Abstract In this paper, a three-dimensional (3-D) geometric model is considered to simul-taneously describe angle of arrival (AoA) of multipath waves in the azimuth and elevationplanes. The model is suitable in a macrocellular environment with a low MS antenna and anelevated base station (BS), where scatterers are distributed in a cylinder with the radius of thecell and the height of the BS. Closed-form expressions for the probability density functions inthe angles are provided as seen from the mobile station (MS). Results show that the azimuthAoA depends on the ratio of the distance between the BS and the MS to the radius of the cellwhereas the elevation AoA depends on the height of the BS, the radius of the cell and thedistance between the BS and the MS. Monte-Carlo simulations are performed to examinethe accuracy of the proposed model.
Keywords Angle of Arrival · 3-D · Localization · Statistics
1 Introduction
Spatial channel models that describe the angle of arrival (AoA) and time of arrival statis-tics of the multipath components are very important for the performance evaluation of amobile communication system using antenna arrays. Therefore, it is necessary to have radiopropagation models that provide the AoA multipath components. Various existing geometric
X. Zhu (B) · H. Zhu · Q. ZhuWireless Communication Key Lab of Jiangsu Province, Nanjing University of Postsand Telecommunications, Nanjing, Chinae-mail: [email protected]
Y. WangInstitute of Science, PLA University of Science and Technology, Nanjing, China
123
848 X. Zhu et al.
models, such as the ring model [1], circular scattering model [2], elliptical scattering model[3], Gaussian scatterer density model [4], etc., have been proposed to meet the requirements.However, these models are all two-dimensional (2-D) in nature. The 2-D models are adequateto describe the angular spreading of the incoming multipath waves in the azimuth plane butthey fail to describe any signal variations in the elevation plane. However, in order to meet thechallenges of modern wireless communication systems, knowing the vertical spreading ofthe signals is particularly important for some scenarios such as 3-D localization and preciseco-channel interference calculation both of which need to consider the statistics of AoA atthe mobile station (MS) or base station (BS). As far as 3-D models, Qu and Yeap proposed ageneral and accurate 3-D model in [5]. They give analytical solutions of the power spectraldensity (psd) of the received signal in the three dimensions and derive a new expressionwhich directly relates the pdf of the AoA of the incoming multipaths in the elevation planeto the psd of the received signal. However, the complexity and computational cost of thisproposed model are considerable when used in some scenarios such as real-time or withlimited computational power. The 3-D geometrically based statistical models are proposedin [6–8]. A uniform distribution of scatterers around the MS on a spheroid surface in [6], or ahemispheroid surface in [7] is assumed, where the closed-form probability density function(pdf) expressions are derived. In [8], a 3-D geometric scattering model for the uplink ofa macrocell mobile environment is proposed, where the scatterers are confined to a circlearound the mobile. Explicit closed-form expressions are derived for the statistics of the AoAof the multipaths in the azimuth and elevation planes. Therefore, all of these models assumethat the scatterers are uniformly distributed within a circle, on a spheroid, or a hemispheroidsurface of a radius R0 around the mobile, and the BS is outside the assumed models. And allassume that each multipath component of the propagation signal undergoes only one bouncetravelling from the transmitter to the receiver. Hence, the selection of the region of the scatter-ers is a critical step in using these models. However, no particular method for determinationof R0 is introduced and only a simple assumption is made, namely, an arbitrary value of R0
around the MS.The purpose of this paper is to derive a simple 3-D AoA model for a macro-cellular envi-
ronment. The scatterers are assumed to be uniformly distributed within the cylinder with aradius R and height h, where R is the coverage radius of the BS and h is the height differencebetween the BS and the MS antennas. The joint pdf of AoA of the multipath components atthe MS is derived. Monte-Carlo simulations confirm the validity of our proposal.
2 Theory and Formulation
Consider a macrocellular environment where a low MS antenna, surrounded by buildingsand other obstacles, communicates with a high BS antenna. The system geometry is depictedin Fig. 1. The BS is marked as B, its projection onto the azimuth plane is B′, and the MS ismarked as M. The azimuth distance between BS and MS is D. Although only one scatterer,marked as P, and its projection onto the azimuth plane marked as P′, are shown in the figure,we assume that there are many scatterers uniformly distributed within the scattering cylinderformed by the coverage radius R of the BS and the height difference h between the BS andthe MS. For analytical simplicity, the MS antenna may be assumed to be located at groundlevel. The scatterers include ground, rooftops, building sides, trees, etc.
Thus, the AoA of the signal from P to M can be expressed as the azimuth angle ϕ
(i.e.� P′MO′) and the elevation angle θ (i.e. � P′MP). Next we will derive the pdf of AoA.
123
Angle of Arrival Statistics for a 3-D Cylinder Model 849
Fig. 1 Scatterers distributeduniformly within the cylinder
O′
OB
B′M
θ ϕ
P
P′r
D R
2.1 The pdf of ϕ
In this paper, it is assumed that the BS is transmitting and the MS is receiving. Since thescatterers are uniformly distributed and each multipath component from the transmitter toreceiver is assumed to undergo only one bounce, the area with the shadowed strip as shownin Fig. 1 in the azimuth plane is proportional to the probability of the AoA at the MS. Thecumulative distribution function (cdf) of the AoA in the azimuth plane can be expressed as
F (ϕ) =∫ ϕ
0
(D cos x +
√R2 − D2 sin2 x
)2dx
2π R2 (1)
The pdf of the AoA in the azimuth plane, is the derivative of F(ϕ) with respect to ϕ andis given by
f (ϕ) = dF (ϕ)
d (ϕ)= r2
2π R2 (2)
where r = Dcosϕ +√
R2 − D2 sin2 ϕ.
2.2 The Joint pdf of ϕ and θ
In the 3-D model, the AoA of the incoming multipath at the MS includes both spherical polarangles ϕ and θ . The joint pdf f (ϕ, θ ) is written in terms of the conditional pdf of θ for agiven ϕ, f (θ |ϕ), and the marginal pdf f (ϕ). Thus
f (ϕ, θ) = f (ϕ) f (θ |ϕ) (3)
The AoA depends on two independent variables ϕ and θ . The expression of f (θ |ϕ) isindependent of ϕ but the region where f (θ |ϕ) �= 0 depends on it (see Fig. 1). Now we willderive the distribution function of θ for a given ϕ, i.e.F(θ |ϕ). For a given ϕ, the scatterersare inside polyhedron MOC-M′O′C′, denoted as polyhedron V, as shown in Fig. 2a. For agiven θ , the conical surface, denoted by surface S, is formed by revolving around MM′ withtip at the MS and rotation angle π
2 − θ . Note that the intersection lines between surface S andpolyhedron V vary with different θ that can vary from 0 to π
2 . When θ is a smaller value, theintersection lines between surface S and polyhedron V are line PM, arc PE and line EM, asshown in Fig. 2a. The probability of the scatterers inside in polyhedron PCOE-M is the ratiobetween VPCOE-M and VMOC-M′O′C′ , where VPCOE-M and VMOC-M′O′C′ are the volumes ofpolyhedrons PCOE-M and MOC-M′O′C′. Note that VPCOE-M depends on θ . Hence, θ needsto be divided into several segments when to derive its pdf.
123
850 X. Zhu et al.
O′
O
B
B′M
θϕ
P
M'
E
C'
C1P2P
3P4P
O′
O
B
B′M
θ
ϕ
PM'
C'
C
O′
O
B
B′M
θ0ϕ
P
M' E
C'
C
K
K'
O′
O
B
B′M
θϕ
P
M' E
C'
C
O ′
O
B
B′M
θϕ
PM' E
C'
C
(a) (b) (c)
(d) (e)
Fig. 2 Angles of Arrival in the azimuth and elevation planes, where −π ≤ ϕ ≤ π ,a 0≤ θ ≤ arctan hR+D ,
b θ = arctan hR+D , c arctan h
R+D < θ < arctan hrϕmax
, d θ = arctan hrϕmax
, and e arctan hrϕmax
< θ < π2
Referring to Fig. 2a, when θ is becoming larger from 0, points E and P will rise untilpoint E arrives at point O′, as shown in Fig. 2b. Since for a certain θ the length of linesegment MO is always longer than that of line segment MC, the length of line segment OEis also always longer than that of line segment CP. Hence, with the increasing of θ point Ewill earlier arrive at point O′, where θ = arctan h
D+R , as shown in Fig. 2b. Therefore, when
0 ≤ θ ≤ arctan hD+R , F (θ |ϕ ) = VPCOE-M
VMOC-M′O′C′ , where
VMOC-M′O′C′ =∫ ϕ
0 r2dx
2· h (4)
In order to calculate VPCOE-M, we first derive VP1P2P3P4-M = 13 SP1P2P3P4-M × r , where
SP1P2P3P4-M = r tanθ · rdϕ. Then
VPCOE-M = 1
3tan θ
ϕ∫
0
r2dx (5)
123
Angle of Arrival Statistics for a 3-D Cylinder Model 851
Therefore,
F (θ |ϕ ) = VPCOE-M
VMOC-M′O′C′= 2 tan θ
∫ ϕ
0 r3dx
3h∫ ϕ
0 r2dx− π ≤ ϕ ≤ π, 0 ≤ θ ≤ arctan
h
R + D
(6)
Then,
F (θ, ϕ) = F (ϕ) F (θ |ϕ) = tan θ
3π R2h
ϕ∫
0
r3dx − π ≤ ϕ ≤ π, 0 ≤ θ ≤ arctanh
R + D
(7)
Hence, the joint pdf of ϕ and θ can be derived as
f (θ, ϕ) = ∂2 F (θ, ϕ)
∂ϕ∂θ= r3 sec2 θ
3π R2h− π ≤ ϕ ≤ π, 0 ≤ θ ≤ arctan
h
R + D(8)
When θ continues to become larger from arctan hD+R , the intersection lines between sur-
face S and polyhedron V are line MP, arc PK, arc KE and line EM, as shown in Fig. 2c.Point P will continue to rise when θ becomes larger, until point P overlaps with point C′,as shown in Fig. 2d, where θ = arctan h
rϕmaxand rϕmax = |MC|for a given ϕmax. Therefore,
when arctan hR+D ≤ θ ≤ arctan h
rϕmax, as shown in Fig. 2c, F (θ |ϕ) = VO′EK-OMK′+VPKK′C-M
VMOC-MOC,
where
VO′EK-OMK′ = VMOK′M′O′K − VM′KE-M = h
2
ϕ0∫
0
r2dϕ − 1
6hϕ0
(h
tan θ
)2
(9)
and
VPKK′C-M = 1
3tan θ
ϕmax∫
ϕ0
r3dx (10)
where r (ϕ0) = htan θ
. Therefore,
F (θ, ϕ) = F (ϕ) F (θ |ϕ) =h2
∫ ϕ00 r2dϕ0 − 1
6 hϕ0( h
tan θ
)2 + 13 tan θ
∫ ϕmaxϕ0
r3dx
πhr2
−π ≤ ϕ ≤ π, arctanh
R + D≤ θ ≤ arctan
h
rϕmax
(11)
and then
f (θ |ϕ) = ∂2 F (θ, ϕ)
∂ϕ∂θ= r3
3π R2h cos2 θ− π ≤ ϕ ≤ π, arctan
h
R + D≤ θ ≤ arctan
h
rϕmax
(12)
When θ varies from arctan hrϕmax
to π2 , the intersection lines between Region S and Volume
M are line MP, arc PE and line EM, where arc PE centers at point M′, as shown in Fig. 2e.
123
852 X. Zhu et al.
Therefore, F (θ |ϕ ) = VPC′O′E-MCOVMOC-M′O′C′ , where
VPC′O′E-MCO = VMOC-M′O′C′ − VM′PE-M = h
2
ϕ∫
0
r2dx − ϕh3
6 tan2 θ(13)
Therefore,
F (θ, ϕ) =∫ ϕ
0 r2dx
2π R2 − ϕh2
6π R2 tan2 θ− π ≤ ϕ ≤ π, arctan
h
rϕmax
≤ θ ≤ π
2(14)
and then
f (θ, ϕ) = h2 cos θ
3π R2 sin3 θ− π ≤ ϕ ≤ π, arctan
h
rϕmax
≤ θ ≤ π
2(15)
Therefore, the joint pdf of ϕ and θ can be expressed as
f (ϕ, θ) =
⎧⎪⎪⎨
⎪⎪⎩
= r3 sec2 θ3π R2h
− π ≤ ϕ ≤ π ≤ ϕ ≤ π, 0 ≤ θ ≤ arctan hR+D
= r3
3π R2h cos2 θ− π ≤ ϕ ≤ π, arctan h
R+D < θ ≤ arctan hrϕmax
= h2 cos θ
3π R2 sin3 θ− π ≤ ϕ ≤ π, arctan h
rϕmax< θ ≤ π
2
(16)
where r = Dcosϕ +√
R2 − D2 sin2 ϕ.The pdf marginal f (ϕ) is can be determined by (2). The pdf f (θ ) is obtained by integrating
the joint density f (ϕ, θ ). And the mean and the standard derivation in the azimuth AoA andelevation AoA can be obtained by their marginal pdfs.
3 Numerical Results and Discussions
In order to examine the accuracy of the proposed method, we performed Monte-Carlo sim-ulations for some different values of D, R, and h. The comparisons in Figs. 3 and 4a, b, cshow that our theoretical pdfs match the simulated results very well.
Figure 3 plots the marginal pdf f (ϕ) curves for different values of D/R. From Fig. 3 wecan see that the AoA at the receiver ranges mostly from about −100◦ to 100◦ for large valueof R/D, i.e. the mobile receiver is far from the base station. The abscissas of the intersectionsof the three curves are around 60◦.
The elevation plane pdf was evaluated by computing the integral (16) numerically and theresults are plotted in Fig. 4a, b, c. Figure 4a shows the elevation pdf as a function of elevationangle with h as a parameter. Notice that when D remains unchanged, the erect angular spreadseen at the MS varies with different h. For example, for h = 30 the erect AoA at the receiverranges mostly from about 0◦ to 20◦ whereas from 0◦ to 50◦ for h = 100. The elevation planepdfs of AoA with R and D as a parameter were respectively shown in Fig. 4b, c. Note thatthe elevation plane pdf depends on R, D and h.
Applications of the proposed model in areas such as cellular networks, smart antennasystems, 3-D localization and precise co-channel interference calculation can be developed.In this paper, an application in mobile location estimation is given. The proposed methodmeasures the marginal pdfs of the AoA of the incoming multipaths from the MS. The measure-ment of the maxima of the pdfs in the azimuth and the elevation plane allows the approximateestimation of the mobile position.
123
Angle of Arrival Statistics for a 3-D Cylinder Model 853
-200 -150 -100 -50 0 50 100 150 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
φ (°)
f (
φ)
R/D=1
R/D=2
R/D=4
Monte carlo simulation
Fig. 3 pdfs in the AOA in the azimuth plane as seen at the MS
The major sources of errors of AoA-based localization technique may come from twoaspects: multipath propagation and measurement bias. For simplicity we neglected the mea-surement errors. We performed simulation with various parameters D, R, and h. In thesesimulations, scatterers were uniformly distributed in a cylinder with radius R and height h.Without loss of generality, R was normalized to unity. Each snapshot was taken for a differentset of D/R and h/R. These two quantities were uniformly distributed in [0.2, 0.9] and [0.1,0.5], respectively. The θ -angle of the MS was derived from the estimation of the maximumof the f(θ ) under the assumption that the received signal at the BS propagates through asingle Line-of-Sight path. In Fig. 5, the results of 10,000 simulations are presented. Eachsnapshot represents the real, θrea, and the estimated, θest, elevation angle from the BS to MS.The optimum case with no estimation errors (θrea = θest) is also presented. An almost linearrelation between the real and the estimated angle is observed. The accuracy of the methoddecreases with increase of θest. In Table 1, the mean value and variance of D/R, h/R, θrea, θest
are presented. Also, the relative eθ = |(θrea − θest)/θrea| estimation error of the θ -angle,is given. In columns 2–5, the statistics of the simulation parameters are presented. The lastcolumn shows the accuracy of the method.
4 Conclusions
A geometrically based 3-D propagation model for mobile communication systems is derivedin this letter. The AoA at the mobile station is analyzed for a 3-D cylinder model, where thegeometric parameters of the model are the radius R of the cell, the height h of the BS, andthe distance D between the BS and the MS. At the MS, the pdf of AoA in azimuth planedepends on the single parameter R/D whereas the pdf in elevation plane depends on R, Dand h. The model presented here is useful in a macrocellular environment where significantmultipath components can arrive both from horizontal and vertical directions at both the BSsand MSs.
123
854 X. Zhu et al.
0 10 20 30 40 50 60 70 80 900
1
2
3
4
5
6
7
θ (°)
p (
θ)
h=30h=50h=100monte carlo simulation
0 10 20 30 40 50 60 70 80 900
0.5
1
1.5
2
2.5
3
3.5
θ (°)
f (θ
)
R=200R=150R=110monte carlo simulation
0 10 20 30 40 50 60 70 80 900
1
2
3
4
5
6
θ (°)
f ( θ
)
D=100D=20D=10monte carlo simulation
(a)
(b)
(c)
Fig. 4 pdfs of AOA in the elevation plane as seen at the MS a for R = 200, D = 100, and h = 30, 50, 100.b For D = 100, h = 50, R = 110, 150, 200. c For R = 200, h = 30, D = 10, 20, 100
123
Angle of Arrival Statistics for a 3-D Cylinder Model 855
0 5 10 15 20 25 300
5
10
15
20
25
30
35
θrea
(deg)
θ est(d
eg)
Real values
Simulation results
Fig. 5 The proposed model application in mobile location estimation
Table 1 System parameters andsimulation results (angles aremeasured in degrees)
D/R h/R θrea θest eθ
Mean 0.67 0.34 11.92 12.33 0.13
Variance 0.14 0.08 16.02 19.36 0.04
Acknowledgements This work is supported by the Natural Science Foundation of China (No.61001078),National Science and Technology Major Project of the Ministry of Science and Technology of China (No.2011ZX03005-004-03, 2010ZX03003-001-02), China Postdoctoral Science Foundation (2010047065), thePostdoctoral Science Foundation of Jiangsu Province (0902005C), and Natural Science Foundation of theHigher Education Institutions of Jiangsu Province (10KJB510016).
References
1. Jakes, W. C. (1994). Microwave mobile communications. Piscataway, NJ: IEEE Press Classic Reissue.2. Petrus, P., Reed, J. H., & Rappaport, T. S. (2002). Geometrical-based statistical macrocell channel
model for mobile environments. IEEE Transactions Communications, 50(3), 495–502.3. Ertel, R. B., & Reed, J. H. (1999). Angle and time of arrival statistics for circular and elliptical
scattering models. IEEE Journal of Selected Areas Communations, 17, 1829–1840.4. Janaswamy, R. (2002). Angle and time of arrival statistics for the Gaussian scatter density model. IEEE
Transactions Wireless Communnications, 1, 488–497.5. Qu, S., & Yeap, T. (1999). A three-dimensional scattering model for fading channels in land mobile
environment. IEEE Transactions on Vehicular Technology, 48, 765–781.6. Janaswamy, R. (2002). Angle of arrival statistics for a 3-D spheroid model. IEEE Transactions on
Vehicular Technology, 51, 1242–1247.7. Olenko, A. Y., Wong, K. T., Qasmi, S. A., & Ahmadi-Shokouh, J. (2006). Analytically derived
uplink/downlink TOA and 2-D-DOA distributions with scatterers in a 3-D hemispheroid surroundingthe mobile. IEEE Transactions on Antennas and Propagation, 54, 2446–2454.
8. Baltzis, K. B., & Sahalos, J. N. (2009). A simple 3-D geometric channel model for macrocell mobilecommunications. Wireless Personal Communications, 51, 329–347.
123
856 X. Zhu et al.
Author Biographies
Xiaorong Zhu received her Ph.D. degree in wireless communicationsin 2008 from Southeast University, Nanjing, China. She has been apostdoctoral in The Chinese University of Hong Kong in 2008 and2009. Now she is an associate professor of Nanjing University of Postsand Telecommunications. Her research interests include wireless net-works, wireless access technology, as well as cognitive radio commu-nications.
Yong Wang is a lecturer in Department of Science, PLA Universityof Science and Technology, Nanjing, China. His research fields includeinformation theory and technologies, coding theory, and wireless andmobile communications. He has been recently focusing on the broad-band mobile communications including wireless internet technology,broadband wireless access systems, intelligent home networks, wire-less sensor networks and cognitive radio networks.
Hongbo Zhu is a professor at the wireless communication key labof Jiangsu Province, Nanjing University of Posts and Telecommunica-tions. His research fields include information theory and technologies,coding theory, and wireless and mobile communications.
123
Angle of Arrival Statistics for a 3-D Cylinder Model 857
Qi Zhu is a professor at the wireless communication key lab of JiangsuProvince, Nanjing University of Posts and Telecommunications. Herresearch interests include wireless networks, wireless access technol-ogy, as well as cognitive radio communications.
