1199011 = 916 1199012 = 516 1199013 = 116and 1199014 = 1199015 = 132 The other parameters used to obtainthe modeled SD-PSD curves in Figure 10 are summarizedin (Table 2 Col 5) As can be observed in Figure 10 40 dBfading bandwidths for the effect of the foliage and vehiclesmovements are 500Hz and 1 kHz approximately respectivelyFade depth due to foliage is less than the variation due topassing vehicles Close agreements between the modeled andmeasured results are shown in Figure 10These close matchescan be seen not only for relatively slow moving scatterers butalso for relatively fast moving scatterers as well
Figure 11 shows the comparison among the modeled SD-PSD obtained from the Fourier transform of the ST-CFpresented in (16) in the nonisotropic scattering scenario thetheoretical SD-PSD of Figure 8 in [18] and themeasured SD-PSD of Figure 11 in [27]The theoretical SD-PSD reproducedhere is based on the two-cylinder model with moving andstationary scatterers for 119889119877 = 119889119879 = 0 The channel mea-surements in [27] for the measured SD-PSD were collectedat 2435GHz in the urban street surface environment and thespacing between two adjacent antenna elements of119879119883 and119877119883was set to zero We have used the half-Gaussian distributionwith average speeds of 2394ms 0798ms and 04ms
Figure 11 Comparison among the modeled SD-PSD with119872 = 5the theoretical SD-PSD of Figure 8 in [18] and the measured SD-PSD of Figure 11 in [27]
and the standard deviation of 3ms 1ms and 05ms tomodel the velocity of the first the second and the remainingmoving scatterers on the communication links respectivelyThe power weight of the 119898th cluster of rays has been set to1199011 = 815 1199012 = 13 1199013 = 115 1199014 = 130 and 1199015 = 130for119872 = 5 The other parameters used to obtain the modeledSD-PSD curve in Figure 11 are summarized in (Table 2 Col6) We note that different from the measured SD-PSDs in[26] themeasured SD-PSD curve in [27] has oscillationsTheoscillating pattern in measurements for the case 119889119877 = 119889119879 = 0may appear because it is not a true SISO scenario whereonly one transmit and one receive antenna are active [27] Ascan be observed in Figure 11 the modeled SD-PSD matchesbetter withmeasured SD-PSD than the theoretical SD-PSD in[18] This may be because of the introduction of the multiplebounced rays into the propagation model or the constraintson the position of local scatterers in the two-cylinder modelproposed in [18]
Figure 12 demonstrates the CDFs of the relative devi-ations between modeledtheoretical SD-PSDs and corre-sponding measured values in Figures 10 and 11 The meanrelative deviation of the modeled SD-PSDs for foliage inFigure 10 modeled SD-PSDs for vehicles in Figure 10 mod-eled SD-PSDs in Figure 11 and theoretical SD-PSDs inFigure 11 is 256 216 39 and 832 respectively Asshown in Figure 12 more than 80 of the relative deviationsof modeled SD-PSDs in Figures 10 and 11 are less than5 and the relative deviations of theoretical SD-PSDs inFigure 11 are explicitly larger than those of modeled SD-PSDs In addition from Figures 10 and 11 we note that themodeled SD-PSD matches better with measured SD-PSDs atthe lower and higher Doppler frequencies than the middleDoppler frequencies This may be because 119879119883 and 119877119883 werenot strictly fixed and not all the scatterers were in motion inmeasurementsThe approximate calculation of the frequencyshift in (A4) and Doppler shift in (10) may be anotherfactor which causes the mismatch at the middle Dopplerfrequencies andmore investigations concerning the accuracyof the proposed model will be addressed in future works
12 Mobile Information Systems
0 5 10 15 20 25 30 350
01
02
03
04
05
06
07
08
09
1
Relative deviation ()
Relat
ive d
evia
tion
CDF
Modeled SD-PSD (foliage)Modeled SD-PSD (vehicles)
Modeled SD-PSDTheoretical SD-PSD
Figure 12 The relative deviation cumulative distribution functions(CDFs) of the modeled and theoretical SD-PSDs in Figures 10 and11
However as shown in Figures 10ndash12 the acceptable matchesconfirm the utility and generality of the proposed model andshow the need for including multiple moving scatterers inpropagation model
5 Conclusion
Without specific constraints on the position of the local mov-ing scatterers a 3D geometrical V2V propagation model thatincludes LoS single bounced and multiple bounced linksbetween the transmitter and receiver was proposed Basedon the geometrical propagation model a 3D reference modelfor narrowband MIMO V2V multipath fading channels wasdeveloped From the reference model the correspondingmathematical expressions and numerical results of ST-CFsT-CFs S-CFs and SD-PSDs were studied for different para-metric sets It has been shown that the maximum bouncesscattering forms scatterer velocity distributions and spacingbetween adjacent antenna elements have significant effect onthe ST-CFs and only the maximum bounces and scatterervelocity distributions have significant effect on the T-CFsFinally the modeled SD-PSD results were compared withthe measured data and the other literaturersquos theoreticalresult The close agreements between the analytically andempirically obtained channel statistics confirmed the utilityand generality of the proposed model
Appendix
A Derivation Process of Frequency Shifts
The derivation process of 119891119902119901119902119899 in (7) is presented in thissection and 119891119901119901119902120588 119891119902119901119902119899 119891119901119901119902119899 and 119891119901119902120588 can be calculatedsimilarly
AOD
Δl
TX qth element
Figure 13 Projection from the coordinate vector of 119902th antennaelement to the AOD direction
According to Figure 1 the coordinate vectors of 119902thantenna element and the AOD direction can be written asrespectively
minus 1) 119889119879 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902 + sin120573119902 sin120573119879119901119902] (A3)
Because we mainly consider the difference of the fre-quency shifts caused by the TPD changes to calculate ST-CFsthe TPD change of ℎ11(119905) is assumed to be zero Eventually119891119902119901119902119899 can be calculated as
119891119902119901119902119899 asymp Δ1198971198910119888= (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(A4)
B Derivation Process of Approximate ST-CFs
In the 2D scattering environment the simplified and approx-imate expressions of (28) can be derived in this section If120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 (27) and (10) can be writtenas
Δ119891119899119898
= 1198910119888 [(119902 minus 119902) 119889119879 cos (120572119902 minus 120572119879119901119902)+ (119901 minus 119901) 119889119877 cos (120572119901 minus 120572119877119901119902)]
Mobile Information Systems 13
119891119901119902119899119898
= 1198910119888 [V119879 cos (120572119879 minus 120572119879119901119902) + V119877 cos (120572119877 minus 120572119877119901119902)
+ ]1119901119902 cos (1205721119901119902 minus 120572119879119901119902) + 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119879119901119902)+ ]119898119901119902 cos (120572119898119901119902 minus 120572119879119901119902) + ]1119901119902 cos (1205721119901119902 minus 120572119877119901119902)+ 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119877119901119902) + ]119898119901119902 cos (120572119898119901119902 minus 120572119877119901119902)] (B1)
If 120572119879 = 120572119902 and 120572119877 = 120572119901 that is the antenna array and itsvelocity have the same directions (28) can be simplified as bysubstituting (B1) into (28)
where 1198960 = 21205871198910119888 is the wave numberIn isotropic scattering environment we use the uniform
distribution in (31) to describe all the random azimuth anglesIn this case (B2) can be written as119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879] cos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877] cos (120572119877 minus 120572119877119901119902)
times1198690 [21198960]1119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times1198690 [21198960]119898119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times119898minus1prod119894=2
1198690 [41198960]119894119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind
The AOD 120572119879119901119902 and AOA 120572119877119901119902 are uniformly and indepen-dently distributed so120572119879119901119902minus120572119877119901119902 can be equal to zero in average
As a consequence the term cos[(120572119879119901119902 minus120572119877119901119902)2] in (B3) can beapproximated by one [13 14] and an approximate ST-CF canbe written as from (B3)
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879]sdotcos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877]sdot cos (120572119877 minus 120572119877119901119902) 1198690 (21198960]1119901119902120591)times1198690 (21198960]119898119901119902120591)119898minus1prod
Eventually the approximate ST-CF in (34) can be derivedfrom (B4) according to the definition of 1198690(∙)Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was funded by the National Natural Science Foun-dation of China (no 91438104 no 61501065 no 61571069and no 61601067) the Fundamental Research Funds forthe Central Universities (no 106112016CDJX160001) and theChongqing Research Program of Basic Research and FrontierTechnology (no CSTC2016JCYJA0021)
References
[1] A Bazzi B M Masini A Zanella and I Thibault ldquoBeaconingfrom connected vehicles IEEE 80211p vs LTE-V2Vrdquo in Pro-ceedings of 2016 IEEE 27th Annual International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRC)pp 1ndash6 Valencia Spain September 2016
[2] VM Rodrigo-Penarrocha J Reig L Rubio and et al ldquoAnalysisof small- scale fading distributions in vehicle-to-vehicle com-municationsrdquoMobile Information Systems vol 2016 Article ID9584815 7 pages 2016
[3] A Bazzi B M Masini A Zanella and G Pasolini ldquoIEEE80211p for cellular offloading in vehicular sensor networksrdquoComputer Communications vol 60 pp 97ndash108 2015
[4] X Cheng MIMO channel modelling and simulation for cellu-lar and mobile-to-mobile communication systems[Phd thesis]Heriot-Watt University 2009
[5] N Avazov and M Patzold ldquoA geometric street scatteringchannel model for car-to-car communication systemsrdquo inProceedings of 4th Annual International Conference onAdvanced
14 Mobile Information Systems
Technologies for Communications ATC2011 pp 224ndash230 vnmAugust 2011
[6] X Zhao X Liang S Li and K Haneda ldquoMobile-to-MobileWideband MIMO Channel Realization by Using a Two-RingGeometry-Based Stochastic Scattering Modelrdquo Wireless Per-sonal Communications vol 84 no 4 pp 2445ndash2465 2015
[7] M Walter D Shutin and U-C Fiebig ldquoDelay-dependentdoppler probability density functions for vehicle-to-vehiclescatter channelsrdquo IEEE Transactions on Antennas and Propaga-tion vol 62 no 4 pp 2238ndash2249 2014
[8] P T Samarasinghe T A Lamahewa T D Abhayapala and RA Kennedy ldquo3D mobile-to-mobile wireless channel modelrdquo inProceedings of 2010 Australian Communications Theory Work-shop AusCTW 2010 pp 30ndash34 aus February 2010
[9] M Riaz S J Nawaz and N M Khan ldquo3D ellipsoidal model formobile-to-mobile radio propagation environmentsrdquo WirelessPersonal Communications vol 72 no 4 pp 2465ndash2479 2013
[10] J Chen and T G Pratt ldquoThree-dimensional geometry-based stochastic modeling and performance of 4times4 space-polarization mobile-to-mobile wideband MIMO channelsrdquo inProceedings of 2013 IEEE Global Communications ConferenceGLOBECOM 2013 pp 3936ndash3941 usa December 2013
[11] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA Novel 3DNonstationary Channel Model Based on the von Mises-FisherScattering DistributionrdquoMobile Information Systems vol 2016Article ID 2161460 2016
[12] A Chelli and M Patzold ldquoThe impact of fixed and movingscatterers on the statistics of MIMO vehicle-to-vehicle chan-nelsrdquo in Proceedings of VTC Spring 2009 - IEEE 69th VehicularTechnology Conference esp April 2009
[13] A Borhani and M Patzold ldquoModeling of vehicle-to-vehiclechannels in the presence of moving scatterersrdquo in Proceedingsof 2012 IEEE Vehicular Technology Conference (VTC Fall) pp1ndash5 Quebec City QC Canada September 2012
[14] A Borhani andM Patzold ldquoCorrelation and spectral propertiesof vehicle-to-vehicle channels in the presence ofmoving scatter-ersrdquo IEEE Transactions on Vehicular Technology vol 62 no 9pp 4228ndash4239 2013
[15] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the IEEE 74th Vehicular TechnologyConference (VTC Fall rsquo11) pp 1ndash6 IEEE San Francisco CalifUSA September 2011
[16] M D Soltani M Alimadadi Y Seyedi and H AmindavarldquoModeling of doppler spectrum in V2V urban canyon oncom-ing environmentrdquo in Proceedings of 2014 7th InternationalSymposium on Telecommunications IST 2014 pp 1155ndash1160 irnSeptember 2014
[17] M D Soltani M Alimadadi and A Mohammadi ldquoModelingof mobile scatterer clusters for doppler spectrum in widebandvehicle-to-vehicle communication channelsrdquo IEEE Communi-cations Letters vol 18 no 4 pp 628ndash631 2014
[18] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014
[19] A G Zajic ldquoModeling impact of moving scatterers on dopplerspectrum in wideband vehicle-to-vehicle channelsrdquo in Proceed-ings of Proc Eur Conf Antennas Propag p 1 Lisbon May 2015
[20] J Zhang X Yin and X Cheng ldquoTheoretical analysis andmeasurements Doppler spectra of vehicular communication
channelsrdquo in Proceedings of 2012 12th International Confer-ence on ITS Telecommunications ITST 2012 pp 98ndash102 twnNovember 2012
[21] M Picone S Busanelli M Amoretti F Zanichelli and GFerrari Advanced Technologies for Intelligent TransportationSystems Springer 2015
[22] V H Pham M H Taieb J Y Chouinard S Roy and H THuynh ldquoOn the double Doppler effect generated by scatterermotionrdquo REV Journal on Electronics and Communications vol1 pp 30ndash37 2011
[23] S Roy H T Huynh and P Fortier ldquoCompound Dopplerspread effects of subscriber motion and scatterer motionrdquo AEU- International Journal of Electronics and Communications vol57 no 4 pp 237ndash246 2003
[24] J B Andersen J O Nielsen G F Pedersen G Bauch and GDietl ldquoDoppler spectrum from moving scatterers in a randomenvironmentrdquo IEEE Transactions on Wireless Communicationsvol 8 no 6 pp 3270ndash3277 2009
[25] H S Rad S Gazor and P Shariatpanahi ldquoNon-fixed scatterersand their effects on MIMOmulticarrier fading communicationchannelsrdquo in Proceedings of 50th Annual IEEE Global Telecom-munications Conference GLOBECOM2007 pp 3765ndash3769 usaNovember 2007
[26] N Naz and D Falconer ldquoTemporal variations characterizationfor fixed wireless at 295 GHzrdquo in Proceedings of 2000 IEEE51st Vehicular Technology Conference Proceedings VTC2000-Springer pp 2178ndash2182 Tokyo Japan 2000
[27] A G Zajic G L Stuber T G Pratt and S T NguyenldquoWideband MIMO mobile-to-mobile channels Geometry-based statistical modeling with experimental verificationrdquo IEEETransactions on Vehicular Technology vol 58 no 2 pp 517ndash5342009
[28] W C Jakes Microwave Mobile Communications Wiley-IEEEPress Piscataway NJ USA 1994
[29] M Patzold B O Hogstad and N Youssef ldquoModeling analysisand simulation of MIMO mobile-to-mobile fading channelsrdquoIEEE Transactions onWireless Communications vol 7 no 2 pp510ndash520 2008
[30] M PatzoldMobile Radio Channels Wiley Chichester UK 2ndedition 2011
[31] R Wang and D Cox ldquoDouble mobility mitigates fading in adhoc wireless networksrdquo in Proceedings of IEEE Antennas andPropagation Society International Symposium pp 306ndash309 SanAntonio TX USA
[32] G Bakhshi K Shahtalebi and H S Rad ldquoA novel full-three-dimensional MIMOmobile-to-mobile channel referencemodelrdquo in Proceedings of 3rd International Conference on SignalProcessing and Communication Systems ICSPCSrsquo2009 usaSeptember 2009
[33] W Zhou X Wang X Wang and W Chen ldquoA modified two-erose- ringmodel formimomobile-to-mobile fading channelsrdquoin Proceedings of 7th International Conference on WirelessCommunications Networking and Mobile Computing WiCOM2011 5 1 pages Wuhan China September 2011
= 119891119902119901119902119899 + 119891119901119901119902119899 minus 119891119902119901119902119899 minus 119891119901119901119902119899 (26)
It is assumed that AOD and AOA of ℎ119898119901119902(119905) and ℎ119898119901119902(119905) areindependent and identically distributed (iid) By substitut-ing (7) and (8) into (26) Δ119891119899
Since the number of local scatterers in the referencemodel is infinite the parameters 120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902 120572119894119901119902and 120573119894119901119902 can be seen as continuous random variables withcorresponding probability density functions (PDFs) Thenthe ST-CF of the 119898B component (25) can be writtenas
Now the complete expression of (16) can be obtainedby substituting (20) and (28) into (16) 119901(120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902)describes the joint distribution of AOD and AOA and itcan be optionally used to present some propagation channelmodels As a result the ST-CF in (28) can provide a suitableplatform to study the statistical properties of some differentchannelmodels such as the random scatteringmodel [13 14]Jakes model [28] one-ring model [23] and two-ring model[29] as described in Section 313Therefore the ST-CF in (28)is a generalized and parametric expression However due tothe complex nature of 119901(120572119879119901119902 120573119879119901119902 120572119877119901119902 120573119877119901119902) it is assumed thatAOD and AOA are independent [13 14 32 33] and azimuthangles and elevation angles in (28) are also independent[18 19 32] The parameters in (28) such as the velocities ofthe multiple moving scatterers and random angles can becalculated as follows
(i) Scatterer Velocity Distributions The Gaussian Laplaceexponential and uniform distributions can be used todescribe the velocity of moving scatterers [13] In fact thescatterer velocity ]119894119901119902 is always positive or equal to zeroWe use the uniform distribution in (29) and half-Gaussiandistribution in (30) to describe the velocity of multiplemoving scatterers
119901 (]119894119901119902) = 1V119894max 0 le ]119894119901119902 le V119894max (29)
where V119894max is the maximum of ]119894119901119902
21205902119901119902119894]] ]119894119901119902 ge 0 (30)
where 120590119901119902119894 is the standard deviation of ]119894119901119902
(ii) Angle Distributions To characterize the statistical anglesin Table 1 we use the uniform distribution in (31) in theisotropic scattering environment and use the von Misesdistribution in (32) and the cosine distribution in (33) inthe nonisotropic scattering environment In addition theinterval of azimuth angles 120572119879119901119902 120572119877119901119902 and 120572119894119901119902 is (minus120587 120587] and theinterval of elevation angles 120573119879119901119902 120573119877119901119902 and 120573119894119901119902 is (minus1205872 1205872]119901 (120574) = 11205742 minus 1205741 1205741 le 120574 le 1205742 (31)
119901 (120572) = exp [119896 cos (120572 minus 120572)]21205871198680 (119896) 1205721 le 120572 le 1205721 + 2120587 (32)
where 1198680(∙) is the zeroth-order modified Bessel function ofthe first kind 120572 is the mean angle and 119896 controls the spreadof angles around the mean The von Mises distribution PDFwith 120572 = 0 is used to describe the azimuth angles
119901 (120573) = 1205874 10038161003816100381610038161205731198981003816100381610038161003816 cos(1205872 120573120573119898) minus 120573119898 le 120573 le 120573119898 (33)
where 120573119898 is the maximum of 120573 The cosine distribution PDFis used to describe the elevation angles
313 Special Cases of 119898119861rsquos ST-CF If 120573119879 = 120573119877 = 120573119879119901119902 =120573119877119901119902 = 120573119894119901119902 = 0 that is the scattering environment is 2D
Mobile Information Systems 7
some different special cases can be derived from the generalexpression of the119898Brsquos ST-CF in (28)
In the 2D scattering environment (28) can be written as(B2) in Appendix B If 119901 = 119901 119902 = 119902 and119898 = 1 (B2) equals(11) in [13] and (6) in [14] regardless of the plus or minussigns under the assumption that the angles 120572119879 and 120572119877 areindependent of each other in [13 14] In isotropic scatteringenvironments some other special cases with closed-formexpressions can be derived as follows
Appendix B shows that an approximate ST-CF of (B2)can be written as
119877119898119901119902119901119902 (119889119879 119889119877 120591) = 212059021198981198690 1198960 [V119879120591 + (119902 minus 119902) 119889119879]times 1198690 1198960 [V119877120591 + (119901 minus 119901) 119889119877] int sdot sdot sdot int 1198690 (21198960]1119901119902120591)times 1198690 (21198960]119898119901119902120591)119898minus1prod
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind and 1198960 = 21205871198910119888 is the wave number
Note that if 119898 approaches to the infinity (34) can bewritten as
lim119898rarrinfin
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 212059021198981198690 [(119902 minus 119902) 1198960119889119879] 1198690 [(119901 minus 119901) 1198960119889119877] 120591 = 00 120591 gt 0
(35)
In the NLoS communication environment with veryhigh-density scatterers (35) implies that the ST-CFapproaches to zero at the nonzero time difference 120591 and hasnothing to do with velocities of scatterers 119879119883 and 119877119883 Thelarge-scale antenna arrays are very suitable to be used in thisenvironment because the antenna element spacing can bereduced to a smaller value
If ]119894119901119902 is constant (34) can be presented as the followingclosed-form expression
The T-CF of the classical F2M scenario with fixed scat-terers is obtained if V119879 = V119894119901119902 = 0 (119894 = 1 2 3 119898)119902 = 119902 and 119901 = 119901 in (36) In this case the ST-CF in(36) is 212059021198981198690(1198960V119877120591) which is known as the Jakes model[28] If V119879 = 0119898 = 1 119902 = 119902 and 119901 = 119901 the ST-CFin (36) results in 212059021198981198690(1198960V1119901119902120591)1198690(1198960V119877120591) which equals the
T-CF of F2M single-ring channel model in the presence ofmoving scatterers reported in (11) of [23] If V119894119901119902 = 0 (119894 =1 2 3 119898) 119902 = 119902 and 119901 = 119901 the ST-CF in (36) resultsin 212059021198981198690(1198960V119879120591)1198690(1198960V119877120591) which equals the T-CFs of theclassical M2M two-ring channel model in the presence offixed scatterers reported in (46) of [29] If V119894119901119902 = 120591 = 0 (119894 =1 2 3 119898) 119902 = 119901 = 2 and 119902 = 119901 = 1 the ST-CF in (36)results in 212059021198981198690(1198960119889119879)1198690(1198960119889119877) which equals the S-CF in (46)of [29]
The velocities ofmoving scatterers such asmoving foliagewalking pedestrians and passing vehicles generally are ran-dom variables If ]119894119901119902 is described by the uniform distributionin (29) (34) can be presented as the following closed-formexpression
times 1198690 1198960 [V119877120591 + (119901 minus 119901) 119889119877] [21198690 (21198960V1max120591)+ 1205871198691 (21198960V1max120591)1198670 (21198960V1max120591) minus 1205871198690 (21198960V1max120591)times 1198671 (21198960V1max120591)] [21198690 (21198960V119898max120591)+ 1205871198691 (21198960V119898max120591)1198670 (21198960V119898max120591) minus 1205871198690 (21198960V119898max120591)sdot 1198671 (21198960V119898max120591)] times 119898minus1prod
119894=2
[21198690 (41198960V119894max120591)+ 1205871198691 (41198960V119894max120591) times 1198670 (41198960V119894max120591)minus 1205871198690 (41198960V119894max120591)1198671 (41198960V119894max120591)]
(37)
where 1198691(∙) denotes the first-order Bessel function of the firstkind 1198670(∙) denotes the zeroth-order Struve function and1198671(∙) denotes the first-order Struve function
If ]119894119901119902 is described by the half-Gaussian distribution in(30) (34) can be presented as the following closed-formexpression
32 Space-Doppler Power Spectral Density The SD-PSD canbe obtained by taking the Fourier transform of the ST-CF in(16)with respect to time difference 120591 From (16) it follows that
8 Mobile Information Systems
Table 2 Parameters used in the numerical simulation
where 120575(∙) is the Dirac delta functionSince the ST-CF of the 119898B component is the multiple
integral as (28) the closed-form expression of SD-PSDcannot be derived In Section 42 we show the SD-PSD of the3DV2Vchannel in the presence ofmultiplemoving scatterersby means of numerical integrations
4 Numerical Results and Validation
This section demonstrates the normalized correlation func-tions and validates the Doppler power spectral density (D-PSD) described in Section 3 through theMATLABnumericalsimulations Unless indicated otherwise the values of thenumerical simulation parameters are summarized in Table 2
41 Numerical Results In this section the numerical curvesof ST-CFs T-CFs and S-CFs influenced by some importantcontributory factors are presented As in the typical urbanenvironments the power weight of the119898th cluster of rays hasbeen set to 1199011 = 1199012 = 12 for119872 = 2 1199011 = 1199012 = 1199013 = 13 for119872 = 3 1199011 = 1199012 = 13 1199013 = 1199014 = 16 for119872 = 4 1199011 = 131199012 = 14 1199013 = 1199014 = 16 and 1199015 = 112 for119872 = 5411 ST-CFs and T-CFs for Different Scattering Scenarios andBounces Figures 2ndash5 demonstrate the ST-CFs and T-CFs forthe different maximum bounces 119872 in the 3D isotropic and
0 05 1 15 2 25 3 35 4066
067
068
069
07
071
072
휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 2 ST-CFs in (16) in the isotropic scattering scenario fordifferent maximum bounces
nonisotropic scattering scenarios The scatterer velocity isuniformly distributed with an average speed of 25ms thatmay be the velocity of passing vehicles The other parametersused to obtain curves in Figures 2ndash5 are summarized in(Table 2 Cols 2 and 3) As shown in Figures 2 and 4 thelarger119872 is the faster the ST-CFs and T-CFs decrease in theisotropic scattering scenario However the descent rates ofST-CFs and T-CFs increase slowly when119872 is larger than 3The similar conclusions can be obtained fromFigures 3 and 5however the ST-CF curves in Figure 3 have oscillationswhichmay be caused by the nonisotropic scattering However thisconclusion differs from [27] which showed that the triple-or higher-order bounced rays had statistical properties verysimilar to those of the double-bounced rays and could beapproximated as double-bounced rays The discrepancy maybe caused by the different communication environmentsconsidered by us and [27] Specifically channel-soundingexperimental campaign in [27] was conducted along surface
Mobile Information Systems 9
066
067
068
069
07
071
072
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 3 ST-CFs in (16) in the nonisotropic scattering scenario fordifferent maximum bounces
065
07
075
08
085
09
095
1
M = 1M = 2M = 3
M = 4M = 5
0 05 1 15 2 25 3 35 4휏 (ms)
|R11
11(휏)|
Figure 4 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropicscattering scenario for different maximum bounces
streets around the Georgia Tech campus and on the Interstatehighways in the Midtown Atlanta metropolitan area whichmay have lower dense scatterers than our communicationenvironments The ST-CF curves in Figure 3 converge rel-atively slower for the larger 119872 that means there are morelocal scatterers inmotionwith randomvelocities and randomdirections As can be observed in Figures 4 and 5 theT-CFs inboth figures are very similar that indicates that the scatteringforms have no significant effect on the T-CFs of the 3D V2Vchannel in the presence of multiple moving scatterers Fur-thermore in Figures 2ndash5 it can be observed that a tendencyto a limit value curve of ST-CFs and T-CFs can be inferred if
065
07
075
08
085
09
095
1
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
|R11
11(휏)|
Figure 5 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the nonisotropicscattering scenario for different maximum bounces
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
066
067
068
069
071
072
1122(d
T=
05휆d
R=
05휆휏
)||R
07
Figure 6 ST-CFs in (16) in the isotropic and nonisotropic scatteringscenarios with119872 = 5 for different scatterer velocity distributions
119872 approaches infinity and the similar inference also can beobtained in (35) for the 2D scattering environment
412 ST-CFs and T-CFs for Different Scattering Scenarios andScatterer Velocity Distributions Figures 6 and 7 demonstratethe ST-CFs and T-CFs for the different scatterer velocitydistributions in the 3D isotropic and nonisotropic scatteringscenarios We use two uniform distributions in the isotropicscattering environment and use the von Mises distributionand the cosine distribution in the nonisotropic scatteringenvironment to describe all the statistical azimuth andelevation angles in Table 1 respectively The average speedof the uniform and half-Gaussian distributions is 25ms
10 Mobile Information Systems
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
065
07
075
08
085
09
095
1
|R11
11(휏)|
Figure 7 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropic andnonisotropic scattering scenarios with119872 = 5 for different scatterervelocity distributions
and the standard deviation of the half-Gaussian distribu-tion is 31ms The other parameters used to obtain curvesin Figures 6 and 7 are summarized in (Table 2 Cols 2and 3) As shown in Figure 6 the ST-CF for the uniformdistributed velocity of scatterers in the isotropic scatteringenvironment decreases fastest and both the nonisotropicscattering and the half-Gaussian distribution can result inswift oscillations of the fading curvesThe comparison resultsindicate that the scatterer velocity distributions have moresignificant effect than the scattering forms on the ST-CFsof the 3D V2V channel in the presence of multiple movingscatterers Figure 7 indicates that not like scattering formsthe scatterer velocity distributions have significant effect onthe T-CFs Generally the realistic propagation environmentis nonisotropic scattering and the scatterer velocity canbe modeled by the half-Gaussian or uniform distributionTherefore the results of Figures 6 and 7 may be helpful inanalysis of the system performance and modeling a specificpropagation channel for V2V communications
413 ST-CFs and S-CFs for Different Scattering Scenariosand Spacing between Adjacent Antenna Elements Figures8 and 9 demonstrate the 3D ST-CFs and S-CFs for thedifferent spacing between adjacent antenna elements of 119879119883and 119877119883 in isotropic and nonisotropic scattering scenariosThe set of the scatterer velocity in Figure 8 is the sameas Figures 2ndash5 and the other parameters used to obtaincurves in Figures 8 and 9 are summarized in (Table 2 Col4) Figures 8 and 9 show that spacing between adjacentantenna elements has significant effect on the 3D ST-CFsand S-CFs Furthermore the 3D ST-CFs in isotropic andnonisotropic scattering scenarios decrease to the bottomwhen 119889119877 = 119889119879 tends to 12120582 and 07120582 respectively Inaddition the 3D S-CF in nonisotropic scattering scenarios isexplicitly less than that in isotropic scattering scenarios for
|R11
22(d
Td
R휏)|
dR = d
T (휆) 휏 (ms)
0025
05075 1
12515 0
12
34
1
095
09
085
08
075
07
065
IsotropicNonisotropic
Figure 8 3D ST-CFs in (16) in the isotropic and nonisotropicscattering scenarios with119872 = 5 120582 is the carrier wavelength
|R11
22(d
Td
R)|
dR (휆)
dT (휆)
1
095
09
085
08
075
07
065
0 025 05 0751 125 15
0025
05075 1
12515
IsotropicNonisotropic
Figure 9 3D S-CFs in (16) with 120591 = 0 in the isotropic andnonisotropic scattering scenarios 120582 is the carrier wavelength
most spacing between adjacent antenna elements as shown inFigure 9Therefore the large-scale antenna arrays employingthe spatial multiplexing are more suitable to be used in thenonisotropic scattering scenarios owing to the high antennaspace utilization The spacing between adjacent antennaelements of antenna arrays is suggested to be more than 120582in the V2V communications especially in the environmentswith low-density scatterers
42 Comparison with Measurements The parametric natureof the proposed channel model makes it adaptable to avariety of propagation environments D-PSD is one of themost important and unique channel characteristics for V2Vcommunication channels To illustrate the validity of theproposed model we compare the modeled SD-PSDs of the3D V2V channel with multiple moving scatterers with themeasured F2F D-PSD in [26] the theoretical V2V SD-PSDin [18] and the measured V2V SD-PSD in [27]
Mobile Information Systems 11
minus2 minus15 minus1 minus05 0 05 1 15 2Doppler frequency f (kHz)
Figure 10 Comparison between the modeled SD-PSDs with119872 = 5and the measured D-PSDs of Figure 4 in [26]
Figure 10 shows the comparison between the modeledSD-PSDs obtained from the Fourier transform of the ST-CF presented in (16) in the isotropic scattering scenario andthe measured D-PSDs of Figure 4 in [26] The outdoor mea-surement experiments in [26] have shown that at millimeterwavelengths fading caused by foliage movement and themotion of nearby vehicles is a significantly deleterious effectin systems with static subscribers Figure 4 in [26] showsthe D-PSD generated by foliage movements and the D-PSDcaused by passing vehicles both in a fixed wireless channelat a carrier frequency of 295GHz We have used the half-Gaussian distribution with an average speed of 04ms andthe standard deviation of 05ms to model the velocity ofmoving foliage scatterers while the uniformdistributionwithan average speed of 6ms has been used tomodel the velocityof passing vehicles scatterers The power weight of the 119898thcluster of rays scattered by the moving foliage or passingvehicles has been set to 1199011 = 916 1199012 = 516 1199013 = 116and 1199014 = 1199015 = 132 The other parameters used to obtainthe modeled SD-PSD curves in Figure 10 are summarizedin (Table 2 Col 5) As can be observed in Figure 10 40 dBfading bandwidths for the effect of the foliage and vehiclesmovements are 500Hz and 1 kHz approximately respectivelyFade depth due to foliage is less than the variation due topassing vehicles Close agreements between the modeled andmeasured results are shown in Figure 10These close matchescan be seen not only for relatively slow moving scatterers butalso for relatively fast moving scatterers as well
Figure 11 shows the comparison among the modeled SD-PSD obtained from the Fourier transform of the ST-CFpresented in (16) in the nonisotropic scattering scenario thetheoretical SD-PSD of Figure 8 in [18] and themeasured SD-PSD of Figure 11 in [27]The theoretical SD-PSD reproducedhere is based on the two-cylinder model with moving andstationary scatterers for 119889119877 = 119889119879 = 0 The channel mea-surements in [27] for the measured SD-PSD were collectedat 2435GHz in the urban street surface environment and thespacing between two adjacent antenna elements of119879119883 and119877119883was set to zero We have used the half-Gaussian distributionwith average speeds of 2394ms 0798ms and 04ms
Figure 11 Comparison among the modeled SD-PSD with119872 = 5the theoretical SD-PSD of Figure 8 in [18] and the measured SD-PSD of Figure 11 in [27]
and the standard deviation of 3ms 1ms and 05ms tomodel the velocity of the first the second and the remainingmoving scatterers on the communication links respectivelyThe power weight of the 119898th cluster of rays has been set to1199011 = 815 1199012 = 13 1199013 = 115 1199014 = 130 and 1199015 = 130for119872 = 5 The other parameters used to obtain the modeledSD-PSD curve in Figure 11 are summarized in (Table 2 Col6) We note that different from the measured SD-PSDs in[26] themeasured SD-PSD curve in [27] has oscillationsTheoscillating pattern in measurements for the case 119889119877 = 119889119879 = 0may appear because it is not a true SISO scenario whereonly one transmit and one receive antenna are active [27] Ascan be observed in Figure 11 the modeled SD-PSD matchesbetter withmeasured SD-PSD than the theoretical SD-PSD in[18] This may be because of the introduction of the multiplebounced rays into the propagation model or the constraintson the position of local scatterers in the two-cylinder modelproposed in [18]
Figure 12 demonstrates the CDFs of the relative devi-ations between modeledtheoretical SD-PSDs and corre-sponding measured values in Figures 10 and 11 The meanrelative deviation of the modeled SD-PSDs for foliage inFigure 10 modeled SD-PSDs for vehicles in Figure 10 mod-eled SD-PSDs in Figure 11 and theoretical SD-PSDs inFigure 11 is 256 216 39 and 832 respectively Asshown in Figure 12 more than 80 of the relative deviationsof modeled SD-PSDs in Figures 10 and 11 are less than5 and the relative deviations of theoretical SD-PSDs inFigure 11 are explicitly larger than those of modeled SD-PSDs In addition from Figures 10 and 11 we note that themodeled SD-PSD matches better with measured SD-PSDs atthe lower and higher Doppler frequencies than the middleDoppler frequencies This may be because 119879119883 and 119877119883 werenot strictly fixed and not all the scatterers were in motion inmeasurementsThe approximate calculation of the frequencyshift in (A4) and Doppler shift in (10) may be anotherfactor which causes the mismatch at the middle Dopplerfrequencies andmore investigations concerning the accuracyof the proposed model will be addressed in future works
12 Mobile Information Systems
0 5 10 15 20 25 30 350
01
02
03
04
05
06
07
08
09
1
Relative deviation ()
Relat
ive d
evia
tion
CDF
Modeled SD-PSD (foliage)Modeled SD-PSD (vehicles)
Modeled SD-PSDTheoretical SD-PSD
Figure 12 The relative deviation cumulative distribution functions(CDFs) of the modeled and theoretical SD-PSDs in Figures 10 and11
However as shown in Figures 10ndash12 the acceptable matchesconfirm the utility and generality of the proposed model andshow the need for including multiple moving scatterers inpropagation model
5 Conclusion
Without specific constraints on the position of the local mov-ing scatterers a 3D geometrical V2V propagation model thatincludes LoS single bounced and multiple bounced linksbetween the transmitter and receiver was proposed Basedon the geometrical propagation model a 3D reference modelfor narrowband MIMO V2V multipath fading channels wasdeveloped From the reference model the correspondingmathematical expressions and numerical results of ST-CFsT-CFs S-CFs and SD-PSDs were studied for different para-metric sets It has been shown that the maximum bouncesscattering forms scatterer velocity distributions and spacingbetween adjacent antenna elements have significant effect onthe ST-CFs and only the maximum bounces and scatterervelocity distributions have significant effect on the T-CFsFinally the modeled SD-PSD results were compared withthe measured data and the other literaturersquos theoreticalresult The close agreements between the analytically andempirically obtained channel statistics confirmed the utilityand generality of the proposed model
Appendix
A Derivation Process of Frequency Shifts
The derivation process of 119891119902119901119902119899 in (7) is presented in thissection and 119891119901119901119902120588 119891119902119901119902119899 119891119901119901119902119899 and 119891119901119902120588 can be calculatedsimilarly
AOD
Δl
TX qth element
Figure 13 Projection from the coordinate vector of 119902th antennaelement to the AOD direction
According to Figure 1 the coordinate vectors of 119902thantenna element and the AOD direction can be written asrespectively
minus 1) 119889119879 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902 + sin120573119902 sin120573119879119901119902] (A3)
Because we mainly consider the difference of the fre-quency shifts caused by the TPD changes to calculate ST-CFsthe TPD change of ℎ11(119905) is assumed to be zero Eventually119891119902119901119902119899 can be calculated as
119891119902119901119902119899 asymp Δ1198971198910119888= (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(A4)
B Derivation Process of Approximate ST-CFs
In the 2D scattering environment the simplified and approx-imate expressions of (28) can be derived in this section If120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 (27) and (10) can be writtenas
Δ119891119899119898
= 1198910119888 [(119902 minus 119902) 119889119879 cos (120572119902 minus 120572119879119901119902)+ (119901 minus 119901) 119889119877 cos (120572119901 minus 120572119877119901119902)]
Mobile Information Systems 13
119891119901119902119899119898
= 1198910119888 [V119879 cos (120572119879 minus 120572119879119901119902) + V119877 cos (120572119877 minus 120572119877119901119902)
+ ]1119901119902 cos (1205721119901119902 minus 120572119879119901119902) + 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119879119901119902)+ ]119898119901119902 cos (120572119898119901119902 minus 120572119879119901119902) + ]1119901119902 cos (1205721119901119902 minus 120572119877119901119902)+ 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119877119901119902) + ]119898119901119902 cos (120572119898119901119902 minus 120572119877119901119902)] (B1)
If 120572119879 = 120572119902 and 120572119877 = 120572119901 that is the antenna array and itsvelocity have the same directions (28) can be simplified as bysubstituting (B1) into (28)
where 1198960 = 21205871198910119888 is the wave numberIn isotropic scattering environment we use the uniform
distribution in (31) to describe all the random azimuth anglesIn this case (B2) can be written as119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879] cos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877] cos (120572119877 minus 120572119877119901119902)
times1198690 [21198960]1119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times1198690 [21198960]119898119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times119898minus1prod119894=2
1198690 [41198960]119894119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind
The AOD 120572119879119901119902 and AOA 120572119877119901119902 are uniformly and indepen-dently distributed so120572119879119901119902minus120572119877119901119902 can be equal to zero in average
As a consequence the term cos[(120572119879119901119902 minus120572119877119901119902)2] in (B3) can beapproximated by one [13 14] and an approximate ST-CF canbe written as from (B3)
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879]sdotcos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877]sdot cos (120572119877 minus 120572119877119901119902) 1198690 (21198960]1119901119902120591)times1198690 (21198960]119898119901119902120591)119898minus1prod
Eventually the approximate ST-CF in (34) can be derivedfrom (B4) according to the definition of 1198690(∙)Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was funded by the National Natural Science Foun-dation of China (no 91438104 no 61501065 no 61571069and no 61601067) the Fundamental Research Funds forthe Central Universities (no 106112016CDJX160001) and theChongqing Research Program of Basic Research and FrontierTechnology (no CSTC2016JCYJA0021)
References
[1] A Bazzi B M Masini A Zanella and I Thibault ldquoBeaconingfrom connected vehicles IEEE 80211p vs LTE-V2Vrdquo in Pro-ceedings of 2016 IEEE 27th Annual International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRC)pp 1ndash6 Valencia Spain September 2016
[2] VM Rodrigo-Penarrocha J Reig L Rubio and et al ldquoAnalysisof small- scale fading distributions in vehicle-to-vehicle com-municationsrdquoMobile Information Systems vol 2016 Article ID9584815 7 pages 2016
[3] A Bazzi B M Masini A Zanella and G Pasolini ldquoIEEE80211p for cellular offloading in vehicular sensor networksrdquoComputer Communications vol 60 pp 97ndash108 2015
[4] X Cheng MIMO channel modelling and simulation for cellu-lar and mobile-to-mobile communication systems[Phd thesis]Heriot-Watt University 2009
[5] N Avazov and M Patzold ldquoA geometric street scatteringchannel model for car-to-car communication systemsrdquo inProceedings of 4th Annual International Conference onAdvanced
14 Mobile Information Systems
Technologies for Communications ATC2011 pp 224ndash230 vnmAugust 2011
[6] X Zhao X Liang S Li and K Haneda ldquoMobile-to-MobileWideband MIMO Channel Realization by Using a Two-RingGeometry-Based Stochastic Scattering Modelrdquo Wireless Per-sonal Communications vol 84 no 4 pp 2445ndash2465 2015
[7] M Walter D Shutin and U-C Fiebig ldquoDelay-dependentdoppler probability density functions for vehicle-to-vehiclescatter channelsrdquo IEEE Transactions on Antennas and Propaga-tion vol 62 no 4 pp 2238ndash2249 2014
[8] P T Samarasinghe T A Lamahewa T D Abhayapala and RA Kennedy ldquo3D mobile-to-mobile wireless channel modelrdquo inProceedings of 2010 Australian Communications Theory Work-shop AusCTW 2010 pp 30ndash34 aus February 2010
[9] M Riaz S J Nawaz and N M Khan ldquo3D ellipsoidal model formobile-to-mobile radio propagation environmentsrdquo WirelessPersonal Communications vol 72 no 4 pp 2465ndash2479 2013
[10] J Chen and T G Pratt ldquoThree-dimensional geometry-based stochastic modeling and performance of 4times4 space-polarization mobile-to-mobile wideband MIMO channelsrdquo inProceedings of 2013 IEEE Global Communications ConferenceGLOBECOM 2013 pp 3936ndash3941 usa December 2013
[11] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA Novel 3DNonstationary Channel Model Based on the von Mises-FisherScattering DistributionrdquoMobile Information Systems vol 2016Article ID 2161460 2016
[12] A Chelli and M Patzold ldquoThe impact of fixed and movingscatterers on the statistics of MIMO vehicle-to-vehicle chan-nelsrdquo in Proceedings of VTC Spring 2009 - IEEE 69th VehicularTechnology Conference esp April 2009
[13] A Borhani and M Patzold ldquoModeling of vehicle-to-vehiclechannels in the presence of moving scatterersrdquo in Proceedingsof 2012 IEEE Vehicular Technology Conference (VTC Fall) pp1ndash5 Quebec City QC Canada September 2012
[14] A Borhani andM Patzold ldquoCorrelation and spectral propertiesof vehicle-to-vehicle channels in the presence ofmoving scatter-ersrdquo IEEE Transactions on Vehicular Technology vol 62 no 9pp 4228ndash4239 2013
[15] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the IEEE 74th Vehicular TechnologyConference (VTC Fall rsquo11) pp 1ndash6 IEEE San Francisco CalifUSA September 2011
[16] M D Soltani M Alimadadi Y Seyedi and H AmindavarldquoModeling of doppler spectrum in V2V urban canyon oncom-ing environmentrdquo in Proceedings of 2014 7th InternationalSymposium on Telecommunications IST 2014 pp 1155ndash1160 irnSeptember 2014
[17] M D Soltani M Alimadadi and A Mohammadi ldquoModelingof mobile scatterer clusters for doppler spectrum in widebandvehicle-to-vehicle communication channelsrdquo IEEE Communi-cations Letters vol 18 no 4 pp 628ndash631 2014
[18] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014
[19] A G Zajic ldquoModeling impact of moving scatterers on dopplerspectrum in wideband vehicle-to-vehicle channelsrdquo in Proceed-ings of Proc Eur Conf Antennas Propag p 1 Lisbon May 2015
[20] J Zhang X Yin and X Cheng ldquoTheoretical analysis andmeasurements Doppler spectra of vehicular communication
channelsrdquo in Proceedings of 2012 12th International Confer-ence on ITS Telecommunications ITST 2012 pp 98ndash102 twnNovember 2012
[21] M Picone S Busanelli M Amoretti F Zanichelli and GFerrari Advanced Technologies for Intelligent TransportationSystems Springer 2015
[22] V H Pham M H Taieb J Y Chouinard S Roy and H THuynh ldquoOn the double Doppler effect generated by scatterermotionrdquo REV Journal on Electronics and Communications vol1 pp 30ndash37 2011
[23] S Roy H T Huynh and P Fortier ldquoCompound Dopplerspread effects of subscriber motion and scatterer motionrdquo AEU- International Journal of Electronics and Communications vol57 no 4 pp 237ndash246 2003
[24] J B Andersen J O Nielsen G F Pedersen G Bauch and GDietl ldquoDoppler spectrum from moving scatterers in a randomenvironmentrdquo IEEE Transactions on Wireless Communicationsvol 8 no 6 pp 3270ndash3277 2009
[25] H S Rad S Gazor and P Shariatpanahi ldquoNon-fixed scatterersand their effects on MIMOmulticarrier fading communicationchannelsrdquo in Proceedings of 50th Annual IEEE Global Telecom-munications Conference GLOBECOM2007 pp 3765ndash3769 usaNovember 2007
[26] N Naz and D Falconer ldquoTemporal variations characterizationfor fixed wireless at 295 GHzrdquo in Proceedings of 2000 IEEE51st Vehicular Technology Conference Proceedings VTC2000-Springer pp 2178ndash2182 Tokyo Japan 2000
[27] A G Zajic G L Stuber T G Pratt and S T NguyenldquoWideband MIMO mobile-to-mobile channels Geometry-based statistical modeling with experimental verificationrdquo IEEETransactions on Vehicular Technology vol 58 no 2 pp 517ndash5342009
[28] W C Jakes Microwave Mobile Communications Wiley-IEEEPress Piscataway NJ USA 1994
[29] M Patzold B O Hogstad and N Youssef ldquoModeling analysisand simulation of MIMO mobile-to-mobile fading channelsrdquoIEEE Transactions onWireless Communications vol 7 no 2 pp510ndash520 2008
[30] M PatzoldMobile Radio Channels Wiley Chichester UK 2ndedition 2011
[31] R Wang and D Cox ldquoDouble mobility mitigates fading in adhoc wireless networksrdquo in Proceedings of IEEE Antennas andPropagation Society International Symposium pp 306ndash309 SanAntonio TX USA
[32] G Bakhshi K Shahtalebi and H S Rad ldquoA novel full-three-dimensional MIMOmobile-to-mobile channel referencemodelrdquo in Proceedings of 3rd International Conference on SignalProcessing and Communication Systems ICSPCSrsquo2009 usaSeptember 2009
[33] W Zhou X Wang X Wang and W Chen ldquoA modified two-erose- ringmodel formimomobile-to-mobile fading channelsrdquoin Proceedings of 7th International Conference on WirelessCommunications Networking and Mobile Computing WiCOM2011 5 1 pages Wuhan China September 2011
some different special cases can be derived from the generalexpression of the119898Brsquos ST-CF in (28)
In the 2D scattering environment (28) can be written as(B2) in Appendix B If 119901 = 119901 119902 = 119902 and119898 = 1 (B2) equals(11) in [13] and (6) in [14] regardless of the plus or minussigns under the assumption that the angles 120572119879 and 120572119877 areindependent of each other in [13 14] In isotropic scatteringenvironments some other special cases with closed-formexpressions can be derived as follows
Appendix B shows that an approximate ST-CF of (B2)can be written as
119877119898119901119902119901119902 (119889119879 119889119877 120591) = 212059021198981198690 1198960 [V119879120591 + (119902 minus 119902) 119889119879]times 1198690 1198960 [V119877120591 + (119901 minus 119901) 119889119877] int sdot sdot sdot int 1198690 (21198960]1119901119902120591)times 1198690 (21198960]119898119901119902120591)119898minus1prod
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind and 1198960 = 21205871198910119888 is the wave number
Note that if 119898 approaches to the infinity (34) can bewritten as
lim119898rarrinfin
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 212059021198981198690 [(119902 minus 119902) 1198960119889119879] 1198690 [(119901 minus 119901) 1198960119889119877] 120591 = 00 120591 gt 0
(35)
In the NLoS communication environment with veryhigh-density scatterers (35) implies that the ST-CFapproaches to zero at the nonzero time difference 120591 and hasnothing to do with velocities of scatterers 119879119883 and 119877119883 Thelarge-scale antenna arrays are very suitable to be used in thisenvironment because the antenna element spacing can bereduced to a smaller value
If ]119894119901119902 is constant (34) can be presented as the followingclosed-form expression
The T-CF of the classical F2M scenario with fixed scat-terers is obtained if V119879 = V119894119901119902 = 0 (119894 = 1 2 3 119898)119902 = 119902 and 119901 = 119901 in (36) In this case the ST-CF in(36) is 212059021198981198690(1198960V119877120591) which is known as the Jakes model[28] If V119879 = 0119898 = 1 119902 = 119902 and 119901 = 119901 the ST-CFin (36) results in 212059021198981198690(1198960V1119901119902120591)1198690(1198960V119877120591) which equals the
T-CF of F2M single-ring channel model in the presence ofmoving scatterers reported in (11) of [23] If V119894119901119902 = 0 (119894 =1 2 3 119898) 119902 = 119902 and 119901 = 119901 the ST-CF in (36) resultsin 212059021198981198690(1198960V119879120591)1198690(1198960V119877120591) which equals the T-CFs of theclassical M2M two-ring channel model in the presence offixed scatterers reported in (46) of [29] If V119894119901119902 = 120591 = 0 (119894 =1 2 3 119898) 119902 = 119901 = 2 and 119902 = 119901 = 1 the ST-CF in (36)results in 212059021198981198690(1198960119889119879)1198690(1198960119889119877) which equals the S-CF in (46)of [29]
The velocities ofmoving scatterers such asmoving foliagewalking pedestrians and passing vehicles generally are ran-dom variables If ]119894119901119902 is described by the uniform distributionin (29) (34) can be presented as the following closed-formexpression
times 1198690 1198960 [V119877120591 + (119901 minus 119901) 119889119877] [21198690 (21198960V1max120591)+ 1205871198691 (21198960V1max120591)1198670 (21198960V1max120591) minus 1205871198690 (21198960V1max120591)times 1198671 (21198960V1max120591)] [21198690 (21198960V119898max120591)+ 1205871198691 (21198960V119898max120591)1198670 (21198960V119898max120591) minus 1205871198690 (21198960V119898max120591)sdot 1198671 (21198960V119898max120591)] times 119898minus1prod
119894=2
[21198690 (41198960V119894max120591)+ 1205871198691 (41198960V119894max120591) times 1198670 (41198960V119894max120591)minus 1205871198690 (41198960V119894max120591)1198671 (41198960V119894max120591)]
(37)
where 1198691(∙) denotes the first-order Bessel function of the firstkind 1198670(∙) denotes the zeroth-order Struve function and1198671(∙) denotes the first-order Struve function
If ]119894119901119902 is described by the half-Gaussian distribution in(30) (34) can be presented as the following closed-formexpression
32 Space-Doppler Power Spectral Density The SD-PSD canbe obtained by taking the Fourier transform of the ST-CF in(16)with respect to time difference 120591 From (16) it follows that
8 Mobile Information Systems
Table 2 Parameters used in the numerical simulation
where 120575(∙) is the Dirac delta functionSince the ST-CF of the 119898B component is the multiple
integral as (28) the closed-form expression of SD-PSDcannot be derived In Section 42 we show the SD-PSD of the3DV2Vchannel in the presence ofmultiplemoving scatterersby means of numerical integrations
4 Numerical Results and Validation
This section demonstrates the normalized correlation func-tions and validates the Doppler power spectral density (D-PSD) described in Section 3 through theMATLABnumericalsimulations Unless indicated otherwise the values of thenumerical simulation parameters are summarized in Table 2
41 Numerical Results In this section the numerical curvesof ST-CFs T-CFs and S-CFs influenced by some importantcontributory factors are presented As in the typical urbanenvironments the power weight of the119898th cluster of rays hasbeen set to 1199011 = 1199012 = 12 for119872 = 2 1199011 = 1199012 = 1199013 = 13 for119872 = 3 1199011 = 1199012 = 13 1199013 = 1199014 = 16 for119872 = 4 1199011 = 131199012 = 14 1199013 = 1199014 = 16 and 1199015 = 112 for119872 = 5411 ST-CFs and T-CFs for Different Scattering Scenarios andBounces Figures 2ndash5 demonstrate the ST-CFs and T-CFs forthe different maximum bounces 119872 in the 3D isotropic and
0 05 1 15 2 25 3 35 4066
067
068
069
07
071
072
휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 2 ST-CFs in (16) in the isotropic scattering scenario fordifferent maximum bounces
nonisotropic scattering scenarios The scatterer velocity isuniformly distributed with an average speed of 25ms thatmay be the velocity of passing vehicles The other parametersused to obtain curves in Figures 2ndash5 are summarized in(Table 2 Cols 2 and 3) As shown in Figures 2 and 4 thelarger119872 is the faster the ST-CFs and T-CFs decrease in theisotropic scattering scenario However the descent rates ofST-CFs and T-CFs increase slowly when119872 is larger than 3The similar conclusions can be obtained fromFigures 3 and 5however the ST-CF curves in Figure 3 have oscillationswhichmay be caused by the nonisotropic scattering However thisconclusion differs from [27] which showed that the triple-or higher-order bounced rays had statistical properties verysimilar to those of the double-bounced rays and could beapproximated as double-bounced rays The discrepancy maybe caused by the different communication environmentsconsidered by us and [27] Specifically channel-soundingexperimental campaign in [27] was conducted along surface
Mobile Information Systems 9
066
067
068
069
07
071
072
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 3 ST-CFs in (16) in the nonisotropic scattering scenario fordifferent maximum bounces
065
07
075
08
085
09
095
1
M = 1M = 2M = 3
M = 4M = 5
0 05 1 15 2 25 3 35 4휏 (ms)
|R11
11(휏)|
Figure 4 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropicscattering scenario for different maximum bounces
streets around the Georgia Tech campus and on the Interstatehighways in the Midtown Atlanta metropolitan area whichmay have lower dense scatterers than our communicationenvironments The ST-CF curves in Figure 3 converge rel-atively slower for the larger 119872 that means there are morelocal scatterers inmotionwith randomvelocities and randomdirections As can be observed in Figures 4 and 5 theT-CFs inboth figures are very similar that indicates that the scatteringforms have no significant effect on the T-CFs of the 3D V2Vchannel in the presence of multiple moving scatterers Fur-thermore in Figures 2ndash5 it can be observed that a tendencyto a limit value curve of ST-CFs and T-CFs can be inferred if
065
07
075
08
085
09
095
1
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
|R11
11(휏)|
Figure 5 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the nonisotropicscattering scenario for different maximum bounces
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
066
067
068
069
071
072
1122(d
T=
05휆d
R=
05휆휏
)||R
07
Figure 6 ST-CFs in (16) in the isotropic and nonisotropic scatteringscenarios with119872 = 5 for different scatterer velocity distributions
119872 approaches infinity and the similar inference also can beobtained in (35) for the 2D scattering environment
412 ST-CFs and T-CFs for Different Scattering Scenarios andScatterer Velocity Distributions Figures 6 and 7 demonstratethe ST-CFs and T-CFs for the different scatterer velocitydistributions in the 3D isotropic and nonisotropic scatteringscenarios We use two uniform distributions in the isotropicscattering environment and use the von Mises distributionand the cosine distribution in the nonisotropic scatteringenvironment to describe all the statistical azimuth andelevation angles in Table 1 respectively The average speedof the uniform and half-Gaussian distributions is 25ms
10 Mobile Information Systems
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
065
07
075
08
085
09
095
1
|R11
11(휏)|
Figure 7 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropic andnonisotropic scattering scenarios with119872 = 5 for different scatterervelocity distributions
and the standard deviation of the half-Gaussian distribu-tion is 31ms The other parameters used to obtain curvesin Figures 6 and 7 are summarized in (Table 2 Cols 2and 3) As shown in Figure 6 the ST-CF for the uniformdistributed velocity of scatterers in the isotropic scatteringenvironment decreases fastest and both the nonisotropicscattering and the half-Gaussian distribution can result inswift oscillations of the fading curvesThe comparison resultsindicate that the scatterer velocity distributions have moresignificant effect than the scattering forms on the ST-CFsof the 3D V2V channel in the presence of multiple movingscatterers Figure 7 indicates that not like scattering formsthe scatterer velocity distributions have significant effect onthe T-CFs Generally the realistic propagation environmentis nonisotropic scattering and the scatterer velocity canbe modeled by the half-Gaussian or uniform distributionTherefore the results of Figures 6 and 7 may be helpful inanalysis of the system performance and modeling a specificpropagation channel for V2V communications
413 ST-CFs and S-CFs for Different Scattering Scenariosand Spacing between Adjacent Antenna Elements Figures8 and 9 demonstrate the 3D ST-CFs and S-CFs for thedifferent spacing between adjacent antenna elements of 119879119883and 119877119883 in isotropic and nonisotropic scattering scenariosThe set of the scatterer velocity in Figure 8 is the sameas Figures 2ndash5 and the other parameters used to obtaincurves in Figures 8 and 9 are summarized in (Table 2 Col4) Figures 8 and 9 show that spacing between adjacentantenna elements has significant effect on the 3D ST-CFsand S-CFs Furthermore the 3D ST-CFs in isotropic andnonisotropic scattering scenarios decrease to the bottomwhen 119889119877 = 119889119879 tends to 12120582 and 07120582 respectively Inaddition the 3D S-CF in nonisotropic scattering scenarios isexplicitly less than that in isotropic scattering scenarios for
|R11
22(d
Td
R휏)|
dR = d
T (휆) 휏 (ms)
0025
05075 1
12515 0
12
34
1
095
09
085
08
075
07
065
IsotropicNonisotropic
Figure 8 3D ST-CFs in (16) in the isotropic and nonisotropicscattering scenarios with119872 = 5 120582 is the carrier wavelength
|R11
22(d
Td
R)|
dR (휆)
dT (휆)
1
095
09
085
08
075
07
065
0 025 05 0751 125 15
0025
05075 1
12515
IsotropicNonisotropic
Figure 9 3D S-CFs in (16) with 120591 = 0 in the isotropic andnonisotropic scattering scenarios 120582 is the carrier wavelength
most spacing between adjacent antenna elements as shown inFigure 9Therefore the large-scale antenna arrays employingthe spatial multiplexing are more suitable to be used in thenonisotropic scattering scenarios owing to the high antennaspace utilization The spacing between adjacent antennaelements of antenna arrays is suggested to be more than 120582in the V2V communications especially in the environmentswith low-density scatterers
42 Comparison with Measurements The parametric natureof the proposed channel model makes it adaptable to avariety of propagation environments D-PSD is one of themost important and unique channel characteristics for V2Vcommunication channels To illustrate the validity of theproposed model we compare the modeled SD-PSDs of the3D V2V channel with multiple moving scatterers with themeasured F2F D-PSD in [26] the theoretical V2V SD-PSDin [18] and the measured V2V SD-PSD in [27]
Mobile Information Systems 11
minus2 minus15 minus1 minus05 0 05 1 15 2Doppler frequency f (kHz)
Figure 10 Comparison between the modeled SD-PSDs with119872 = 5and the measured D-PSDs of Figure 4 in [26]
Figure 10 shows the comparison between the modeledSD-PSDs obtained from the Fourier transform of the ST-CF presented in (16) in the isotropic scattering scenario andthe measured D-PSDs of Figure 4 in [26] The outdoor mea-surement experiments in [26] have shown that at millimeterwavelengths fading caused by foliage movement and themotion of nearby vehicles is a significantly deleterious effectin systems with static subscribers Figure 4 in [26] showsthe D-PSD generated by foliage movements and the D-PSDcaused by passing vehicles both in a fixed wireless channelat a carrier frequency of 295GHz We have used the half-Gaussian distribution with an average speed of 04ms andthe standard deviation of 05ms to model the velocity ofmoving foliage scatterers while the uniformdistributionwithan average speed of 6ms has been used tomodel the velocityof passing vehicles scatterers The power weight of the 119898thcluster of rays scattered by the moving foliage or passingvehicles has been set to 1199011 = 916 1199012 = 516 1199013 = 116and 1199014 = 1199015 = 132 The other parameters used to obtainthe modeled SD-PSD curves in Figure 10 are summarizedin (Table 2 Col 5) As can be observed in Figure 10 40 dBfading bandwidths for the effect of the foliage and vehiclesmovements are 500Hz and 1 kHz approximately respectivelyFade depth due to foliage is less than the variation due topassing vehicles Close agreements between the modeled andmeasured results are shown in Figure 10These close matchescan be seen not only for relatively slow moving scatterers butalso for relatively fast moving scatterers as well
Figure 11 shows the comparison among the modeled SD-PSD obtained from the Fourier transform of the ST-CFpresented in (16) in the nonisotropic scattering scenario thetheoretical SD-PSD of Figure 8 in [18] and themeasured SD-PSD of Figure 11 in [27]The theoretical SD-PSD reproducedhere is based on the two-cylinder model with moving andstationary scatterers for 119889119877 = 119889119879 = 0 The channel mea-surements in [27] for the measured SD-PSD were collectedat 2435GHz in the urban street surface environment and thespacing between two adjacent antenna elements of119879119883 and119877119883was set to zero We have used the half-Gaussian distributionwith average speeds of 2394ms 0798ms and 04ms
Figure 11 Comparison among the modeled SD-PSD with119872 = 5the theoretical SD-PSD of Figure 8 in [18] and the measured SD-PSD of Figure 11 in [27]
and the standard deviation of 3ms 1ms and 05ms tomodel the velocity of the first the second and the remainingmoving scatterers on the communication links respectivelyThe power weight of the 119898th cluster of rays has been set to1199011 = 815 1199012 = 13 1199013 = 115 1199014 = 130 and 1199015 = 130for119872 = 5 The other parameters used to obtain the modeledSD-PSD curve in Figure 11 are summarized in (Table 2 Col6) We note that different from the measured SD-PSDs in[26] themeasured SD-PSD curve in [27] has oscillationsTheoscillating pattern in measurements for the case 119889119877 = 119889119879 = 0may appear because it is not a true SISO scenario whereonly one transmit and one receive antenna are active [27] Ascan be observed in Figure 11 the modeled SD-PSD matchesbetter withmeasured SD-PSD than the theoretical SD-PSD in[18] This may be because of the introduction of the multiplebounced rays into the propagation model or the constraintson the position of local scatterers in the two-cylinder modelproposed in [18]
Figure 12 demonstrates the CDFs of the relative devi-ations between modeledtheoretical SD-PSDs and corre-sponding measured values in Figures 10 and 11 The meanrelative deviation of the modeled SD-PSDs for foliage inFigure 10 modeled SD-PSDs for vehicles in Figure 10 mod-eled SD-PSDs in Figure 11 and theoretical SD-PSDs inFigure 11 is 256 216 39 and 832 respectively Asshown in Figure 12 more than 80 of the relative deviationsof modeled SD-PSDs in Figures 10 and 11 are less than5 and the relative deviations of theoretical SD-PSDs inFigure 11 are explicitly larger than those of modeled SD-PSDs In addition from Figures 10 and 11 we note that themodeled SD-PSD matches better with measured SD-PSDs atthe lower and higher Doppler frequencies than the middleDoppler frequencies This may be because 119879119883 and 119877119883 werenot strictly fixed and not all the scatterers were in motion inmeasurementsThe approximate calculation of the frequencyshift in (A4) and Doppler shift in (10) may be anotherfactor which causes the mismatch at the middle Dopplerfrequencies andmore investigations concerning the accuracyof the proposed model will be addressed in future works
12 Mobile Information Systems
0 5 10 15 20 25 30 350
01
02
03
04
05
06
07
08
09
1
Relative deviation ()
Relat
ive d
evia
tion
CDF
Modeled SD-PSD (foliage)Modeled SD-PSD (vehicles)
Modeled SD-PSDTheoretical SD-PSD
Figure 12 The relative deviation cumulative distribution functions(CDFs) of the modeled and theoretical SD-PSDs in Figures 10 and11
However as shown in Figures 10ndash12 the acceptable matchesconfirm the utility and generality of the proposed model andshow the need for including multiple moving scatterers inpropagation model
5 Conclusion
Without specific constraints on the position of the local mov-ing scatterers a 3D geometrical V2V propagation model thatincludes LoS single bounced and multiple bounced linksbetween the transmitter and receiver was proposed Basedon the geometrical propagation model a 3D reference modelfor narrowband MIMO V2V multipath fading channels wasdeveloped From the reference model the correspondingmathematical expressions and numerical results of ST-CFsT-CFs S-CFs and SD-PSDs were studied for different para-metric sets It has been shown that the maximum bouncesscattering forms scatterer velocity distributions and spacingbetween adjacent antenna elements have significant effect onthe ST-CFs and only the maximum bounces and scatterervelocity distributions have significant effect on the T-CFsFinally the modeled SD-PSD results were compared withthe measured data and the other literaturersquos theoreticalresult The close agreements between the analytically andempirically obtained channel statistics confirmed the utilityand generality of the proposed model
Appendix
A Derivation Process of Frequency Shifts
The derivation process of 119891119902119901119902119899 in (7) is presented in thissection and 119891119901119901119902120588 119891119902119901119902119899 119891119901119901119902119899 and 119891119901119902120588 can be calculatedsimilarly
AOD
Δl
TX qth element
Figure 13 Projection from the coordinate vector of 119902th antennaelement to the AOD direction
According to Figure 1 the coordinate vectors of 119902thantenna element and the AOD direction can be written asrespectively
minus 1) 119889119879 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902 + sin120573119902 sin120573119879119901119902] (A3)
Because we mainly consider the difference of the fre-quency shifts caused by the TPD changes to calculate ST-CFsthe TPD change of ℎ11(119905) is assumed to be zero Eventually119891119902119901119902119899 can be calculated as
119891119902119901119902119899 asymp Δ1198971198910119888= (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(A4)
B Derivation Process of Approximate ST-CFs
In the 2D scattering environment the simplified and approx-imate expressions of (28) can be derived in this section If120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 (27) and (10) can be writtenas
Δ119891119899119898
= 1198910119888 [(119902 minus 119902) 119889119879 cos (120572119902 minus 120572119879119901119902)+ (119901 minus 119901) 119889119877 cos (120572119901 minus 120572119877119901119902)]
Mobile Information Systems 13
119891119901119902119899119898
= 1198910119888 [V119879 cos (120572119879 minus 120572119879119901119902) + V119877 cos (120572119877 minus 120572119877119901119902)
+ ]1119901119902 cos (1205721119901119902 minus 120572119879119901119902) + 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119879119901119902)+ ]119898119901119902 cos (120572119898119901119902 minus 120572119879119901119902) + ]1119901119902 cos (1205721119901119902 minus 120572119877119901119902)+ 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119877119901119902) + ]119898119901119902 cos (120572119898119901119902 minus 120572119877119901119902)] (B1)
If 120572119879 = 120572119902 and 120572119877 = 120572119901 that is the antenna array and itsvelocity have the same directions (28) can be simplified as bysubstituting (B1) into (28)
where 1198960 = 21205871198910119888 is the wave numberIn isotropic scattering environment we use the uniform
distribution in (31) to describe all the random azimuth anglesIn this case (B2) can be written as119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879] cos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877] cos (120572119877 minus 120572119877119901119902)
times1198690 [21198960]1119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times1198690 [21198960]119898119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times119898minus1prod119894=2
1198690 [41198960]119894119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind
The AOD 120572119879119901119902 and AOA 120572119877119901119902 are uniformly and indepen-dently distributed so120572119879119901119902minus120572119877119901119902 can be equal to zero in average
As a consequence the term cos[(120572119879119901119902 minus120572119877119901119902)2] in (B3) can beapproximated by one [13 14] and an approximate ST-CF canbe written as from (B3)
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879]sdotcos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877]sdot cos (120572119877 minus 120572119877119901119902) 1198690 (21198960]1119901119902120591)times1198690 (21198960]119898119901119902120591)119898minus1prod
Eventually the approximate ST-CF in (34) can be derivedfrom (B4) according to the definition of 1198690(∙)Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was funded by the National Natural Science Foun-dation of China (no 91438104 no 61501065 no 61571069and no 61601067) the Fundamental Research Funds forthe Central Universities (no 106112016CDJX160001) and theChongqing Research Program of Basic Research and FrontierTechnology (no CSTC2016JCYJA0021)
References
[1] A Bazzi B M Masini A Zanella and I Thibault ldquoBeaconingfrom connected vehicles IEEE 80211p vs LTE-V2Vrdquo in Pro-ceedings of 2016 IEEE 27th Annual International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRC)pp 1ndash6 Valencia Spain September 2016
[2] VM Rodrigo-Penarrocha J Reig L Rubio and et al ldquoAnalysisof small- scale fading distributions in vehicle-to-vehicle com-municationsrdquoMobile Information Systems vol 2016 Article ID9584815 7 pages 2016
[3] A Bazzi B M Masini A Zanella and G Pasolini ldquoIEEE80211p for cellular offloading in vehicular sensor networksrdquoComputer Communications vol 60 pp 97ndash108 2015
[4] X Cheng MIMO channel modelling and simulation for cellu-lar and mobile-to-mobile communication systems[Phd thesis]Heriot-Watt University 2009
[5] N Avazov and M Patzold ldquoA geometric street scatteringchannel model for car-to-car communication systemsrdquo inProceedings of 4th Annual International Conference onAdvanced
14 Mobile Information Systems
Technologies for Communications ATC2011 pp 224ndash230 vnmAugust 2011
[6] X Zhao X Liang S Li and K Haneda ldquoMobile-to-MobileWideband MIMO Channel Realization by Using a Two-RingGeometry-Based Stochastic Scattering Modelrdquo Wireless Per-sonal Communications vol 84 no 4 pp 2445ndash2465 2015
[7] M Walter D Shutin and U-C Fiebig ldquoDelay-dependentdoppler probability density functions for vehicle-to-vehiclescatter channelsrdquo IEEE Transactions on Antennas and Propaga-tion vol 62 no 4 pp 2238ndash2249 2014
[8] P T Samarasinghe T A Lamahewa T D Abhayapala and RA Kennedy ldquo3D mobile-to-mobile wireless channel modelrdquo inProceedings of 2010 Australian Communications Theory Work-shop AusCTW 2010 pp 30ndash34 aus February 2010
[9] M Riaz S J Nawaz and N M Khan ldquo3D ellipsoidal model formobile-to-mobile radio propagation environmentsrdquo WirelessPersonal Communications vol 72 no 4 pp 2465ndash2479 2013
[10] J Chen and T G Pratt ldquoThree-dimensional geometry-based stochastic modeling and performance of 4times4 space-polarization mobile-to-mobile wideband MIMO channelsrdquo inProceedings of 2013 IEEE Global Communications ConferenceGLOBECOM 2013 pp 3936ndash3941 usa December 2013
[11] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA Novel 3DNonstationary Channel Model Based on the von Mises-FisherScattering DistributionrdquoMobile Information Systems vol 2016Article ID 2161460 2016
[12] A Chelli and M Patzold ldquoThe impact of fixed and movingscatterers on the statistics of MIMO vehicle-to-vehicle chan-nelsrdquo in Proceedings of VTC Spring 2009 - IEEE 69th VehicularTechnology Conference esp April 2009
[13] A Borhani and M Patzold ldquoModeling of vehicle-to-vehiclechannels in the presence of moving scatterersrdquo in Proceedingsof 2012 IEEE Vehicular Technology Conference (VTC Fall) pp1ndash5 Quebec City QC Canada September 2012
[14] A Borhani andM Patzold ldquoCorrelation and spectral propertiesof vehicle-to-vehicle channels in the presence ofmoving scatter-ersrdquo IEEE Transactions on Vehicular Technology vol 62 no 9pp 4228ndash4239 2013
[15] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the IEEE 74th Vehicular TechnologyConference (VTC Fall rsquo11) pp 1ndash6 IEEE San Francisco CalifUSA September 2011
[16] M D Soltani M Alimadadi Y Seyedi and H AmindavarldquoModeling of doppler spectrum in V2V urban canyon oncom-ing environmentrdquo in Proceedings of 2014 7th InternationalSymposium on Telecommunications IST 2014 pp 1155ndash1160 irnSeptember 2014
[17] M D Soltani M Alimadadi and A Mohammadi ldquoModelingof mobile scatterer clusters for doppler spectrum in widebandvehicle-to-vehicle communication channelsrdquo IEEE Communi-cations Letters vol 18 no 4 pp 628ndash631 2014
[18] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014
[19] A G Zajic ldquoModeling impact of moving scatterers on dopplerspectrum in wideband vehicle-to-vehicle channelsrdquo in Proceed-ings of Proc Eur Conf Antennas Propag p 1 Lisbon May 2015
[20] J Zhang X Yin and X Cheng ldquoTheoretical analysis andmeasurements Doppler spectra of vehicular communication
channelsrdquo in Proceedings of 2012 12th International Confer-ence on ITS Telecommunications ITST 2012 pp 98ndash102 twnNovember 2012
[21] M Picone S Busanelli M Amoretti F Zanichelli and GFerrari Advanced Technologies for Intelligent TransportationSystems Springer 2015
[22] V H Pham M H Taieb J Y Chouinard S Roy and H THuynh ldquoOn the double Doppler effect generated by scatterermotionrdquo REV Journal on Electronics and Communications vol1 pp 30ndash37 2011
[23] S Roy H T Huynh and P Fortier ldquoCompound Dopplerspread effects of subscriber motion and scatterer motionrdquo AEU- International Journal of Electronics and Communications vol57 no 4 pp 237ndash246 2003
[24] J B Andersen J O Nielsen G F Pedersen G Bauch and GDietl ldquoDoppler spectrum from moving scatterers in a randomenvironmentrdquo IEEE Transactions on Wireless Communicationsvol 8 no 6 pp 3270ndash3277 2009
[25] H S Rad S Gazor and P Shariatpanahi ldquoNon-fixed scatterersand their effects on MIMOmulticarrier fading communicationchannelsrdquo in Proceedings of 50th Annual IEEE Global Telecom-munications Conference GLOBECOM2007 pp 3765ndash3769 usaNovember 2007
[26] N Naz and D Falconer ldquoTemporal variations characterizationfor fixed wireless at 295 GHzrdquo in Proceedings of 2000 IEEE51st Vehicular Technology Conference Proceedings VTC2000-Springer pp 2178ndash2182 Tokyo Japan 2000
[27] A G Zajic G L Stuber T G Pratt and S T NguyenldquoWideband MIMO mobile-to-mobile channels Geometry-based statistical modeling with experimental verificationrdquo IEEETransactions on Vehicular Technology vol 58 no 2 pp 517ndash5342009
[28] W C Jakes Microwave Mobile Communications Wiley-IEEEPress Piscataway NJ USA 1994
[29] M Patzold B O Hogstad and N Youssef ldquoModeling analysisand simulation of MIMO mobile-to-mobile fading channelsrdquoIEEE Transactions onWireless Communications vol 7 no 2 pp510ndash520 2008
[30] M PatzoldMobile Radio Channels Wiley Chichester UK 2ndedition 2011
[31] R Wang and D Cox ldquoDouble mobility mitigates fading in adhoc wireless networksrdquo in Proceedings of IEEE Antennas andPropagation Society International Symposium pp 306ndash309 SanAntonio TX USA
[32] G Bakhshi K Shahtalebi and H S Rad ldquoA novel full-three-dimensional MIMOmobile-to-mobile channel referencemodelrdquo in Proceedings of 3rd International Conference on SignalProcessing and Communication Systems ICSPCSrsquo2009 usaSeptember 2009
[33] W Zhou X Wang X Wang and W Chen ldquoA modified two-erose- ringmodel formimomobile-to-mobile fading channelsrdquoin Proceedings of 7th International Conference on WirelessCommunications Networking and Mobile Computing WiCOM2011 5 1 pages Wuhan China September 2011
where 120575(∙) is the Dirac delta functionSince the ST-CF of the 119898B component is the multiple
integral as (28) the closed-form expression of SD-PSDcannot be derived In Section 42 we show the SD-PSD of the3DV2Vchannel in the presence ofmultiplemoving scatterersby means of numerical integrations
4 Numerical Results and Validation
This section demonstrates the normalized correlation func-tions and validates the Doppler power spectral density (D-PSD) described in Section 3 through theMATLABnumericalsimulations Unless indicated otherwise the values of thenumerical simulation parameters are summarized in Table 2
41 Numerical Results In this section the numerical curvesof ST-CFs T-CFs and S-CFs influenced by some importantcontributory factors are presented As in the typical urbanenvironments the power weight of the119898th cluster of rays hasbeen set to 1199011 = 1199012 = 12 for119872 = 2 1199011 = 1199012 = 1199013 = 13 for119872 = 3 1199011 = 1199012 = 13 1199013 = 1199014 = 16 for119872 = 4 1199011 = 131199012 = 14 1199013 = 1199014 = 16 and 1199015 = 112 for119872 = 5411 ST-CFs and T-CFs for Different Scattering Scenarios andBounces Figures 2ndash5 demonstrate the ST-CFs and T-CFs forthe different maximum bounces 119872 in the 3D isotropic and
0 05 1 15 2 25 3 35 4066
067
068
069
07
071
072
휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 2 ST-CFs in (16) in the isotropic scattering scenario fordifferent maximum bounces
nonisotropic scattering scenarios The scatterer velocity isuniformly distributed with an average speed of 25ms thatmay be the velocity of passing vehicles The other parametersused to obtain curves in Figures 2ndash5 are summarized in(Table 2 Cols 2 and 3) As shown in Figures 2 and 4 thelarger119872 is the faster the ST-CFs and T-CFs decrease in theisotropic scattering scenario However the descent rates ofST-CFs and T-CFs increase slowly when119872 is larger than 3The similar conclusions can be obtained fromFigures 3 and 5however the ST-CF curves in Figure 3 have oscillationswhichmay be caused by the nonisotropic scattering However thisconclusion differs from [27] which showed that the triple-or higher-order bounced rays had statistical properties verysimilar to those of the double-bounced rays and could beapproximated as double-bounced rays The discrepancy maybe caused by the different communication environmentsconsidered by us and [27] Specifically channel-soundingexperimental campaign in [27] was conducted along surface
Mobile Information Systems 9
066
067
068
069
07
071
072
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
1122(d
T=
05휆d
R=
05휆휏
)||R
Figure 3 ST-CFs in (16) in the nonisotropic scattering scenario fordifferent maximum bounces
065
07
075
08
085
09
095
1
M = 1M = 2M = 3
M = 4M = 5
0 05 1 15 2 25 3 35 4휏 (ms)
|R11
11(휏)|
Figure 4 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropicscattering scenario for different maximum bounces
streets around the Georgia Tech campus and on the Interstatehighways in the Midtown Atlanta metropolitan area whichmay have lower dense scatterers than our communicationenvironments The ST-CF curves in Figure 3 converge rel-atively slower for the larger 119872 that means there are morelocal scatterers inmotionwith randomvelocities and randomdirections As can be observed in Figures 4 and 5 theT-CFs inboth figures are very similar that indicates that the scatteringforms have no significant effect on the T-CFs of the 3D V2Vchannel in the presence of multiple moving scatterers Fur-thermore in Figures 2ndash5 it can be observed that a tendencyto a limit value curve of ST-CFs and T-CFs can be inferred if
065
07
075
08
085
09
095
1
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
|R11
11(휏)|
Figure 5 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the nonisotropicscattering scenario for different maximum bounces
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
066
067
068
069
071
072
1122(d
T=
05휆d
R=
05휆휏
)||R
07
Figure 6 ST-CFs in (16) in the isotropic and nonisotropic scatteringscenarios with119872 = 5 for different scatterer velocity distributions
119872 approaches infinity and the similar inference also can beobtained in (35) for the 2D scattering environment
412 ST-CFs and T-CFs for Different Scattering Scenarios andScatterer Velocity Distributions Figures 6 and 7 demonstratethe ST-CFs and T-CFs for the different scatterer velocitydistributions in the 3D isotropic and nonisotropic scatteringscenarios We use two uniform distributions in the isotropicscattering environment and use the von Mises distributionand the cosine distribution in the nonisotropic scatteringenvironment to describe all the statistical azimuth andelevation angles in Table 1 respectively The average speedof the uniform and half-Gaussian distributions is 25ms
10 Mobile Information Systems
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
065
07
075
08
085
09
095
1
|R11
11(휏)|
Figure 7 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropic andnonisotropic scattering scenarios with119872 = 5 for different scatterervelocity distributions
and the standard deviation of the half-Gaussian distribu-tion is 31ms The other parameters used to obtain curvesin Figures 6 and 7 are summarized in (Table 2 Cols 2and 3) As shown in Figure 6 the ST-CF for the uniformdistributed velocity of scatterers in the isotropic scatteringenvironment decreases fastest and both the nonisotropicscattering and the half-Gaussian distribution can result inswift oscillations of the fading curvesThe comparison resultsindicate that the scatterer velocity distributions have moresignificant effect than the scattering forms on the ST-CFsof the 3D V2V channel in the presence of multiple movingscatterers Figure 7 indicates that not like scattering formsthe scatterer velocity distributions have significant effect onthe T-CFs Generally the realistic propagation environmentis nonisotropic scattering and the scatterer velocity canbe modeled by the half-Gaussian or uniform distributionTherefore the results of Figures 6 and 7 may be helpful inanalysis of the system performance and modeling a specificpropagation channel for V2V communications
413 ST-CFs and S-CFs for Different Scattering Scenariosand Spacing between Adjacent Antenna Elements Figures8 and 9 demonstrate the 3D ST-CFs and S-CFs for thedifferent spacing between adjacent antenna elements of 119879119883and 119877119883 in isotropic and nonisotropic scattering scenariosThe set of the scatterer velocity in Figure 8 is the sameas Figures 2ndash5 and the other parameters used to obtaincurves in Figures 8 and 9 are summarized in (Table 2 Col4) Figures 8 and 9 show that spacing between adjacentantenna elements has significant effect on the 3D ST-CFsand S-CFs Furthermore the 3D ST-CFs in isotropic andnonisotropic scattering scenarios decrease to the bottomwhen 119889119877 = 119889119879 tends to 12120582 and 07120582 respectively Inaddition the 3D S-CF in nonisotropic scattering scenarios isexplicitly less than that in isotropic scattering scenarios for
|R11
22(d
Td
R휏)|
dR = d
T (휆) 휏 (ms)
0025
05075 1
12515 0
12
34
1
095
09
085
08
075
07
065
IsotropicNonisotropic
Figure 8 3D ST-CFs in (16) in the isotropic and nonisotropicscattering scenarios with119872 = 5 120582 is the carrier wavelength
|R11
22(d
Td
R)|
dR (휆)
dT (휆)
1
095
09
085
08
075
07
065
0 025 05 0751 125 15
0025
05075 1
12515
IsotropicNonisotropic
Figure 9 3D S-CFs in (16) with 120591 = 0 in the isotropic andnonisotropic scattering scenarios 120582 is the carrier wavelength
most spacing between adjacent antenna elements as shown inFigure 9Therefore the large-scale antenna arrays employingthe spatial multiplexing are more suitable to be used in thenonisotropic scattering scenarios owing to the high antennaspace utilization The spacing between adjacent antennaelements of antenna arrays is suggested to be more than 120582in the V2V communications especially in the environmentswith low-density scatterers
42 Comparison with Measurements The parametric natureof the proposed channel model makes it adaptable to avariety of propagation environments D-PSD is one of themost important and unique channel characteristics for V2Vcommunication channels To illustrate the validity of theproposed model we compare the modeled SD-PSDs of the3D V2V channel with multiple moving scatterers with themeasured F2F D-PSD in [26] the theoretical V2V SD-PSDin [18] and the measured V2V SD-PSD in [27]
Mobile Information Systems 11
minus2 minus15 minus1 minus05 0 05 1 15 2Doppler frequency f (kHz)
Figure 10 Comparison between the modeled SD-PSDs with119872 = 5and the measured D-PSDs of Figure 4 in [26]
Figure 10 shows the comparison between the modeledSD-PSDs obtained from the Fourier transform of the ST-CF presented in (16) in the isotropic scattering scenario andthe measured D-PSDs of Figure 4 in [26] The outdoor mea-surement experiments in [26] have shown that at millimeterwavelengths fading caused by foliage movement and themotion of nearby vehicles is a significantly deleterious effectin systems with static subscribers Figure 4 in [26] showsthe D-PSD generated by foliage movements and the D-PSDcaused by passing vehicles both in a fixed wireless channelat a carrier frequency of 295GHz We have used the half-Gaussian distribution with an average speed of 04ms andthe standard deviation of 05ms to model the velocity ofmoving foliage scatterers while the uniformdistributionwithan average speed of 6ms has been used tomodel the velocityof passing vehicles scatterers The power weight of the 119898thcluster of rays scattered by the moving foliage or passingvehicles has been set to 1199011 = 916 1199012 = 516 1199013 = 116and 1199014 = 1199015 = 132 The other parameters used to obtainthe modeled SD-PSD curves in Figure 10 are summarizedin (Table 2 Col 5) As can be observed in Figure 10 40 dBfading bandwidths for the effect of the foliage and vehiclesmovements are 500Hz and 1 kHz approximately respectivelyFade depth due to foliage is less than the variation due topassing vehicles Close agreements between the modeled andmeasured results are shown in Figure 10These close matchescan be seen not only for relatively slow moving scatterers butalso for relatively fast moving scatterers as well
Figure 11 shows the comparison among the modeled SD-PSD obtained from the Fourier transform of the ST-CFpresented in (16) in the nonisotropic scattering scenario thetheoretical SD-PSD of Figure 8 in [18] and themeasured SD-PSD of Figure 11 in [27]The theoretical SD-PSD reproducedhere is based on the two-cylinder model with moving andstationary scatterers for 119889119877 = 119889119879 = 0 The channel mea-surements in [27] for the measured SD-PSD were collectedat 2435GHz in the urban street surface environment and thespacing between two adjacent antenna elements of119879119883 and119877119883was set to zero We have used the half-Gaussian distributionwith average speeds of 2394ms 0798ms and 04ms
Figure 11 Comparison among the modeled SD-PSD with119872 = 5the theoretical SD-PSD of Figure 8 in [18] and the measured SD-PSD of Figure 11 in [27]
and the standard deviation of 3ms 1ms and 05ms tomodel the velocity of the first the second and the remainingmoving scatterers on the communication links respectivelyThe power weight of the 119898th cluster of rays has been set to1199011 = 815 1199012 = 13 1199013 = 115 1199014 = 130 and 1199015 = 130for119872 = 5 The other parameters used to obtain the modeledSD-PSD curve in Figure 11 are summarized in (Table 2 Col6) We note that different from the measured SD-PSDs in[26] themeasured SD-PSD curve in [27] has oscillationsTheoscillating pattern in measurements for the case 119889119877 = 119889119879 = 0may appear because it is not a true SISO scenario whereonly one transmit and one receive antenna are active [27] Ascan be observed in Figure 11 the modeled SD-PSD matchesbetter withmeasured SD-PSD than the theoretical SD-PSD in[18] This may be because of the introduction of the multiplebounced rays into the propagation model or the constraintson the position of local scatterers in the two-cylinder modelproposed in [18]
Figure 12 demonstrates the CDFs of the relative devi-ations between modeledtheoretical SD-PSDs and corre-sponding measured values in Figures 10 and 11 The meanrelative deviation of the modeled SD-PSDs for foliage inFigure 10 modeled SD-PSDs for vehicles in Figure 10 mod-eled SD-PSDs in Figure 11 and theoretical SD-PSDs inFigure 11 is 256 216 39 and 832 respectively Asshown in Figure 12 more than 80 of the relative deviationsof modeled SD-PSDs in Figures 10 and 11 are less than5 and the relative deviations of theoretical SD-PSDs inFigure 11 are explicitly larger than those of modeled SD-PSDs In addition from Figures 10 and 11 we note that themodeled SD-PSD matches better with measured SD-PSDs atthe lower and higher Doppler frequencies than the middleDoppler frequencies This may be because 119879119883 and 119877119883 werenot strictly fixed and not all the scatterers were in motion inmeasurementsThe approximate calculation of the frequencyshift in (A4) and Doppler shift in (10) may be anotherfactor which causes the mismatch at the middle Dopplerfrequencies andmore investigations concerning the accuracyof the proposed model will be addressed in future works
12 Mobile Information Systems
0 5 10 15 20 25 30 350
01
02
03
04
05
06
07
08
09
1
Relative deviation ()
Relat
ive d
evia
tion
CDF
Modeled SD-PSD (foliage)Modeled SD-PSD (vehicles)
Modeled SD-PSDTheoretical SD-PSD
Figure 12 The relative deviation cumulative distribution functions(CDFs) of the modeled and theoretical SD-PSDs in Figures 10 and11
However as shown in Figures 10ndash12 the acceptable matchesconfirm the utility and generality of the proposed model andshow the need for including multiple moving scatterers inpropagation model
5 Conclusion
Without specific constraints on the position of the local mov-ing scatterers a 3D geometrical V2V propagation model thatincludes LoS single bounced and multiple bounced linksbetween the transmitter and receiver was proposed Basedon the geometrical propagation model a 3D reference modelfor narrowband MIMO V2V multipath fading channels wasdeveloped From the reference model the correspondingmathematical expressions and numerical results of ST-CFsT-CFs S-CFs and SD-PSDs were studied for different para-metric sets It has been shown that the maximum bouncesscattering forms scatterer velocity distributions and spacingbetween adjacent antenna elements have significant effect onthe ST-CFs and only the maximum bounces and scatterervelocity distributions have significant effect on the T-CFsFinally the modeled SD-PSD results were compared withthe measured data and the other literaturersquos theoreticalresult The close agreements between the analytically andempirically obtained channel statistics confirmed the utilityand generality of the proposed model
Appendix
A Derivation Process of Frequency Shifts
The derivation process of 119891119902119901119902119899 in (7) is presented in thissection and 119891119901119901119902120588 119891119902119901119902119899 119891119901119901119902119899 and 119891119901119902120588 can be calculatedsimilarly
AOD
Δl
TX qth element
Figure 13 Projection from the coordinate vector of 119902th antennaelement to the AOD direction
According to Figure 1 the coordinate vectors of 119902thantenna element and the AOD direction can be written asrespectively
minus 1) 119889119879 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902 + sin120573119902 sin120573119879119901119902] (A3)
Because we mainly consider the difference of the fre-quency shifts caused by the TPD changes to calculate ST-CFsthe TPD change of ℎ11(119905) is assumed to be zero Eventually119891119902119901119902119899 can be calculated as
119891119902119901119902119899 asymp Δ1198971198910119888= (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(A4)
B Derivation Process of Approximate ST-CFs
In the 2D scattering environment the simplified and approx-imate expressions of (28) can be derived in this section If120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 (27) and (10) can be writtenas
Δ119891119899119898
= 1198910119888 [(119902 minus 119902) 119889119879 cos (120572119902 minus 120572119879119901119902)+ (119901 minus 119901) 119889119877 cos (120572119901 minus 120572119877119901119902)]
Mobile Information Systems 13
119891119901119902119899119898
= 1198910119888 [V119879 cos (120572119879 minus 120572119879119901119902) + V119877 cos (120572119877 minus 120572119877119901119902)
+ ]1119901119902 cos (1205721119901119902 minus 120572119879119901119902) + 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119879119901119902)+ ]119898119901119902 cos (120572119898119901119902 minus 120572119879119901119902) + ]1119901119902 cos (1205721119901119902 minus 120572119877119901119902)+ 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119877119901119902) + ]119898119901119902 cos (120572119898119901119902 minus 120572119877119901119902)] (B1)
If 120572119879 = 120572119902 and 120572119877 = 120572119901 that is the antenna array and itsvelocity have the same directions (28) can be simplified as bysubstituting (B1) into (28)
where 1198960 = 21205871198910119888 is the wave numberIn isotropic scattering environment we use the uniform
distribution in (31) to describe all the random azimuth anglesIn this case (B2) can be written as119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879] cos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877] cos (120572119877 minus 120572119877119901119902)
times1198690 [21198960]1119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times1198690 [21198960]119898119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times119898minus1prod119894=2
1198690 [41198960]119894119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind
The AOD 120572119879119901119902 and AOA 120572119877119901119902 are uniformly and indepen-dently distributed so120572119879119901119902minus120572119877119901119902 can be equal to zero in average
As a consequence the term cos[(120572119879119901119902 minus120572119877119901119902)2] in (B3) can beapproximated by one [13 14] and an approximate ST-CF canbe written as from (B3)
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879]sdotcos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877]sdot cos (120572119877 minus 120572119877119901119902) 1198690 (21198960]1119901119902120591)times1198690 (21198960]119898119901119902120591)119898minus1prod
Eventually the approximate ST-CF in (34) can be derivedfrom (B4) according to the definition of 1198690(∙)Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was funded by the National Natural Science Foun-dation of China (no 91438104 no 61501065 no 61571069and no 61601067) the Fundamental Research Funds forthe Central Universities (no 106112016CDJX160001) and theChongqing Research Program of Basic Research and FrontierTechnology (no CSTC2016JCYJA0021)
References
[1] A Bazzi B M Masini A Zanella and I Thibault ldquoBeaconingfrom connected vehicles IEEE 80211p vs LTE-V2Vrdquo in Pro-ceedings of 2016 IEEE 27th Annual International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRC)pp 1ndash6 Valencia Spain September 2016
[2] VM Rodrigo-Penarrocha J Reig L Rubio and et al ldquoAnalysisof small- scale fading distributions in vehicle-to-vehicle com-municationsrdquoMobile Information Systems vol 2016 Article ID9584815 7 pages 2016
[3] A Bazzi B M Masini A Zanella and G Pasolini ldquoIEEE80211p for cellular offloading in vehicular sensor networksrdquoComputer Communications vol 60 pp 97ndash108 2015
[4] X Cheng MIMO channel modelling and simulation for cellu-lar and mobile-to-mobile communication systems[Phd thesis]Heriot-Watt University 2009
[5] N Avazov and M Patzold ldquoA geometric street scatteringchannel model for car-to-car communication systemsrdquo inProceedings of 4th Annual International Conference onAdvanced
14 Mobile Information Systems
Technologies for Communications ATC2011 pp 224ndash230 vnmAugust 2011
[6] X Zhao X Liang S Li and K Haneda ldquoMobile-to-MobileWideband MIMO Channel Realization by Using a Two-RingGeometry-Based Stochastic Scattering Modelrdquo Wireless Per-sonal Communications vol 84 no 4 pp 2445ndash2465 2015
[7] M Walter D Shutin and U-C Fiebig ldquoDelay-dependentdoppler probability density functions for vehicle-to-vehiclescatter channelsrdquo IEEE Transactions on Antennas and Propaga-tion vol 62 no 4 pp 2238ndash2249 2014
[8] P T Samarasinghe T A Lamahewa T D Abhayapala and RA Kennedy ldquo3D mobile-to-mobile wireless channel modelrdquo inProceedings of 2010 Australian Communications Theory Work-shop AusCTW 2010 pp 30ndash34 aus February 2010
[9] M Riaz S J Nawaz and N M Khan ldquo3D ellipsoidal model formobile-to-mobile radio propagation environmentsrdquo WirelessPersonal Communications vol 72 no 4 pp 2465ndash2479 2013
[10] J Chen and T G Pratt ldquoThree-dimensional geometry-based stochastic modeling and performance of 4times4 space-polarization mobile-to-mobile wideband MIMO channelsrdquo inProceedings of 2013 IEEE Global Communications ConferenceGLOBECOM 2013 pp 3936ndash3941 usa December 2013
[11] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA Novel 3DNonstationary Channel Model Based on the von Mises-FisherScattering DistributionrdquoMobile Information Systems vol 2016Article ID 2161460 2016
[12] A Chelli and M Patzold ldquoThe impact of fixed and movingscatterers on the statistics of MIMO vehicle-to-vehicle chan-nelsrdquo in Proceedings of VTC Spring 2009 - IEEE 69th VehicularTechnology Conference esp April 2009
[13] A Borhani and M Patzold ldquoModeling of vehicle-to-vehiclechannels in the presence of moving scatterersrdquo in Proceedingsof 2012 IEEE Vehicular Technology Conference (VTC Fall) pp1ndash5 Quebec City QC Canada September 2012
[14] A Borhani andM Patzold ldquoCorrelation and spectral propertiesof vehicle-to-vehicle channels in the presence ofmoving scatter-ersrdquo IEEE Transactions on Vehicular Technology vol 62 no 9pp 4228ndash4239 2013
[15] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the IEEE 74th Vehicular TechnologyConference (VTC Fall rsquo11) pp 1ndash6 IEEE San Francisco CalifUSA September 2011
[16] M D Soltani M Alimadadi Y Seyedi and H AmindavarldquoModeling of doppler spectrum in V2V urban canyon oncom-ing environmentrdquo in Proceedings of 2014 7th InternationalSymposium on Telecommunications IST 2014 pp 1155ndash1160 irnSeptember 2014
[17] M D Soltani M Alimadadi and A Mohammadi ldquoModelingof mobile scatterer clusters for doppler spectrum in widebandvehicle-to-vehicle communication channelsrdquo IEEE Communi-cations Letters vol 18 no 4 pp 628ndash631 2014
[18] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014
[19] A G Zajic ldquoModeling impact of moving scatterers on dopplerspectrum in wideband vehicle-to-vehicle channelsrdquo in Proceed-ings of Proc Eur Conf Antennas Propag p 1 Lisbon May 2015
[20] J Zhang X Yin and X Cheng ldquoTheoretical analysis andmeasurements Doppler spectra of vehicular communication
channelsrdquo in Proceedings of 2012 12th International Confer-ence on ITS Telecommunications ITST 2012 pp 98ndash102 twnNovember 2012
[21] M Picone S Busanelli M Amoretti F Zanichelli and GFerrari Advanced Technologies for Intelligent TransportationSystems Springer 2015
[22] V H Pham M H Taieb J Y Chouinard S Roy and H THuynh ldquoOn the double Doppler effect generated by scatterermotionrdquo REV Journal on Electronics and Communications vol1 pp 30ndash37 2011
[23] S Roy H T Huynh and P Fortier ldquoCompound Dopplerspread effects of subscriber motion and scatterer motionrdquo AEU- International Journal of Electronics and Communications vol57 no 4 pp 237ndash246 2003
[24] J B Andersen J O Nielsen G F Pedersen G Bauch and GDietl ldquoDoppler spectrum from moving scatterers in a randomenvironmentrdquo IEEE Transactions on Wireless Communicationsvol 8 no 6 pp 3270ndash3277 2009
[25] H S Rad S Gazor and P Shariatpanahi ldquoNon-fixed scatterersand their effects on MIMOmulticarrier fading communicationchannelsrdquo in Proceedings of 50th Annual IEEE Global Telecom-munications Conference GLOBECOM2007 pp 3765ndash3769 usaNovember 2007
[26] N Naz and D Falconer ldquoTemporal variations characterizationfor fixed wireless at 295 GHzrdquo in Proceedings of 2000 IEEE51st Vehicular Technology Conference Proceedings VTC2000-Springer pp 2178ndash2182 Tokyo Japan 2000
[27] A G Zajic G L Stuber T G Pratt and S T NguyenldquoWideband MIMO mobile-to-mobile channels Geometry-based statistical modeling with experimental verificationrdquo IEEETransactions on Vehicular Technology vol 58 no 2 pp 517ndash5342009
[28] W C Jakes Microwave Mobile Communications Wiley-IEEEPress Piscataway NJ USA 1994
[29] M Patzold B O Hogstad and N Youssef ldquoModeling analysisand simulation of MIMO mobile-to-mobile fading channelsrdquoIEEE Transactions onWireless Communications vol 7 no 2 pp510ndash520 2008
[30] M PatzoldMobile Radio Channels Wiley Chichester UK 2ndedition 2011
[31] R Wang and D Cox ldquoDouble mobility mitigates fading in adhoc wireless networksrdquo in Proceedings of IEEE Antennas andPropagation Society International Symposium pp 306ndash309 SanAntonio TX USA
[32] G Bakhshi K Shahtalebi and H S Rad ldquoA novel full-three-dimensional MIMOmobile-to-mobile channel referencemodelrdquo in Proceedings of 3rd International Conference on SignalProcessing and Communication Systems ICSPCSrsquo2009 usaSeptember 2009
[33] W Zhou X Wang X Wang and W Chen ldquoA modified two-erose- ringmodel formimomobile-to-mobile fading channelsrdquoin Proceedings of 7th International Conference on WirelessCommunications Networking and Mobile Computing WiCOM2011 5 1 pages Wuhan China September 2011
Figure 3 ST-CFs in (16) in the nonisotropic scattering scenario fordifferent maximum bounces
065
07
075
08
085
09
095
1
M = 1M = 2M = 3
M = 4M = 5
0 05 1 15 2 25 3 35 4휏 (ms)
|R11
11(휏)|
Figure 4 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropicscattering scenario for different maximum bounces
streets around the Georgia Tech campus and on the Interstatehighways in the Midtown Atlanta metropolitan area whichmay have lower dense scatterers than our communicationenvironments The ST-CF curves in Figure 3 converge rel-atively slower for the larger 119872 that means there are morelocal scatterers inmotionwith randomvelocities and randomdirections As can be observed in Figures 4 and 5 theT-CFs inboth figures are very similar that indicates that the scatteringforms have no significant effect on the T-CFs of the 3D V2Vchannel in the presence of multiple moving scatterers Fur-thermore in Figures 2ndash5 it can be observed that a tendencyto a limit value curve of ST-CFs and T-CFs can be inferred if
065
07
075
08
085
09
095
1
0 05 1 15 2 25 3 35 4휏 (ms)
M = 1M = 2M = 3
M = 4M = 5
|R11
11(휏)|
Figure 5 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the nonisotropicscattering scenario for different maximum bounces
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
066
067
068
069
071
072
1122(d
T=
05휆d
R=
05휆휏
)||R
07
Figure 6 ST-CFs in (16) in the isotropic and nonisotropic scatteringscenarios with119872 = 5 for different scatterer velocity distributions
119872 approaches infinity and the similar inference also can beobtained in (35) for the 2D scattering environment
412 ST-CFs and T-CFs for Different Scattering Scenarios andScatterer Velocity Distributions Figures 6 and 7 demonstratethe ST-CFs and T-CFs for the different scatterer velocitydistributions in the 3D isotropic and nonisotropic scatteringscenarios We use two uniform distributions in the isotropicscattering environment and use the von Mises distributionand the cosine distribution in the nonisotropic scatteringenvironment to describe all the statistical azimuth andelevation angles in Table 1 respectively The average speedof the uniform and half-Gaussian distributions is 25ms
10 Mobile Information Systems
Isotropic uniformIsotropic Gaussian
Nonisotropic uniformNonisotropic Gaussian
0 05 1 15 2 25 3 35 4휏 (ms)
065
07
075
08
085
09
095
1
|R11
11(휏)|
Figure 7 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropic andnonisotropic scattering scenarios with119872 = 5 for different scatterervelocity distributions
and the standard deviation of the half-Gaussian distribu-tion is 31ms The other parameters used to obtain curvesin Figures 6 and 7 are summarized in (Table 2 Cols 2and 3) As shown in Figure 6 the ST-CF for the uniformdistributed velocity of scatterers in the isotropic scatteringenvironment decreases fastest and both the nonisotropicscattering and the half-Gaussian distribution can result inswift oscillations of the fading curvesThe comparison resultsindicate that the scatterer velocity distributions have moresignificant effect than the scattering forms on the ST-CFsof the 3D V2V channel in the presence of multiple movingscatterers Figure 7 indicates that not like scattering formsthe scatterer velocity distributions have significant effect onthe T-CFs Generally the realistic propagation environmentis nonisotropic scattering and the scatterer velocity canbe modeled by the half-Gaussian or uniform distributionTherefore the results of Figures 6 and 7 may be helpful inanalysis of the system performance and modeling a specificpropagation channel for V2V communications
413 ST-CFs and S-CFs for Different Scattering Scenariosand Spacing between Adjacent Antenna Elements Figures8 and 9 demonstrate the 3D ST-CFs and S-CFs for thedifferent spacing between adjacent antenna elements of 119879119883and 119877119883 in isotropic and nonisotropic scattering scenariosThe set of the scatterer velocity in Figure 8 is the sameas Figures 2ndash5 and the other parameters used to obtaincurves in Figures 8 and 9 are summarized in (Table 2 Col4) Figures 8 and 9 show that spacing between adjacentantenna elements has significant effect on the 3D ST-CFsand S-CFs Furthermore the 3D ST-CFs in isotropic andnonisotropic scattering scenarios decrease to the bottomwhen 119889119877 = 119889119879 tends to 12120582 and 07120582 respectively Inaddition the 3D S-CF in nonisotropic scattering scenarios isexplicitly less than that in isotropic scattering scenarios for
|R11
22(d
Td
R휏)|
dR = d
T (휆) 휏 (ms)
0025
05075 1
12515 0
12
34
1
095
09
085
08
075
07
065
IsotropicNonisotropic
Figure 8 3D ST-CFs in (16) in the isotropic and nonisotropicscattering scenarios with119872 = 5 120582 is the carrier wavelength
|R11
22(d
Td
R)|
dR (휆)
dT (휆)
1
095
09
085
08
075
07
065
0 025 05 0751 125 15
0025
05075 1
12515
IsotropicNonisotropic
Figure 9 3D S-CFs in (16) with 120591 = 0 in the isotropic andnonisotropic scattering scenarios 120582 is the carrier wavelength
most spacing between adjacent antenna elements as shown inFigure 9Therefore the large-scale antenna arrays employingthe spatial multiplexing are more suitable to be used in thenonisotropic scattering scenarios owing to the high antennaspace utilization The spacing between adjacent antennaelements of antenna arrays is suggested to be more than 120582in the V2V communications especially in the environmentswith low-density scatterers
42 Comparison with Measurements The parametric natureof the proposed channel model makes it adaptable to avariety of propagation environments D-PSD is one of themost important and unique channel characteristics for V2Vcommunication channels To illustrate the validity of theproposed model we compare the modeled SD-PSDs of the3D V2V channel with multiple moving scatterers with themeasured F2F D-PSD in [26] the theoretical V2V SD-PSDin [18] and the measured V2V SD-PSD in [27]
Mobile Information Systems 11
minus2 minus15 minus1 minus05 0 05 1 15 2Doppler frequency f (kHz)
Figure 10 Comparison between the modeled SD-PSDs with119872 = 5and the measured D-PSDs of Figure 4 in [26]
Figure 10 shows the comparison between the modeledSD-PSDs obtained from the Fourier transform of the ST-CF presented in (16) in the isotropic scattering scenario andthe measured D-PSDs of Figure 4 in [26] The outdoor mea-surement experiments in [26] have shown that at millimeterwavelengths fading caused by foliage movement and themotion of nearby vehicles is a significantly deleterious effectin systems with static subscribers Figure 4 in [26] showsthe D-PSD generated by foliage movements and the D-PSDcaused by passing vehicles both in a fixed wireless channelat a carrier frequency of 295GHz We have used the half-Gaussian distribution with an average speed of 04ms andthe standard deviation of 05ms to model the velocity ofmoving foliage scatterers while the uniformdistributionwithan average speed of 6ms has been used tomodel the velocityof passing vehicles scatterers The power weight of the 119898thcluster of rays scattered by the moving foliage or passingvehicles has been set to 1199011 = 916 1199012 = 516 1199013 = 116and 1199014 = 1199015 = 132 The other parameters used to obtainthe modeled SD-PSD curves in Figure 10 are summarizedin (Table 2 Col 5) As can be observed in Figure 10 40 dBfading bandwidths for the effect of the foliage and vehiclesmovements are 500Hz and 1 kHz approximately respectivelyFade depth due to foliage is less than the variation due topassing vehicles Close agreements between the modeled andmeasured results are shown in Figure 10These close matchescan be seen not only for relatively slow moving scatterers butalso for relatively fast moving scatterers as well
Figure 11 shows the comparison among the modeled SD-PSD obtained from the Fourier transform of the ST-CFpresented in (16) in the nonisotropic scattering scenario thetheoretical SD-PSD of Figure 8 in [18] and themeasured SD-PSD of Figure 11 in [27]The theoretical SD-PSD reproducedhere is based on the two-cylinder model with moving andstationary scatterers for 119889119877 = 119889119879 = 0 The channel mea-surements in [27] for the measured SD-PSD were collectedat 2435GHz in the urban street surface environment and thespacing between two adjacent antenna elements of119879119883 and119877119883was set to zero We have used the half-Gaussian distributionwith average speeds of 2394ms 0798ms and 04ms
Figure 11 Comparison among the modeled SD-PSD with119872 = 5the theoretical SD-PSD of Figure 8 in [18] and the measured SD-PSD of Figure 11 in [27]
and the standard deviation of 3ms 1ms and 05ms tomodel the velocity of the first the second and the remainingmoving scatterers on the communication links respectivelyThe power weight of the 119898th cluster of rays has been set to1199011 = 815 1199012 = 13 1199013 = 115 1199014 = 130 and 1199015 = 130for119872 = 5 The other parameters used to obtain the modeledSD-PSD curve in Figure 11 are summarized in (Table 2 Col6) We note that different from the measured SD-PSDs in[26] themeasured SD-PSD curve in [27] has oscillationsTheoscillating pattern in measurements for the case 119889119877 = 119889119879 = 0may appear because it is not a true SISO scenario whereonly one transmit and one receive antenna are active [27] Ascan be observed in Figure 11 the modeled SD-PSD matchesbetter withmeasured SD-PSD than the theoretical SD-PSD in[18] This may be because of the introduction of the multiplebounced rays into the propagation model or the constraintson the position of local scatterers in the two-cylinder modelproposed in [18]
Figure 12 demonstrates the CDFs of the relative devi-ations between modeledtheoretical SD-PSDs and corre-sponding measured values in Figures 10 and 11 The meanrelative deviation of the modeled SD-PSDs for foliage inFigure 10 modeled SD-PSDs for vehicles in Figure 10 mod-eled SD-PSDs in Figure 11 and theoretical SD-PSDs inFigure 11 is 256 216 39 and 832 respectively Asshown in Figure 12 more than 80 of the relative deviationsof modeled SD-PSDs in Figures 10 and 11 are less than5 and the relative deviations of theoretical SD-PSDs inFigure 11 are explicitly larger than those of modeled SD-PSDs In addition from Figures 10 and 11 we note that themodeled SD-PSD matches better with measured SD-PSDs atthe lower and higher Doppler frequencies than the middleDoppler frequencies This may be because 119879119883 and 119877119883 werenot strictly fixed and not all the scatterers were in motion inmeasurementsThe approximate calculation of the frequencyshift in (A4) and Doppler shift in (10) may be anotherfactor which causes the mismatch at the middle Dopplerfrequencies andmore investigations concerning the accuracyof the proposed model will be addressed in future works
12 Mobile Information Systems
0 5 10 15 20 25 30 350
01
02
03
04
05
06
07
08
09
1
Relative deviation ()
Relat
ive d
evia
tion
CDF
Modeled SD-PSD (foliage)Modeled SD-PSD (vehicles)
Modeled SD-PSDTheoretical SD-PSD
Figure 12 The relative deviation cumulative distribution functions(CDFs) of the modeled and theoretical SD-PSDs in Figures 10 and11
However as shown in Figures 10ndash12 the acceptable matchesconfirm the utility and generality of the proposed model andshow the need for including multiple moving scatterers inpropagation model
5 Conclusion
Without specific constraints on the position of the local mov-ing scatterers a 3D geometrical V2V propagation model thatincludes LoS single bounced and multiple bounced linksbetween the transmitter and receiver was proposed Basedon the geometrical propagation model a 3D reference modelfor narrowband MIMO V2V multipath fading channels wasdeveloped From the reference model the correspondingmathematical expressions and numerical results of ST-CFsT-CFs S-CFs and SD-PSDs were studied for different para-metric sets It has been shown that the maximum bouncesscattering forms scatterer velocity distributions and spacingbetween adjacent antenna elements have significant effect onthe ST-CFs and only the maximum bounces and scatterervelocity distributions have significant effect on the T-CFsFinally the modeled SD-PSD results were compared withthe measured data and the other literaturersquos theoreticalresult The close agreements between the analytically andempirically obtained channel statistics confirmed the utilityand generality of the proposed model
Appendix
A Derivation Process of Frequency Shifts
The derivation process of 119891119902119901119902119899 in (7) is presented in thissection and 119891119901119901119902120588 119891119902119901119902119899 119891119901119901119902119899 and 119891119901119902120588 can be calculatedsimilarly
AOD
Δl
TX qth element
Figure 13 Projection from the coordinate vector of 119902th antennaelement to the AOD direction
According to Figure 1 the coordinate vectors of 119902thantenna element and the AOD direction can be written asrespectively
minus 1) 119889119879 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902 + sin120573119902 sin120573119879119901119902] (A3)
Because we mainly consider the difference of the fre-quency shifts caused by the TPD changes to calculate ST-CFsthe TPD change of ℎ11(119905) is assumed to be zero Eventually119891119902119901119902119899 can be calculated as
119891119902119901119902119899 asymp Δ1198971198910119888= (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(A4)
B Derivation Process of Approximate ST-CFs
In the 2D scattering environment the simplified and approx-imate expressions of (28) can be derived in this section If120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 (27) and (10) can be writtenas
Δ119891119899119898
= 1198910119888 [(119902 minus 119902) 119889119879 cos (120572119902 minus 120572119879119901119902)+ (119901 minus 119901) 119889119877 cos (120572119901 minus 120572119877119901119902)]
Mobile Information Systems 13
119891119901119902119899119898
= 1198910119888 [V119879 cos (120572119879 minus 120572119879119901119902) + V119877 cos (120572119877 minus 120572119877119901119902)
+ ]1119901119902 cos (1205721119901119902 minus 120572119879119901119902) + 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119879119901119902)+ ]119898119901119902 cos (120572119898119901119902 minus 120572119879119901119902) + ]1119901119902 cos (1205721119901119902 minus 120572119877119901119902)+ 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119877119901119902) + ]119898119901119902 cos (120572119898119901119902 minus 120572119877119901119902)] (B1)
If 120572119879 = 120572119902 and 120572119877 = 120572119901 that is the antenna array and itsvelocity have the same directions (28) can be simplified as bysubstituting (B1) into (28)
where 1198960 = 21205871198910119888 is the wave numberIn isotropic scattering environment we use the uniform
distribution in (31) to describe all the random azimuth anglesIn this case (B2) can be written as119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879] cos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877] cos (120572119877 minus 120572119877119901119902)
times1198690 [21198960]1119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times1198690 [21198960]119898119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times119898minus1prod119894=2
1198690 [41198960]119894119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind
The AOD 120572119879119901119902 and AOA 120572119877119901119902 are uniformly and indepen-dently distributed so120572119879119901119902minus120572119877119901119902 can be equal to zero in average
As a consequence the term cos[(120572119879119901119902 minus120572119877119901119902)2] in (B3) can beapproximated by one [13 14] and an approximate ST-CF canbe written as from (B3)
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879]sdotcos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877]sdot cos (120572119877 minus 120572119877119901119902) 1198690 (21198960]1119901119902120591)times1198690 (21198960]119898119901119902120591)119898minus1prod
Eventually the approximate ST-CF in (34) can be derivedfrom (B4) according to the definition of 1198690(∙)Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was funded by the National Natural Science Foun-dation of China (no 91438104 no 61501065 no 61571069and no 61601067) the Fundamental Research Funds forthe Central Universities (no 106112016CDJX160001) and theChongqing Research Program of Basic Research and FrontierTechnology (no CSTC2016JCYJA0021)
References
[1] A Bazzi B M Masini A Zanella and I Thibault ldquoBeaconingfrom connected vehicles IEEE 80211p vs LTE-V2Vrdquo in Pro-ceedings of 2016 IEEE 27th Annual International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRC)pp 1ndash6 Valencia Spain September 2016
[2] VM Rodrigo-Penarrocha J Reig L Rubio and et al ldquoAnalysisof small- scale fading distributions in vehicle-to-vehicle com-municationsrdquoMobile Information Systems vol 2016 Article ID9584815 7 pages 2016
[3] A Bazzi B M Masini A Zanella and G Pasolini ldquoIEEE80211p for cellular offloading in vehicular sensor networksrdquoComputer Communications vol 60 pp 97ndash108 2015
[4] X Cheng MIMO channel modelling and simulation for cellu-lar and mobile-to-mobile communication systems[Phd thesis]Heriot-Watt University 2009
[5] N Avazov and M Patzold ldquoA geometric street scatteringchannel model for car-to-car communication systemsrdquo inProceedings of 4th Annual International Conference onAdvanced
14 Mobile Information Systems
Technologies for Communications ATC2011 pp 224ndash230 vnmAugust 2011
[6] X Zhao X Liang S Li and K Haneda ldquoMobile-to-MobileWideband MIMO Channel Realization by Using a Two-RingGeometry-Based Stochastic Scattering Modelrdquo Wireless Per-sonal Communications vol 84 no 4 pp 2445ndash2465 2015
[7] M Walter D Shutin and U-C Fiebig ldquoDelay-dependentdoppler probability density functions for vehicle-to-vehiclescatter channelsrdquo IEEE Transactions on Antennas and Propaga-tion vol 62 no 4 pp 2238ndash2249 2014
[8] P T Samarasinghe T A Lamahewa T D Abhayapala and RA Kennedy ldquo3D mobile-to-mobile wireless channel modelrdquo inProceedings of 2010 Australian Communications Theory Work-shop AusCTW 2010 pp 30ndash34 aus February 2010
[9] M Riaz S J Nawaz and N M Khan ldquo3D ellipsoidal model formobile-to-mobile radio propagation environmentsrdquo WirelessPersonal Communications vol 72 no 4 pp 2465ndash2479 2013
[10] J Chen and T G Pratt ldquoThree-dimensional geometry-based stochastic modeling and performance of 4times4 space-polarization mobile-to-mobile wideband MIMO channelsrdquo inProceedings of 2013 IEEE Global Communications ConferenceGLOBECOM 2013 pp 3936ndash3941 usa December 2013
[11] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA Novel 3DNonstationary Channel Model Based on the von Mises-FisherScattering DistributionrdquoMobile Information Systems vol 2016Article ID 2161460 2016
[12] A Chelli and M Patzold ldquoThe impact of fixed and movingscatterers on the statistics of MIMO vehicle-to-vehicle chan-nelsrdquo in Proceedings of VTC Spring 2009 - IEEE 69th VehicularTechnology Conference esp April 2009
[13] A Borhani and M Patzold ldquoModeling of vehicle-to-vehiclechannels in the presence of moving scatterersrdquo in Proceedingsof 2012 IEEE Vehicular Technology Conference (VTC Fall) pp1ndash5 Quebec City QC Canada September 2012
[14] A Borhani andM Patzold ldquoCorrelation and spectral propertiesof vehicle-to-vehicle channels in the presence ofmoving scatter-ersrdquo IEEE Transactions on Vehicular Technology vol 62 no 9pp 4228ndash4239 2013
[15] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the IEEE 74th Vehicular TechnologyConference (VTC Fall rsquo11) pp 1ndash6 IEEE San Francisco CalifUSA September 2011
[16] M D Soltani M Alimadadi Y Seyedi and H AmindavarldquoModeling of doppler spectrum in V2V urban canyon oncom-ing environmentrdquo in Proceedings of 2014 7th InternationalSymposium on Telecommunications IST 2014 pp 1155ndash1160 irnSeptember 2014
[17] M D Soltani M Alimadadi and A Mohammadi ldquoModelingof mobile scatterer clusters for doppler spectrum in widebandvehicle-to-vehicle communication channelsrdquo IEEE Communi-cations Letters vol 18 no 4 pp 628ndash631 2014
[18] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014
[19] A G Zajic ldquoModeling impact of moving scatterers on dopplerspectrum in wideband vehicle-to-vehicle channelsrdquo in Proceed-ings of Proc Eur Conf Antennas Propag p 1 Lisbon May 2015
[20] J Zhang X Yin and X Cheng ldquoTheoretical analysis andmeasurements Doppler spectra of vehicular communication
channelsrdquo in Proceedings of 2012 12th International Confer-ence on ITS Telecommunications ITST 2012 pp 98ndash102 twnNovember 2012
[21] M Picone S Busanelli M Amoretti F Zanichelli and GFerrari Advanced Technologies for Intelligent TransportationSystems Springer 2015
[22] V H Pham M H Taieb J Y Chouinard S Roy and H THuynh ldquoOn the double Doppler effect generated by scatterermotionrdquo REV Journal on Electronics and Communications vol1 pp 30ndash37 2011
[23] S Roy H T Huynh and P Fortier ldquoCompound Dopplerspread effects of subscriber motion and scatterer motionrdquo AEU- International Journal of Electronics and Communications vol57 no 4 pp 237ndash246 2003
[24] J B Andersen J O Nielsen G F Pedersen G Bauch and GDietl ldquoDoppler spectrum from moving scatterers in a randomenvironmentrdquo IEEE Transactions on Wireless Communicationsvol 8 no 6 pp 3270ndash3277 2009
[25] H S Rad S Gazor and P Shariatpanahi ldquoNon-fixed scatterersand their effects on MIMOmulticarrier fading communicationchannelsrdquo in Proceedings of 50th Annual IEEE Global Telecom-munications Conference GLOBECOM2007 pp 3765ndash3769 usaNovember 2007
[26] N Naz and D Falconer ldquoTemporal variations characterizationfor fixed wireless at 295 GHzrdquo in Proceedings of 2000 IEEE51st Vehicular Technology Conference Proceedings VTC2000-Springer pp 2178ndash2182 Tokyo Japan 2000
[27] A G Zajic G L Stuber T G Pratt and S T NguyenldquoWideband MIMO mobile-to-mobile channels Geometry-based statistical modeling with experimental verificationrdquo IEEETransactions on Vehicular Technology vol 58 no 2 pp 517ndash5342009
[28] W C Jakes Microwave Mobile Communications Wiley-IEEEPress Piscataway NJ USA 1994
[29] M Patzold B O Hogstad and N Youssef ldquoModeling analysisand simulation of MIMO mobile-to-mobile fading channelsrdquoIEEE Transactions onWireless Communications vol 7 no 2 pp510ndash520 2008
[30] M PatzoldMobile Radio Channels Wiley Chichester UK 2ndedition 2011
[31] R Wang and D Cox ldquoDouble mobility mitigates fading in adhoc wireless networksrdquo in Proceedings of IEEE Antennas andPropagation Society International Symposium pp 306ndash309 SanAntonio TX USA
[32] G Bakhshi K Shahtalebi and H S Rad ldquoA novel full-three-dimensional MIMOmobile-to-mobile channel referencemodelrdquo in Proceedings of 3rd International Conference on SignalProcessing and Communication Systems ICSPCSrsquo2009 usaSeptember 2009
[33] W Zhou X Wang X Wang and W Chen ldquoA modified two-erose- ringmodel formimomobile-to-mobile fading channelsrdquoin Proceedings of 7th International Conference on WirelessCommunications Networking and Mobile Computing WiCOM2011 5 1 pages Wuhan China September 2011
Figure 7 T-CFs in (16) with 119901 = 119901 and 119902 = 119902 in the isotropic andnonisotropic scattering scenarios with119872 = 5 for different scatterervelocity distributions
and the standard deviation of the half-Gaussian distribu-tion is 31ms The other parameters used to obtain curvesin Figures 6 and 7 are summarized in (Table 2 Cols 2and 3) As shown in Figure 6 the ST-CF for the uniformdistributed velocity of scatterers in the isotropic scatteringenvironment decreases fastest and both the nonisotropicscattering and the half-Gaussian distribution can result inswift oscillations of the fading curvesThe comparison resultsindicate that the scatterer velocity distributions have moresignificant effect than the scattering forms on the ST-CFsof the 3D V2V channel in the presence of multiple movingscatterers Figure 7 indicates that not like scattering formsthe scatterer velocity distributions have significant effect onthe T-CFs Generally the realistic propagation environmentis nonisotropic scattering and the scatterer velocity canbe modeled by the half-Gaussian or uniform distributionTherefore the results of Figures 6 and 7 may be helpful inanalysis of the system performance and modeling a specificpropagation channel for V2V communications
413 ST-CFs and S-CFs for Different Scattering Scenariosand Spacing between Adjacent Antenna Elements Figures8 and 9 demonstrate the 3D ST-CFs and S-CFs for thedifferent spacing between adjacent antenna elements of 119879119883and 119877119883 in isotropic and nonisotropic scattering scenariosThe set of the scatterer velocity in Figure 8 is the sameas Figures 2ndash5 and the other parameters used to obtaincurves in Figures 8 and 9 are summarized in (Table 2 Col4) Figures 8 and 9 show that spacing between adjacentantenna elements has significant effect on the 3D ST-CFsand S-CFs Furthermore the 3D ST-CFs in isotropic andnonisotropic scattering scenarios decrease to the bottomwhen 119889119877 = 119889119879 tends to 12120582 and 07120582 respectively Inaddition the 3D S-CF in nonisotropic scattering scenarios isexplicitly less than that in isotropic scattering scenarios for
|R11
22(d
Td
R휏)|
dR = d
T (휆) 휏 (ms)
0025
05075 1
12515 0
12
34
1
095
09
085
08
075
07
065
IsotropicNonisotropic
Figure 8 3D ST-CFs in (16) in the isotropic and nonisotropicscattering scenarios with119872 = 5 120582 is the carrier wavelength
|R11
22(d
Td
R)|
dR (휆)
dT (휆)
1
095
09
085
08
075
07
065
0 025 05 0751 125 15
0025
05075 1
12515
IsotropicNonisotropic
Figure 9 3D S-CFs in (16) with 120591 = 0 in the isotropic andnonisotropic scattering scenarios 120582 is the carrier wavelength
most spacing between adjacent antenna elements as shown inFigure 9Therefore the large-scale antenna arrays employingthe spatial multiplexing are more suitable to be used in thenonisotropic scattering scenarios owing to the high antennaspace utilization The spacing between adjacent antennaelements of antenna arrays is suggested to be more than 120582in the V2V communications especially in the environmentswith low-density scatterers
42 Comparison with Measurements The parametric natureof the proposed channel model makes it adaptable to avariety of propagation environments D-PSD is one of themost important and unique channel characteristics for V2Vcommunication channels To illustrate the validity of theproposed model we compare the modeled SD-PSDs of the3D V2V channel with multiple moving scatterers with themeasured F2F D-PSD in [26] the theoretical V2V SD-PSDin [18] and the measured V2V SD-PSD in [27]
Mobile Information Systems 11
minus2 minus15 minus1 minus05 0 05 1 15 2Doppler frequency f (kHz)
Figure 10 Comparison between the modeled SD-PSDs with119872 = 5and the measured D-PSDs of Figure 4 in [26]
Figure 10 shows the comparison between the modeledSD-PSDs obtained from the Fourier transform of the ST-CF presented in (16) in the isotropic scattering scenario andthe measured D-PSDs of Figure 4 in [26] The outdoor mea-surement experiments in [26] have shown that at millimeterwavelengths fading caused by foliage movement and themotion of nearby vehicles is a significantly deleterious effectin systems with static subscribers Figure 4 in [26] showsthe D-PSD generated by foliage movements and the D-PSDcaused by passing vehicles both in a fixed wireless channelat a carrier frequency of 295GHz We have used the half-Gaussian distribution with an average speed of 04ms andthe standard deviation of 05ms to model the velocity ofmoving foliage scatterers while the uniformdistributionwithan average speed of 6ms has been used tomodel the velocityof passing vehicles scatterers The power weight of the 119898thcluster of rays scattered by the moving foliage or passingvehicles has been set to 1199011 = 916 1199012 = 516 1199013 = 116and 1199014 = 1199015 = 132 The other parameters used to obtainthe modeled SD-PSD curves in Figure 10 are summarizedin (Table 2 Col 5) As can be observed in Figure 10 40 dBfading bandwidths for the effect of the foliage and vehiclesmovements are 500Hz and 1 kHz approximately respectivelyFade depth due to foliage is less than the variation due topassing vehicles Close agreements between the modeled andmeasured results are shown in Figure 10These close matchescan be seen not only for relatively slow moving scatterers butalso for relatively fast moving scatterers as well
Figure 11 shows the comparison among the modeled SD-PSD obtained from the Fourier transform of the ST-CFpresented in (16) in the nonisotropic scattering scenario thetheoretical SD-PSD of Figure 8 in [18] and themeasured SD-PSD of Figure 11 in [27]The theoretical SD-PSD reproducedhere is based on the two-cylinder model with moving andstationary scatterers for 119889119877 = 119889119879 = 0 The channel mea-surements in [27] for the measured SD-PSD were collectedat 2435GHz in the urban street surface environment and thespacing between two adjacent antenna elements of119879119883 and119877119883was set to zero We have used the half-Gaussian distributionwith average speeds of 2394ms 0798ms and 04ms
Figure 11 Comparison among the modeled SD-PSD with119872 = 5the theoretical SD-PSD of Figure 8 in [18] and the measured SD-PSD of Figure 11 in [27]
and the standard deviation of 3ms 1ms and 05ms tomodel the velocity of the first the second and the remainingmoving scatterers on the communication links respectivelyThe power weight of the 119898th cluster of rays has been set to1199011 = 815 1199012 = 13 1199013 = 115 1199014 = 130 and 1199015 = 130for119872 = 5 The other parameters used to obtain the modeledSD-PSD curve in Figure 11 are summarized in (Table 2 Col6) We note that different from the measured SD-PSDs in[26] themeasured SD-PSD curve in [27] has oscillationsTheoscillating pattern in measurements for the case 119889119877 = 119889119879 = 0may appear because it is not a true SISO scenario whereonly one transmit and one receive antenna are active [27] Ascan be observed in Figure 11 the modeled SD-PSD matchesbetter withmeasured SD-PSD than the theoretical SD-PSD in[18] This may be because of the introduction of the multiplebounced rays into the propagation model or the constraintson the position of local scatterers in the two-cylinder modelproposed in [18]
Figure 12 demonstrates the CDFs of the relative devi-ations between modeledtheoretical SD-PSDs and corre-sponding measured values in Figures 10 and 11 The meanrelative deviation of the modeled SD-PSDs for foliage inFigure 10 modeled SD-PSDs for vehicles in Figure 10 mod-eled SD-PSDs in Figure 11 and theoretical SD-PSDs inFigure 11 is 256 216 39 and 832 respectively Asshown in Figure 12 more than 80 of the relative deviationsof modeled SD-PSDs in Figures 10 and 11 are less than5 and the relative deviations of theoretical SD-PSDs inFigure 11 are explicitly larger than those of modeled SD-PSDs In addition from Figures 10 and 11 we note that themodeled SD-PSD matches better with measured SD-PSDs atthe lower and higher Doppler frequencies than the middleDoppler frequencies This may be because 119879119883 and 119877119883 werenot strictly fixed and not all the scatterers were in motion inmeasurementsThe approximate calculation of the frequencyshift in (A4) and Doppler shift in (10) may be anotherfactor which causes the mismatch at the middle Dopplerfrequencies andmore investigations concerning the accuracyof the proposed model will be addressed in future works
12 Mobile Information Systems
0 5 10 15 20 25 30 350
01
02
03
04
05
06
07
08
09
1
Relative deviation ()
Relat
ive d
evia
tion
CDF
Modeled SD-PSD (foliage)Modeled SD-PSD (vehicles)
Modeled SD-PSDTheoretical SD-PSD
Figure 12 The relative deviation cumulative distribution functions(CDFs) of the modeled and theoretical SD-PSDs in Figures 10 and11
However as shown in Figures 10ndash12 the acceptable matchesconfirm the utility and generality of the proposed model andshow the need for including multiple moving scatterers inpropagation model
5 Conclusion
Without specific constraints on the position of the local mov-ing scatterers a 3D geometrical V2V propagation model thatincludes LoS single bounced and multiple bounced linksbetween the transmitter and receiver was proposed Basedon the geometrical propagation model a 3D reference modelfor narrowband MIMO V2V multipath fading channels wasdeveloped From the reference model the correspondingmathematical expressions and numerical results of ST-CFsT-CFs S-CFs and SD-PSDs were studied for different para-metric sets It has been shown that the maximum bouncesscattering forms scatterer velocity distributions and spacingbetween adjacent antenna elements have significant effect onthe ST-CFs and only the maximum bounces and scatterervelocity distributions have significant effect on the T-CFsFinally the modeled SD-PSD results were compared withthe measured data and the other literaturersquos theoreticalresult The close agreements between the analytically andempirically obtained channel statistics confirmed the utilityand generality of the proposed model
Appendix
A Derivation Process of Frequency Shifts
The derivation process of 119891119902119901119902119899 in (7) is presented in thissection and 119891119901119901119902120588 119891119902119901119902119899 119891119901119901119902119899 and 119891119901119902120588 can be calculatedsimilarly
AOD
Δl
TX qth element
Figure 13 Projection from the coordinate vector of 119902th antennaelement to the AOD direction
According to Figure 1 the coordinate vectors of 119902thantenna element and the AOD direction can be written asrespectively
minus 1) 119889119879 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902 + sin120573119902 sin120573119879119901119902] (A3)
Because we mainly consider the difference of the fre-quency shifts caused by the TPD changes to calculate ST-CFsthe TPD change of ℎ11(119905) is assumed to be zero Eventually119891119902119901119902119899 can be calculated as
119891119902119901119902119899 asymp Δ1198971198910119888= (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(A4)
B Derivation Process of Approximate ST-CFs
In the 2D scattering environment the simplified and approx-imate expressions of (28) can be derived in this section If120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 (27) and (10) can be writtenas
Δ119891119899119898
= 1198910119888 [(119902 minus 119902) 119889119879 cos (120572119902 minus 120572119879119901119902)+ (119901 minus 119901) 119889119877 cos (120572119901 minus 120572119877119901119902)]
Mobile Information Systems 13
119891119901119902119899119898
= 1198910119888 [V119879 cos (120572119879 minus 120572119879119901119902) + V119877 cos (120572119877 minus 120572119877119901119902)
+ ]1119901119902 cos (1205721119901119902 minus 120572119879119901119902) + 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119879119901119902)+ ]119898119901119902 cos (120572119898119901119902 minus 120572119879119901119902) + ]1119901119902 cos (1205721119901119902 minus 120572119877119901119902)+ 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119877119901119902) + ]119898119901119902 cos (120572119898119901119902 minus 120572119877119901119902)] (B1)
If 120572119879 = 120572119902 and 120572119877 = 120572119901 that is the antenna array and itsvelocity have the same directions (28) can be simplified as bysubstituting (B1) into (28)
where 1198960 = 21205871198910119888 is the wave numberIn isotropic scattering environment we use the uniform
distribution in (31) to describe all the random azimuth anglesIn this case (B2) can be written as119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879] cos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877] cos (120572119877 minus 120572119877119901119902)
times1198690 [21198960]1119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times1198690 [21198960]119898119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times119898minus1prod119894=2
1198690 [41198960]119894119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind
The AOD 120572119879119901119902 and AOA 120572119877119901119902 are uniformly and indepen-dently distributed so120572119879119901119902minus120572119877119901119902 can be equal to zero in average
As a consequence the term cos[(120572119879119901119902 minus120572119877119901119902)2] in (B3) can beapproximated by one [13 14] and an approximate ST-CF canbe written as from (B3)
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879]sdotcos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877]sdot cos (120572119877 minus 120572119877119901119902) 1198690 (21198960]1119901119902120591)times1198690 (21198960]119898119901119902120591)119898minus1prod
Eventually the approximate ST-CF in (34) can be derivedfrom (B4) according to the definition of 1198690(∙)Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was funded by the National Natural Science Foun-dation of China (no 91438104 no 61501065 no 61571069and no 61601067) the Fundamental Research Funds forthe Central Universities (no 106112016CDJX160001) and theChongqing Research Program of Basic Research and FrontierTechnology (no CSTC2016JCYJA0021)
References
[1] A Bazzi B M Masini A Zanella and I Thibault ldquoBeaconingfrom connected vehicles IEEE 80211p vs LTE-V2Vrdquo in Pro-ceedings of 2016 IEEE 27th Annual International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRC)pp 1ndash6 Valencia Spain September 2016
[2] VM Rodrigo-Penarrocha J Reig L Rubio and et al ldquoAnalysisof small- scale fading distributions in vehicle-to-vehicle com-municationsrdquoMobile Information Systems vol 2016 Article ID9584815 7 pages 2016
[3] A Bazzi B M Masini A Zanella and G Pasolini ldquoIEEE80211p for cellular offloading in vehicular sensor networksrdquoComputer Communications vol 60 pp 97ndash108 2015
[4] X Cheng MIMO channel modelling and simulation for cellu-lar and mobile-to-mobile communication systems[Phd thesis]Heriot-Watt University 2009
[5] N Avazov and M Patzold ldquoA geometric street scatteringchannel model for car-to-car communication systemsrdquo inProceedings of 4th Annual International Conference onAdvanced
14 Mobile Information Systems
Technologies for Communications ATC2011 pp 224ndash230 vnmAugust 2011
[6] X Zhao X Liang S Li and K Haneda ldquoMobile-to-MobileWideband MIMO Channel Realization by Using a Two-RingGeometry-Based Stochastic Scattering Modelrdquo Wireless Per-sonal Communications vol 84 no 4 pp 2445ndash2465 2015
[7] M Walter D Shutin and U-C Fiebig ldquoDelay-dependentdoppler probability density functions for vehicle-to-vehiclescatter channelsrdquo IEEE Transactions on Antennas and Propaga-tion vol 62 no 4 pp 2238ndash2249 2014
[8] P T Samarasinghe T A Lamahewa T D Abhayapala and RA Kennedy ldquo3D mobile-to-mobile wireless channel modelrdquo inProceedings of 2010 Australian Communications Theory Work-shop AusCTW 2010 pp 30ndash34 aus February 2010
[9] M Riaz S J Nawaz and N M Khan ldquo3D ellipsoidal model formobile-to-mobile radio propagation environmentsrdquo WirelessPersonal Communications vol 72 no 4 pp 2465ndash2479 2013
[10] J Chen and T G Pratt ldquoThree-dimensional geometry-based stochastic modeling and performance of 4times4 space-polarization mobile-to-mobile wideband MIMO channelsrdquo inProceedings of 2013 IEEE Global Communications ConferenceGLOBECOM 2013 pp 3936ndash3941 usa December 2013
[11] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA Novel 3DNonstationary Channel Model Based on the von Mises-FisherScattering DistributionrdquoMobile Information Systems vol 2016Article ID 2161460 2016
[12] A Chelli and M Patzold ldquoThe impact of fixed and movingscatterers on the statistics of MIMO vehicle-to-vehicle chan-nelsrdquo in Proceedings of VTC Spring 2009 - IEEE 69th VehicularTechnology Conference esp April 2009
[13] A Borhani and M Patzold ldquoModeling of vehicle-to-vehiclechannels in the presence of moving scatterersrdquo in Proceedingsof 2012 IEEE Vehicular Technology Conference (VTC Fall) pp1ndash5 Quebec City QC Canada September 2012
[14] A Borhani andM Patzold ldquoCorrelation and spectral propertiesof vehicle-to-vehicle channels in the presence ofmoving scatter-ersrdquo IEEE Transactions on Vehicular Technology vol 62 no 9pp 4228ndash4239 2013
[15] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the IEEE 74th Vehicular TechnologyConference (VTC Fall rsquo11) pp 1ndash6 IEEE San Francisco CalifUSA September 2011
[16] M D Soltani M Alimadadi Y Seyedi and H AmindavarldquoModeling of doppler spectrum in V2V urban canyon oncom-ing environmentrdquo in Proceedings of 2014 7th InternationalSymposium on Telecommunications IST 2014 pp 1155ndash1160 irnSeptember 2014
[17] M D Soltani M Alimadadi and A Mohammadi ldquoModelingof mobile scatterer clusters for doppler spectrum in widebandvehicle-to-vehicle communication channelsrdquo IEEE Communi-cations Letters vol 18 no 4 pp 628ndash631 2014
[18] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014
[19] A G Zajic ldquoModeling impact of moving scatterers on dopplerspectrum in wideband vehicle-to-vehicle channelsrdquo in Proceed-ings of Proc Eur Conf Antennas Propag p 1 Lisbon May 2015
[20] J Zhang X Yin and X Cheng ldquoTheoretical analysis andmeasurements Doppler spectra of vehicular communication
channelsrdquo in Proceedings of 2012 12th International Confer-ence on ITS Telecommunications ITST 2012 pp 98ndash102 twnNovember 2012
[21] M Picone S Busanelli M Amoretti F Zanichelli and GFerrari Advanced Technologies for Intelligent TransportationSystems Springer 2015
[22] V H Pham M H Taieb J Y Chouinard S Roy and H THuynh ldquoOn the double Doppler effect generated by scatterermotionrdquo REV Journal on Electronics and Communications vol1 pp 30ndash37 2011
[23] S Roy H T Huynh and P Fortier ldquoCompound Dopplerspread effects of subscriber motion and scatterer motionrdquo AEU- International Journal of Electronics and Communications vol57 no 4 pp 237ndash246 2003
[24] J B Andersen J O Nielsen G F Pedersen G Bauch and GDietl ldquoDoppler spectrum from moving scatterers in a randomenvironmentrdquo IEEE Transactions on Wireless Communicationsvol 8 no 6 pp 3270ndash3277 2009
[25] H S Rad S Gazor and P Shariatpanahi ldquoNon-fixed scatterersand their effects on MIMOmulticarrier fading communicationchannelsrdquo in Proceedings of 50th Annual IEEE Global Telecom-munications Conference GLOBECOM2007 pp 3765ndash3769 usaNovember 2007
[26] N Naz and D Falconer ldquoTemporal variations characterizationfor fixed wireless at 295 GHzrdquo in Proceedings of 2000 IEEE51st Vehicular Technology Conference Proceedings VTC2000-Springer pp 2178ndash2182 Tokyo Japan 2000
[27] A G Zajic G L Stuber T G Pratt and S T NguyenldquoWideband MIMO mobile-to-mobile channels Geometry-based statistical modeling with experimental verificationrdquo IEEETransactions on Vehicular Technology vol 58 no 2 pp 517ndash5342009
[28] W C Jakes Microwave Mobile Communications Wiley-IEEEPress Piscataway NJ USA 1994
[29] M Patzold B O Hogstad and N Youssef ldquoModeling analysisand simulation of MIMO mobile-to-mobile fading channelsrdquoIEEE Transactions onWireless Communications vol 7 no 2 pp510ndash520 2008
[30] M PatzoldMobile Radio Channels Wiley Chichester UK 2ndedition 2011
[31] R Wang and D Cox ldquoDouble mobility mitigates fading in adhoc wireless networksrdquo in Proceedings of IEEE Antennas andPropagation Society International Symposium pp 306ndash309 SanAntonio TX USA
[32] G Bakhshi K Shahtalebi and H S Rad ldquoA novel full-three-dimensional MIMOmobile-to-mobile channel referencemodelrdquo in Proceedings of 3rd International Conference on SignalProcessing and Communication Systems ICSPCSrsquo2009 usaSeptember 2009
[33] W Zhou X Wang X Wang and W Chen ldquoA modified two-erose- ringmodel formimomobile-to-mobile fading channelsrdquoin Proceedings of 7th International Conference on WirelessCommunications Networking and Mobile Computing WiCOM2011 5 1 pages Wuhan China September 2011
Figure 10 Comparison between the modeled SD-PSDs with119872 = 5and the measured D-PSDs of Figure 4 in [26]
Figure 10 shows the comparison between the modeledSD-PSDs obtained from the Fourier transform of the ST-CF presented in (16) in the isotropic scattering scenario andthe measured D-PSDs of Figure 4 in [26] The outdoor mea-surement experiments in [26] have shown that at millimeterwavelengths fading caused by foliage movement and themotion of nearby vehicles is a significantly deleterious effectin systems with static subscribers Figure 4 in [26] showsthe D-PSD generated by foliage movements and the D-PSDcaused by passing vehicles both in a fixed wireless channelat a carrier frequency of 295GHz We have used the half-Gaussian distribution with an average speed of 04ms andthe standard deviation of 05ms to model the velocity ofmoving foliage scatterers while the uniformdistributionwithan average speed of 6ms has been used tomodel the velocityof passing vehicles scatterers The power weight of the 119898thcluster of rays scattered by the moving foliage or passingvehicles has been set to 1199011 = 916 1199012 = 516 1199013 = 116and 1199014 = 1199015 = 132 The other parameters used to obtainthe modeled SD-PSD curves in Figure 10 are summarizedin (Table 2 Col 5) As can be observed in Figure 10 40 dBfading bandwidths for the effect of the foliage and vehiclesmovements are 500Hz and 1 kHz approximately respectivelyFade depth due to foliage is less than the variation due topassing vehicles Close agreements between the modeled andmeasured results are shown in Figure 10These close matchescan be seen not only for relatively slow moving scatterers butalso for relatively fast moving scatterers as well
Figure 11 shows the comparison among the modeled SD-PSD obtained from the Fourier transform of the ST-CFpresented in (16) in the nonisotropic scattering scenario thetheoretical SD-PSD of Figure 8 in [18] and themeasured SD-PSD of Figure 11 in [27]The theoretical SD-PSD reproducedhere is based on the two-cylinder model with moving andstationary scatterers for 119889119877 = 119889119879 = 0 The channel mea-surements in [27] for the measured SD-PSD were collectedat 2435GHz in the urban street surface environment and thespacing between two adjacent antenna elements of119879119883 and119877119883was set to zero We have used the half-Gaussian distributionwith average speeds of 2394ms 0798ms and 04ms
Figure 11 Comparison among the modeled SD-PSD with119872 = 5the theoretical SD-PSD of Figure 8 in [18] and the measured SD-PSD of Figure 11 in [27]
and the standard deviation of 3ms 1ms and 05ms tomodel the velocity of the first the second and the remainingmoving scatterers on the communication links respectivelyThe power weight of the 119898th cluster of rays has been set to1199011 = 815 1199012 = 13 1199013 = 115 1199014 = 130 and 1199015 = 130for119872 = 5 The other parameters used to obtain the modeledSD-PSD curve in Figure 11 are summarized in (Table 2 Col6) We note that different from the measured SD-PSDs in[26] themeasured SD-PSD curve in [27] has oscillationsTheoscillating pattern in measurements for the case 119889119877 = 119889119879 = 0may appear because it is not a true SISO scenario whereonly one transmit and one receive antenna are active [27] Ascan be observed in Figure 11 the modeled SD-PSD matchesbetter withmeasured SD-PSD than the theoretical SD-PSD in[18] This may be because of the introduction of the multiplebounced rays into the propagation model or the constraintson the position of local scatterers in the two-cylinder modelproposed in [18]
Figure 12 demonstrates the CDFs of the relative devi-ations between modeledtheoretical SD-PSDs and corre-sponding measured values in Figures 10 and 11 The meanrelative deviation of the modeled SD-PSDs for foliage inFigure 10 modeled SD-PSDs for vehicles in Figure 10 mod-eled SD-PSDs in Figure 11 and theoretical SD-PSDs inFigure 11 is 256 216 39 and 832 respectively Asshown in Figure 12 more than 80 of the relative deviationsof modeled SD-PSDs in Figures 10 and 11 are less than5 and the relative deviations of theoretical SD-PSDs inFigure 11 are explicitly larger than those of modeled SD-PSDs In addition from Figures 10 and 11 we note that themodeled SD-PSD matches better with measured SD-PSDs atthe lower and higher Doppler frequencies than the middleDoppler frequencies This may be because 119879119883 and 119877119883 werenot strictly fixed and not all the scatterers were in motion inmeasurementsThe approximate calculation of the frequencyshift in (A4) and Doppler shift in (10) may be anotherfactor which causes the mismatch at the middle Dopplerfrequencies andmore investigations concerning the accuracyof the proposed model will be addressed in future works
12 Mobile Information Systems
0 5 10 15 20 25 30 350
01
02
03
04
05
06
07
08
09
1
Relative deviation ()
Relat
ive d
evia
tion
CDF
Modeled SD-PSD (foliage)Modeled SD-PSD (vehicles)
Modeled SD-PSDTheoretical SD-PSD
Figure 12 The relative deviation cumulative distribution functions(CDFs) of the modeled and theoretical SD-PSDs in Figures 10 and11
However as shown in Figures 10ndash12 the acceptable matchesconfirm the utility and generality of the proposed model andshow the need for including multiple moving scatterers inpropagation model
5 Conclusion
Without specific constraints on the position of the local mov-ing scatterers a 3D geometrical V2V propagation model thatincludes LoS single bounced and multiple bounced linksbetween the transmitter and receiver was proposed Basedon the geometrical propagation model a 3D reference modelfor narrowband MIMO V2V multipath fading channels wasdeveloped From the reference model the correspondingmathematical expressions and numerical results of ST-CFsT-CFs S-CFs and SD-PSDs were studied for different para-metric sets It has been shown that the maximum bouncesscattering forms scatterer velocity distributions and spacingbetween adjacent antenna elements have significant effect onthe ST-CFs and only the maximum bounces and scatterervelocity distributions have significant effect on the T-CFsFinally the modeled SD-PSD results were compared withthe measured data and the other literaturersquos theoreticalresult The close agreements between the analytically andempirically obtained channel statistics confirmed the utilityand generality of the proposed model
Appendix
A Derivation Process of Frequency Shifts
The derivation process of 119891119902119901119902119899 in (7) is presented in thissection and 119891119901119901119902120588 119891119902119901119902119899 119891119901119901119902119899 and 119891119901119902120588 can be calculatedsimilarly
AOD
Δl
TX qth element
Figure 13 Projection from the coordinate vector of 119902th antennaelement to the AOD direction
According to Figure 1 the coordinate vectors of 119902thantenna element and the AOD direction can be written asrespectively
minus 1) 119889119879 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902 + sin120573119902 sin120573119879119901119902] (A3)
Because we mainly consider the difference of the fre-quency shifts caused by the TPD changes to calculate ST-CFsthe TPD change of ℎ11(119905) is assumed to be zero Eventually119891119902119901119902119899 can be calculated as
119891119902119901119902119899 asymp Δ1198971198910119888= (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(A4)
B Derivation Process of Approximate ST-CFs
In the 2D scattering environment the simplified and approx-imate expressions of (28) can be derived in this section If120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 (27) and (10) can be writtenas
Δ119891119899119898
= 1198910119888 [(119902 minus 119902) 119889119879 cos (120572119902 minus 120572119879119901119902)+ (119901 minus 119901) 119889119877 cos (120572119901 minus 120572119877119901119902)]
Mobile Information Systems 13
119891119901119902119899119898
= 1198910119888 [V119879 cos (120572119879 minus 120572119879119901119902) + V119877 cos (120572119877 minus 120572119877119901119902)
+ ]1119901119902 cos (1205721119901119902 minus 120572119879119901119902) + 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119879119901119902)+ ]119898119901119902 cos (120572119898119901119902 minus 120572119879119901119902) + ]1119901119902 cos (1205721119901119902 minus 120572119877119901119902)+ 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119877119901119902) + ]119898119901119902 cos (120572119898119901119902 minus 120572119877119901119902)] (B1)
If 120572119879 = 120572119902 and 120572119877 = 120572119901 that is the antenna array and itsvelocity have the same directions (28) can be simplified as bysubstituting (B1) into (28)
where 1198960 = 21205871198910119888 is the wave numberIn isotropic scattering environment we use the uniform
distribution in (31) to describe all the random azimuth anglesIn this case (B2) can be written as119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879] cos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877] cos (120572119877 minus 120572119877119901119902)
times1198690 [21198960]1119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times1198690 [21198960]119898119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times119898minus1prod119894=2
1198690 [41198960]119894119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind
The AOD 120572119879119901119902 and AOA 120572119877119901119902 are uniformly and indepen-dently distributed so120572119879119901119902minus120572119877119901119902 can be equal to zero in average
As a consequence the term cos[(120572119879119901119902 minus120572119877119901119902)2] in (B3) can beapproximated by one [13 14] and an approximate ST-CF canbe written as from (B3)
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879]sdotcos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877]sdot cos (120572119877 minus 120572119877119901119902) 1198690 (21198960]1119901119902120591)times1198690 (21198960]119898119901119902120591)119898minus1prod
Eventually the approximate ST-CF in (34) can be derivedfrom (B4) according to the definition of 1198690(∙)Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was funded by the National Natural Science Foun-dation of China (no 91438104 no 61501065 no 61571069and no 61601067) the Fundamental Research Funds forthe Central Universities (no 106112016CDJX160001) and theChongqing Research Program of Basic Research and FrontierTechnology (no CSTC2016JCYJA0021)
References
[1] A Bazzi B M Masini A Zanella and I Thibault ldquoBeaconingfrom connected vehicles IEEE 80211p vs LTE-V2Vrdquo in Pro-ceedings of 2016 IEEE 27th Annual International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRC)pp 1ndash6 Valencia Spain September 2016
[2] VM Rodrigo-Penarrocha J Reig L Rubio and et al ldquoAnalysisof small- scale fading distributions in vehicle-to-vehicle com-municationsrdquoMobile Information Systems vol 2016 Article ID9584815 7 pages 2016
[3] A Bazzi B M Masini A Zanella and G Pasolini ldquoIEEE80211p for cellular offloading in vehicular sensor networksrdquoComputer Communications vol 60 pp 97ndash108 2015
[4] X Cheng MIMO channel modelling and simulation for cellu-lar and mobile-to-mobile communication systems[Phd thesis]Heriot-Watt University 2009
[5] N Avazov and M Patzold ldquoA geometric street scatteringchannel model for car-to-car communication systemsrdquo inProceedings of 4th Annual International Conference onAdvanced
14 Mobile Information Systems
Technologies for Communications ATC2011 pp 224ndash230 vnmAugust 2011
[6] X Zhao X Liang S Li and K Haneda ldquoMobile-to-MobileWideband MIMO Channel Realization by Using a Two-RingGeometry-Based Stochastic Scattering Modelrdquo Wireless Per-sonal Communications vol 84 no 4 pp 2445ndash2465 2015
[7] M Walter D Shutin and U-C Fiebig ldquoDelay-dependentdoppler probability density functions for vehicle-to-vehiclescatter channelsrdquo IEEE Transactions on Antennas and Propaga-tion vol 62 no 4 pp 2238ndash2249 2014
[8] P T Samarasinghe T A Lamahewa T D Abhayapala and RA Kennedy ldquo3D mobile-to-mobile wireless channel modelrdquo inProceedings of 2010 Australian Communications Theory Work-shop AusCTW 2010 pp 30ndash34 aus February 2010
[9] M Riaz S J Nawaz and N M Khan ldquo3D ellipsoidal model formobile-to-mobile radio propagation environmentsrdquo WirelessPersonal Communications vol 72 no 4 pp 2465ndash2479 2013
[10] J Chen and T G Pratt ldquoThree-dimensional geometry-based stochastic modeling and performance of 4times4 space-polarization mobile-to-mobile wideband MIMO channelsrdquo inProceedings of 2013 IEEE Global Communications ConferenceGLOBECOM 2013 pp 3936ndash3941 usa December 2013
[11] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA Novel 3DNonstationary Channel Model Based on the von Mises-FisherScattering DistributionrdquoMobile Information Systems vol 2016Article ID 2161460 2016
[12] A Chelli and M Patzold ldquoThe impact of fixed and movingscatterers on the statistics of MIMO vehicle-to-vehicle chan-nelsrdquo in Proceedings of VTC Spring 2009 - IEEE 69th VehicularTechnology Conference esp April 2009
[13] A Borhani and M Patzold ldquoModeling of vehicle-to-vehiclechannels in the presence of moving scatterersrdquo in Proceedingsof 2012 IEEE Vehicular Technology Conference (VTC Fall) pp1ndash5 Quebec City QC Canada September 2012
[14] A Borhani andM Patzold ldquoCorrelation and spectral propertiesof vehicle-to-vehicle channels in the presence ofmoving scatter-ersrdquo IEEE Transactions on Vehicular Technology vol 62 no 9pp 4228ndash4239 2013
[15] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the IEEE 74th Vehicular TechnologyConference (VTC Fall rsquo11) pp 1ndash6 IEEE San Francisco CalifUSA September 2011
[16] M D Soltani M Alimadadi Y Seyedi and H AmindavarldquoModeling of doppler spectrum in V2V urban canyon oncom-ing environmentrdquo in Proceedings of 2014 7th InternationalSymposium on Telecommunications IST 2014 pp 1155ndash1160 irnSeptember 2014
[17] M D Soltani M Alimadadi and A Mohammadi ldquoModelingof mobile scatterer clusters for doppler spectrum in widebandvehicle-to-vehicle communication channelsrdquo IEEE Communi-cations Letters vol 18 no 4 pp 628ndash631 2014
[18] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014
[19] A G Zajic ldquoModeling impact of moving scatterers on dopplerspectrum in wideband vehicle-to-vehicle channelsrdquo in Proceed-ings of Proc Eur Conf Antennas Propag p 1 Lisbon May 2015
[20] J Zhang X Yin and X Cheng ldquoTheoretical analysis andmeasurements Doppler spectra of vehicular communication
channelsrdquo in Proceedings of 2012 12th International Confer-ence on ITS Telecommunications ITST 2012 pp 98ndash102 twnNovember 2012
[21] M Picone S Busanelli M Amoretti F Zanichelli and GFerrari Advanced Technologies for Intelligent TransportationSystems Springer 2015
[22] V H Pham M H Taieb J Y Chouinard S Roy and H THuynh ldquoOn the double Doppler effect generated by scatterermotionrdquo REV Journal on Electronics and Communications vol1 pp 30ndash37 2011
[23] S Roy H T Huynh and P Fortier ldquoCompound Dopplerspread effects of subscriber motion and scatterer motionrdquo AEU- International Journal of Electronics and Communications vol57 no 4 pp 237ndash246 2003
[24] J B Andersen J O Nielsen G F Pedersen G Bauch and GDietl ldquoDoppler spectrum from moving scatterers in a randomenvironmentrdquo IEEE Transactions on Wireless Communicationsvol 8 no 6 pp 3270ndash3277 2009
[25] H S Rad S Gazor and P Shariatpanahi ldquoNon-fixed scatterersand their effects on MIMOmulticarrier fading communicationchannelsrdquo in Proceedings of 50th Annual IEEE Global Telecom-munications Conference GLOBECOM2007 pp 3765ndash3769 usaNovember 2007
[26] N Naz and D Falconer ldquoTemporal variations characterizationfor fixed wireless at 295 GHzrdquo in Proceedings of 2000 IEEE51st Vehicular Technology Conference Proceedings VTC2000-Springer pp 2178ndash2182 Tokyo Japan 2000
[27] A G Zajic G L Stuber T G Pratt and S T NguyenldquoWideband MIMO mobile-to-mobile channels Geometry-based statistical modeling with experimental verificationrdquo IEEETransactions on Vehicular Technology vol 58 no 2 pp 517ndash5342009
[28] W C Jakes Microwave Mobile Communications Wiley-IEEEPress Piscataway NJ USA 1994
[29] M Patzold B O Hogstad and N Youssef ldquoModeling analysisand simulation of MIMO mobile-to-mobile fading channelsrdquoIEEE Transactions onWireless Communications vol 7 no 2 pp510ndash520 2008
[30] M PatzoldMobile Radio Channels Wiley Chichester UK 2ndedition 2011
[31] R Wang and D Cox ldquoDouble mobility mitigates fading in adhoc wireless networksrdquo in Proceedings of IEEE Antennas andPropagation Society International Symposium pp 306ndash309 SanAntonio TX USA
[32] G Bakhshi K Shahtalebi and H S Rad ldquoA novel full-three-dimensional MIMOmobile-to-mobile channel referencemodelrdquo in Proceedings of 3rd International Conference on SignalProcessing and Communication Systems ICSPCSrsquo2009 usaSeptember 2009
[33] W Zhou X Wang X Wang and W Chen ldquoA modified two-erose- ringmodel formimomobile-to-mobile fading channelsrdquoin Proceedings of 7th International Conference on WirelessCommunications Networking and Mobile Computing WiCOM2011 5 1 pages Wuhan China September 2011
Figure 12 The relative deviation cumulative distribution functions(CDFs) of the modeled and theoretical SD-PSDs in Figures 10 and11
However as shown in Figures 10ndash12 the acceptable matchesconfirm the utility and generality of the proposed model andshow the need for including multiple moving scatterers inpropagation model
5 Conclusion
Without specific constraints on the position of the local mov-ing scatterers a 3D geometrical V2V propagation model thatincludes LoS single bounced and multiple bounced linksbetween the transmitter and receiver was proposed Basedon the geometrical propagation model a 3D reference modelfor narrowband MIMO V2V multipath fading channels wasdeveloped From the reference model the correspondingmathematical expressions and numerical results of ST-CFsT-CFs S-CFs and SD-PSDs were studied for different para-metric sets It has been shown that the maximum bouncesscattering forms scatterer velocity distributions and spacingbetween adjacent antenna elements have significant effect onthe ST-CFs and only the maximum bounces and scatterervelocity distributions have significant effect on the T-CFsFinally the modeled SD-PSD results were compared withthe measured data and the other literaturersquos theoreticalresult The close agreements between the analytically andempirically obtained channel statistics confirmed the utilityand generality of the proposed model
Appendix
A Derivation Process of Frequency Shifts
The derivation process of 119891119902119901119902119899 in (7) is presented in thissection and 119891119901119901119902120588 119891119902119901119902119899 119891119901119901119902119899 and 119891119901119902120588 can be calculatedsimilarly
AOD
Δl
TX qth element
Figure 13 Projection from the coordinate vector of 119902th antennaelement to the AOD direction
According to Figure 1 the coordinate vectors of 119902thantenna element and the AOD direction can be written asrespectively
minus 1) 119889119879 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902 + sin120573119902 sin120573119879119901119902] (A3)
Because we mainly consider the difference of the fre-quency shifts caused by the TPD changes to calculate ST-CFsthe TPD change of ℎ11(119905) is assumed to be zero Eventually119891119902119901119902119899 can be calculated as
119891119902119901119902119899 asymp Δ1198971198910119888= (119902 minus 1) 1198891198791198910119888 [cos (120572119902 minus 120572119879119901119902) cos120573119902 cos120573119879119901119902+ sin120573119902 sin120573119879119901119902]
(A4)
B Derivation Process of Approximate ST-CFs
In the 2D scattering environment the simplified and approx-imate expressions of (28) can be derived in this section If120573119879 = 120573119877 = 120573119879119901119902 = 120573119877119901119902 = 120573119894119901119902 = 0 (27) and (10) can be writtenas
Δ119891119899119898
= 1198910119888 [(119902 minus 119902) 119889119879 cos (120572119902 minus 120572119879119901119902)+ (119901 minus 119901) 119889119877 cos (120572119901 minus 120572119877119901119902)]
Mobile Information Systems 13
119891119901119902119899119898
= 1198910119888 [V119879 cos (120572119879 minus 120572119879119901119902) + V119877 cos (120572119877 minus 120572119877119901119902)
+ ]1119901119902 cos (1205721119901119902 minus 120572119879119901119902) + 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119879119901119902)+ ]119898119901119902 cos (120572119898119901119902 minus 120572119879119901119902) + ]1119901119902 cos (1205721119901119902 minus 120572119877119901119902)+ 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119877119901119902) + ]119898119901119902 cos (120572119898119901119902 minus 120572119877119901119902)] (B1)
If 120572119879 = 120572119902 and 120572119877 = 120572119901 that is the antenna array and itsvelocity have the same directions (28) can be simplified as bysubstituting (B1) into (28)
where 1198960 = 21205871198910119888 is the wave numberIn isotropic scattering environment we use the uniform
distribution in (31) to describe all the random azimuth anglesIn this case (B2) can be written as119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879] cos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877] cos (120572119877 minus 120572119877119901119902)
times1198690 [21198960]1119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times1198690 [21198960]119898119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times119898minus1prod119894=2
1198690 [41198960]119894119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind
The AOD 120572119879119901119902 and AOA 120572119877119901119902 are uniformly and indepen-dently distributed so120572119879119901119902minus120572119877119901119902 can be equal to zero in average
As a consequence the term cos[(120572119879119901119902 minus120572119877119901119902)2] in (B3) can beapproximated by one [13 14] and an approximate ST-CF canbe written as from (B3)
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879]sdotcos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877]sdot cos (120572119877 minus 120572119877119901119902) 1198690 (21198960]1119901119902120591)times1198690 (21198960]119898119901119902120591)119898minus1prod
Eventually the approximate ST-CF in (34) can be derivedfrom (B4) according to the definition of 1198690(∙)Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was funded by the National Natural Science Foun-dation of China (no 91438104 no 61501065 no 61571069and no 61601067) the Fundamental Research Funds forthe Central Universities (no 106112016CDJX160001) and theChongqing Research Program of Basic Research and FrontierTechnology (no CSTC2016JCYJA0021)
References
[1] A Bazzi B M Masini A Zanella and I Thibault ldquoBeaconingfrom connected vehicles IEEE 80211p vs LTE-V2Vrdquo in Pro-ceedings of 2016 IEEE 27th Annual International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRC)pp 1ndash6 Valencia Spain September 2016
[2] VM Rodrigo-Penarrocha J Reig L Rubio and et al ldquoAnalysisof small- scale fading distributions in vehicle-to-vehicle com-municationsrdquoMobile Information Systems vol 2016 Article ID9584815 7 pages 2016
[3] A Bazzi B M Masini A Zanella and G Pasolini ldquoIEEE80211p for cellular offloading in vehicular sensor networksrdquoComputer Communications vol 60 pp 97ndash108 2015
[4] X Cheng MIMO channel modelling and simulation for cellu-lar and mobile-to-mobile communication systems[Phd thesis]Heriot-Watt University 2009
[5] N Avazov and M Patzold ldquoA geometric street scatteringchannel model for car-to-car communication systemsrdquo inProceedings of 4th Annual International Conference onAdvanced
14 Mobile Information Systems
Technologies for Communications ATC2011 pp 224ndash230 vnmAugust 2011
[6] X Zhao X Liang S Li and K Haneda ldquoMobile-to-MobileWideband MIMO Channel Realization by Using a Two-RingGeometry-Based Stochastic Scattering Modelrdquo Wireless Per-sonal Communications vol 84 no 4 pp 2445ndash2465 2015
[7] M Walter D Shutin and U-C Fiebig ldquoDelay-dependentdoppler probability density functions for vehicle-to-vehiclescatter channelsrdquo IEEE Transactions on Antennas and Propaga-tion vol 62 no 4 pp 2238ndash2249 2014
[8] P T Samarasinghe T A Lamahewa T D Abhayapala and RA Kennedy ldquo3D mobile-to-mobile wireless channel modelrdquo inProceedings of 2010 Australian Communications Theory Work-shop AusCTW 2010 pp 30ndash34 aus February 2010
[9] M Riaz S J Nawaz and N M Khan ldquo3D ellipsoidal model formobile-to-mobile radio propagation environmentsrdquo WirelessPersonal Communications vol 72 no 4 pp 2465ndash2479 2013
[10] J Chen and T G Pratt ldquoThree-dimensional geometry-based stochastic modeling and performance of 4times4 space-polarization mobile-to-mobile wideband MIMO channelsrdquo inProceedings of 2013 IEEE Global Communications ConferenceGLOBECOM 2013 pp 3936ndash3941 usa December 2013
[11] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA Novel 3DNonstationary Channel Model Based on the von Mises-FisherScattering DistributionrdquoMobile Information Systems vol 2016Article ID 2161460 2016
[12] A Chelli and M Patzold ldquoThe impact of fixed and movingscatterers on the statistics of MIMO vehicle-to-vehicle chan-nelsrdquo in Proceedings of VTC Spring 2009 - IEEE 69th VehicularTechnology Conference esp April 2009
[13] A Borhani and M Patzold ldquoModeling of vehicle-to-vehiclechannels in the presence of moving scatterersrdquo in Proceedingsof 2012 IEEE Vehicular Technology Conference (VTC Fall) pp1ndash5 Quebec City QC Canada September 2012
[14] A Borhani andM Patzold ldquoCorrelation and spectral propertiesof vehicle-to-vehicle channels in the presence ofmoving scatter-ersrdquo IEEE Transactions on Vehicular Technology vol 62 no 9pp 4228ndash4239 2013
[15] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the IEEE 74th Vehicular TechnologyConference (VTC Fall rsquo11) pp 1ndash6 IEEE San Francisco CalifUSA September 2011
[16] M D Soltani M Alimadadi Y Seyedi and H AmindavarldquoModeling of doppler spectrum in V2V urban canyon oncom-ing environmentrdquo in Proceedings of 2014 7th InternationalSymposium on Telecommunications IST 2014 pp 1155ndash1160 irnSeptember 2014
[17] M D Soltani M Alimadadi and A Mohammadi ldquoModelingof mobile scatterer clusters for doppler spectrum in widebandvehicle-to-vehicle communication channelsrdquo IEEE Communi-cations Letters vol 18 no 4 pp 628ndash631 2014
[18] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014
[19] A G Zajic ldquoModeling impact of moving scatterers on dopplerspectrum in wideband vehicle-to-vehicle channelsrdquo in Proceed-ings of Proc Eur Conf Antennas Propag p 1 Lisbon May 2015
[20] J Zhang X Yin and X Cheng ldquoTheoretical analysis andmeasurements Doppler spectra of vehicular communication
channelsrdquo in Proceedings of 2012 12th International Confer-ence on ITS Telecommunications ITST 2012 pp 98ndash102 twnNovember 2012
[21] M Picone S Busanelli M Amoretti F Zanichelli and GFerrari Advanced Technologies for Intelligent TransportationSystems Springer 2015
[22] V H Pham M H Taieb J Y Chouinard S Roy and H THuynh ldquoOn the double Doppler effect generated by scatterermotionrdquo REV Journal on Electronics and Communications vol1 pp 30ndash37 2011
[23] S Roy H T Huynh and P Fortier ldquoCompound Dopplerspread effects of subscriber motion and scatterer motionrdquo AEU- International Journal of Electronics and Communications vol57 no 4 pp 237ndash246 2003
[24] J B Andersen J O Nielsen G F Pedersen G Bauch and GDietl ldquoDoppler spectrum from moving scatterers in a randomenvironmentrdquo IEEE Transactions on Wireless Communicationsvol 8 no 6 pp 3270ndash3277 2009
[25] H S Rad S Gazor and P Shariatpanahi ldquoNon-fixed scatterersand their effects on MIMOmulticarrier fading communicationchannelsrdquo in Proceedings of 50th Annual IEEE Global Telecom-munications Conference GLOBECOM2007 pp 3765ndash3769 usaNovember 2007
[26] N Naz and D Falconer ldquoTemporal variations characterizationfor fixed wireless at 295 GHzrdquo in Proceedings of 2000 IEEE51st Vehicular Technology Conference Proceedings VTC2000-Springer pp 2178ndash2182 Tokyo Japan 2000
[27] A G Zajic G L Stuber T G Pratt and S T NguyenldquoWideband MIMO mobile-to-mobile channels Geometry-based statistical modeling with experimental verificationrdquo IEEETransactions on Vehicular Technology vol 58 no 2 pp 517ndash5342009
[28] W C Jakes Microwave Mobile Communications Wiley-IEEEPress Piscataway NJ USA 1994
[29] M Patzold B O Hogstad and N Youssef ldquoModeling analysisand simulation of MIMO mobile-to-mobile fading channelsrdquoIEEE Transactions onWireless Communications vol 7 no 2 pp510ndash520 2008
[30] M PatzoldMobile Radio Channels Wiley Chichester UK 2ndedition 2011
[31] R Wang and D Cox ldquoDouble mobility mitigates fading in adhoc wireless networksrdquo in Proceedings of IEEE Antennas andPropagation Society International Symposium pp 306ndash309 SanAntonio TX USA
[32] G Bakhshi K Shahtalebi and H S Rad ldquoA novel full-three-dimensional MIMOmobile-to-mobile channel referencemodelrdquo in Proceedings of 3rd International Conference on SignalProcessing and Communication Systems ICSPCSrsquo2009 usaSeptember 2009
[33] W Zhou X Wang X Wang and W Chen ldquoA modified two-erose- ringmodel formimomobile-to-mobile fading channelsrdquoin Proceedings of 7th International Conference on WirelessCommunications Networking and Mobile Computing WiCOM2011 5 1 pages Wuhan China September 2011
= 1198910119888 [V119879 cos (120572119879 minus 120572119879119901119902) + V119877 cos (120572119877 minus 120572119877119901119902)
+ ]1119901119902 cos (1205721119901119902 minus 120572119879119901119902) + 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119879119901119902)+ ]119898119901119902 cos (120572119898119901119902 minus 120572119879119901119902) + ]1119901119902 cos (1205721119901119902 minus 120572119877119901119902)+ 2119898minus1sum119894=2
]119894119901119902 cos (120572119894119901119902 minus 120572119877119901119902) + ]119898119901119902 cos (120572119898119901119902 minus 120572119877119901119902)] (B1)
If 120572119879 = 120572119902 and 120572119877 = 120572119901 that is the antenna array and itsvelocity have the same directions (28) can be simplified as bysubstituting (B1) into (28)
where 1198960 = 21205871198910119888 is the wave numberIn isotropic scattering environment we use the uniform
distribution in (31) to describe all the random azimuth anglesIn this case (B2) can be written as119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879] cos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877] cos (120572119877 minus 120572119877119901119902)
times1198690 [21198960]1119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times1198690 [21198960]119898119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
times119898minus1prod119894=2
1198690 [41198960]119894119901119902120591 cos(120572119879119901119902 minus 1205721198771199011199022 )]
where 1198690(∙) denotes the zeroth-order Bessel function of thefirst kind
The AOD 120572119879119901119902 and AOA 120572119877119901119902 are uniformly and indepen-dently distributed so120572119879119901119902minus120572119877119901119902 can be equal to zero in average
As a consequence the term cos[(120572119879119901119902 minus120572119877119901119902)2] in (B3) can beapproximated by one [13 14] and an approximate ST-CF canbe written as from (B3)
119877119898119901119902119901119902 (119889119879 119889119877 120591)= 120590211989821205872 int sdot sdot sdot intintint exp 1198951198960 [V119879120591 + (119902 minus 119902) 119889119879]sdotcos (120572119879 minus 120572119879119901119902)+1198951198960 [V119877120591 + (119901 minus 119901) 119889119877]sdot cos (120572119877 minus 120572119877119901119902) 1198690 (21198960]1119901119902120591)times1198690 (21198960]119898119901119902120591)119898minus1prod
Eventually the approximate ST-CF in (34) can be derivedfrom (B4) according to the definition of 1198690(∙)Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This work was funded by the National Natural Science Foun-dation of China (no 91438104 no 61501065 no 61571069and no 61601067) the Fundamental Research Funds forthe Central Universities (no 106112016CDJX160001) and theChongqing Research Program of Basic Research and FrontierTechnology (no CSTC2016JCYJA0021)
References
[1] A Bazzi B M Masini A Zanella and I Thibault ldquoBeaconingfrom connected vehicles IEEE 80211p vs LTE-V2Vrdquo in Pro-ceedings of 2016 IEEE 27th Annual International Symposium onPersonal Indoor and Mobile Radio Communications (PIMRC)pp 1ndash6 Valencia Spain September 2016
[2] VM Rodrigo-Penarrocha J Reig L Rubio and et al ldquoAnalysisof small- scale fading distributions in vehicle-to-vehicle com-municationsrdquoMobile Information Systems vol 2016 Article ID9584815 7 pages 2016
[3] A Bazzi B M Masini A Zanella and G Pasolini ldquoIEEE80211p for cellular offloading in vehicular sensor networksrdquoComputer Communications vol 60 pp 97ndash108 2015
[4] X Cheng MIMO channel modelling and simulation for cellu-lar and mobile-to-mobile communication systems[Phd thesis]Heriot-Watt University 2009
[5] N Avazov and M Patzold ldquoA geometric street scatteringchannel model for car-to-car communication systemsrdquo inProceedings of 4th Annual International Conference onAdvanced
14 Mobile Information Systems
Technologies for Communications ATC2011 pp 224ndash230 vnmAugust 2011
[6] X Zhao X Liang S Li and K Haneda ldquoMobile-to-MobileWideband MIMO Channel Realization by Using a Two-RingGeometry-Based Stochastic Scattering Modelrdquo Wireless Per-sonal Communications vol 84 no 4 pp 2445ndash2465 2015
[7] M Walter D Shutin and U-C Fiebig ldquoDelay-dependentdoppler probability density functions for vehicle-to-vehiclescatter channelsrdquo IEEE Transactions on Antennas and Propaga-tion vol 62 no 4 pp 2238ndash2249 2014
[8] P T Samarasinghe T A Lamahewa T D Abhayapala and RA Kennedy ldquo3D mobile-to-mobile wireless channel modelrdquo inProceedings of 2010 Australian Communications Theory Work-shop AusCTW 2010 pp 30ndash34 aus February 2010
[9] M Riaz S J Nawaz and N M Khan ldquo3D ellipsoidal model formobile-to-mobile radio propagation environmentsrdquo WirelessPersonal Communications vol 72 no 4 pp 2465ndash2479 2013
[10] J Chen and T G Pratt ldquoThree-dimensional geometry-based stochastic modeling and performance of 4times4 space-polarization mobile-to-mobile wideband MIMO channelsrdquo inProceedings of 2013 IEEE Global Communications ConferenceGLOBECOM 2013 pp 3936ndash3941 usa December 2013
[11] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA Novel 3DNonstationary Channel Model Based on the von Mises-FisherScattering DistributionrdquoMobile Information Systems vol 2016Article ID 2161460 2016
[12] A Chelli and M Patzold ldquoThe impact of fixed and movingscatterers on the statistics of MIMO vehicle-to-vehicle chan-nelsrdquo in Proceedings of VTC Spring 2009 - IEEE 69th VehicularTechnology Conference esp April 2009
[13] A Borhani and M Patzold ldquoModeling of vehicle-to-vehiclechannels in the presence of moving scatterersrdquo in Proceedingsof 2012 IEEE Vehicular Technology Conference (VTC Fall) pp1ndash5 Quebec City QC Canada September 2012
[14] A Borhani andM Patzold ldquoCorrelation and spectral propertiesof vehicle-to-vehicle channels in the presence ofmoving scatter-ersrdquo IEEE Transactions on Vehicular Technology vol 62 no 9pp 4228ndash4239 2013
[15] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the IEEE 74th Vehicular TechnologyConference (VTC Fall rsquo11) pp 1ndash6 IEEE San Francisco CalifUSA September 2011
[16] M D Soltani M Alimadadi Y Seyedi and H AmindavarldquoModeling of doppler spectrum in V2V urban canyon oncom-ing environmentrdquo in Proceedings of 2014 7th InternationalSymposium on Telecommunications IST 2014 pp 1155ndash1160 irnSeptember 2014
[17] M D Soltani M Alimadadi and A Mohammadi ldquoModelingof mobile scatterer clusters for doppler spectrum in widebandvehicle-to-vehicle communication channelsrdquo IEEE Communi-cations Letters vol 18 no 4 pp 628ndash631 2014
[18] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014
[19] A G Zajic ldquoModeling impact of moving scatterers on dopplerspectrum in wideband vehicle-to-vehicle channelsrdquo in Proceed-ings of Proc Eur Conf Antennas Propag p 1 Lisbon May 2015
[20] J Zhang X Yin and X Cheng ldquoTheoretical analysis andmeasurements Doppler spectra of vehicular communication
channelsrdquo in Proceedings of 2012 12th International Confer-ence on ITS Telecommunications ITST 2012 pp 98ndash102 twnNovember 2012
[21] M Picone S Busanelli M Amoretti F Zanichelli and GFerrari Advanced Technologies for Intelligent TransportationSystems Springer 2015
[22] V H Pham M H Taieb J Y Chouinard S Roy and H THuynh ldquoOn the double Doppler effect generated by scatterermotionrdquo REV Journal on Electronics and Communications vol1 pp 30ndash37 2011
[23] S Roy H T Huynh and P Fortier ldquoCompound Dopplerspread effects of subscriber motion and scatterer motionrdquo AEU- International Journal of Electronics and Communications vol57 no 4 pp 237ndash246 2003
[24] J B Andersen J O Nielsen G F Pedersen G Bauch and GDietl ldquoDoppler spectrum from moving scatterers in a randomenvironmentrdquo IEEE Transactions on Wireless Communicationsvol 8 no 6 pp 3270ndash3277 2009
[25] H S Rad S Gazor and P Shariatpanahi ldquoNon-fixed scatterersand their effects on MIMOmulticarrier fading communicationchannelsrdquo in Proceedings of 50th Annual IEEE Global Telecom-munications Conference GLOBECOM2007 pp 3765ndash3769 usaNovember 2007
[26] N Naz and D Falconer ldquoTemporal variations characterizationfor fixed wireless at 295 GHzrdquo in Proceedings of 2000 IEEE51st Vehicular Technology Conference Proceedings VTC2000-Springer pp 2178ndash2182 Tokyo Japan 2000
[27] A G Zajic G L Stuber T G Pratt and S T NguyenldquoWideband MIMO mobile-to-mobile channels Geometry-based statistical modeling with experimental verificationrdquo IEEETransactions on Vehicular Technology vol 58 no 2 pp 517ndash5342009
[28] W C Jakes Microwave Mobile Communications Wiley-IEEEPress Piscataway NJ USA 1994
[29] M Patzold B O Hogstad and N Youssef ldquoModeling analysisand simulation of MIMO mobile-to-mobile fading channelsrdquoIEEE Transactions onWireless Communications vol 7 no 2 pp510ndash520 2008
[30] M PatzoldMobile Radio Channels Wiley Chichester UK 2ndedition 2011
[31] R Wang and D Cox ldquoDouble mobility mitigates fading in adhoc wireless networksrdquo in Proceedings of IEEE Antennas andPropagation Society International Symposium pp 306ndash309 SanAntonio TX USA
[32] G Bakhshi K Shahtalebi and H S Rad ldquoA novel full-three-dimensional MIMOmobile-to-mobile channel referencemodelrdquo in Proceedings of 3rd International Conference on SignalProcessing and Communication Systems ICSPCSrsquo2009 usaSeptember 2009
[33] W Zhou X Wang X Wang and W Chen ldquoA modified two-erose- ringmodel formimomobile-to-mobile fading channelsrdquoin Proceedings of 7th International Conference on WirelessCommunications Networking and Mobile Computing WiCOM2011 5 1 pages Wuhan China September 2011
Technologies for Communications ATC2011 pp 224ndash230 vnmAugust 2011
[6] X Zhao X Liang S Li and K Haneda ldquoMobile-to-MobileWideband MIMO Channel Realization by Using a Two-RingGeometry-Based Stochastic Scattering Modelrdquo Wireless Per-sonal Communications vol 84 no 4 pp 2445ndash2465 2015
[7] M Walter D Shutin and U-C Fiebig ldquoDelay-dependentdoppler probability density functions for vehicle-to-vehiclescatter channelsrdquo IEEE Transactions on Antennas and Propaga-tion vol 62 no 4 pp 2238ndash2249 2014
[8] P T Samarasinghe T A Lamahewa T D Abhayapala and RA Kennedy ldquo3D mobile-to-mobile wireless channel modelrdquo inProceedings of 2010 Australian Communications Theory Work-shop AusCTW 2010 pp 30ndash34 aus February 2010
[9] M Riaz S J Nawaz and N M Khan ldquo3D ellipsoidal model formobile-to-mobile radio propagation environmentsrdquo WirelessPersonal Communications vol 72 no 4 pp 2465ndash2479 2013
[10] J Chen and T G Pratt ldquoThree-dimensional geometry-based stochastic modeling and performance of 4times4 space-polarization mobile-to-mobile wideband MIMO channelsrdquo inProceedings of 2013 IEEE Global Communications ConferenceGLOBECOM 2013 pp 3936ndash3941 usa December 2013
[11] Y Bi J Zhang M Zeng M Liu and X Xu ldquoA Novel 3DNonstationary Channel Model Based on the von Mises-FisherScattering DistributionrdquoMobile Information Systems vol 2016Article ID 2161460 2016
[12] A Chelli and M Patzold ldquoThe impact of fixed and movingscatterers on the statistics of MIMO vehicle-to-vehicle chan-nelsrdquo in Proceedings of VTC Spring 2009 - IEEE 69th VehicularTechnology Conference esp April 2009
[13] A Borhani and M Patzold ldquoModeling of vehicle-to-vehiclechannels in the presence of moving scatterersrdquo in Proceedingsof 2012 IEEE Vehicular Technology Conference (VTC Fall) pp1ndash5 Quebec City QC Canada September 2012
[14] A Borhani andM Patzold ldquoCorrelation and spectral propertiesof vehicle-to-vehicle channels in the presence ofmoving scatter-ersrdquo IEEE Transactions on Vehicular Technology vol 62 no 9pp 4228ndash4239 2013
[15] A Chelli and M Patzold ldquoA non-stationary MIMO vehicle-to-vehicle channel model derived from the geometrical streetmodelrdquo in Proceedings of the IEEE 74th Vehicular TechnologyConference (VTC Fall rsquo11) pp 1ndash6 IEEE San Francisco CalifUSA September 2011
[16] M D Soltani M Alimadadi Y Seyedi and H AmindavarldquoModeling of doppler spectrum in V2V urban canyon oncom-ing environmentrdquo in Proceedings of 2014 7th InternationalSymposium on Telecommunications IST 2014 pp 1155ndash1160 irnSeptember 2014
[17] M D Soltani M Alimadadi and A Mohammadi ldquoModelingof mobile scatterer clusters for doppler spectrum in widebandvehicle-to-vehicle communication channelsrdquo IEEE Communi-cations Letters vol 18 no 4 pp 628ndash631 2014
[18] A G Zajic ldquoImpact of moving scatterers on vehicle-to-vehiclenarrow-band channel characteristicsrdquo IEEE Transactions onVehicular Technology vol 63 no 7 pp 3094ndash3106 2014
[19] A G Zajic ldquoModeling impact of moving scatterers on dopplerspectrum in wideband vehicle-to-vehicle channelsrdquo in Proceed-ings of Proc Eur Conf Antennas Propag p 1 Lisbon May 2015
[20] J Zhang X Yin and X Cheng ldquoTheoretical analysis andmeasurements Doppler spectra of vehicular communication
channelsrdquo in Proceedings of 2012 12th International Confer-ence on ITS Telecommunications ITST 2012 pp 98ndash102 twnNovember 2012
[21] M Picone S Busanelli M Amoretti F Zanichelli and GFerrari Advanced Technologies for Intelligent TransportationSystems Springer 2015
[22] V H Pham M H Taieb J Y Chouinard S Roy and H THuynh ldquoOn the double Doppler effect generated by scatterermotionrdquo REV Journal on Electronics and Communications vol1 pp 30ndash37 2011
[23] S Roy H T Huynh and P Fortier ldquoCompound Dopplerspread effects of subscriber motion and scatterer motionrdquo AEU- International Journal of Electronics and Communications vol57 no 4 pp 237ndash246 2003
[24] J B Andersen J O Nielsen G F Pedersen G Bauch and GDietl ldquoDoppler spectrum from moving scatterers in a randomenvironmentrdquo IEEE Transactions on Wireless Communicationsvol 8 no 6 pp 3270ndash3277 2009
[25] H S Rad S Gazor and P Shariatpanahi ldquoNon-fixed scatterersand their effects on MIMOmulticarrier fading communicationchannelsrdquo in Proceedings of 50th Annual IEEE Global Telecom-munications Conference GLOBECOM2007 pp 3765ndash3769 usaNovember 2007
[26] N Naz and D Falconer ldquoTemporal variations characterizationfor fixed wireless at 295 GHzrdquo in Proceedings of 2000 IEEE51st Vehicular Technology Conference Proceedings VTC2000-Springer pp 2178ndash2182 Tokyo Japan 2000
[27] A G Zajic G L Stuber T G Pratt and S T NguyenldquoWideband MIMO mobile-to-mobile channels Geometry-based statistical modeling with experimental verificationrdquo IEEETransactions on Vehicular Technology vol 58 no 2 pp 517ndash5342009
[28] W C Jakes Microwave Mobile Communications Wiley-IEEEPress Piscataway NJ USA 1994
[29] M Patzold B O Hogstad and N Youssef ldquoModeling analysisand simulation of MIMO mobile-to-mobile fading channelsrdquoIEEE Transactions onWireless Communications vol 7 no 2 pp510ndash520 2008
[30] M PatzoldMobile Radio Channels Wiley Chichester UK 2ndedition 2011
[31] R Wang and D Cox ldquoDouble mobility mitigates fading in adhoc wireless networksrdquo in Proceedings of IEEE Antennas andPropagation Society International Symposium pp 306ndash309 SanAntonio TX USA
[32] G Bakhshi K Shahtalebi and H S Rad ldquoA novel full-three-dimensional MIMOmobile-to-mobile channel referencemodelrdquo in Proceedings of 3rd International Conference on SignalProcessing and Communication Systems ICSPCSrsquo2009 usaSeptember 2009
[33] W Zhou X Wang X Wang and W Chen ldquoA modified two-erose- ringmodel formimomobile-to-mobile fading channelsrdquoin Proceedings of 7th International Conference on WirelessCommunications Networking and Mobile Computing WiCOM2011 5 1 pages Wuhan China September 2011
