1 School of Information and Communications Engineering Key Lab of Universal Wireless Communications (UWC)Ministry of Education Beijing University of Posts and Telecommunications Beijing 100876 China
2 School of Electrical and Information Engineering Hunan University Changsha Hunan 410082 China3 School of Electronic Engineering and Computer Science Queen Mary University of London London E1 4NS UK
Correspondence should be addressed to Xing Zhang hszhangbupteducn
Received 20 April 2014 Accepted 28 July 2014 Published 28 August 2014
Academic Editor Shensheng Tang
Copyright copy 2014 Xing Zhang 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
Considering the effect of imperfect channel state information (CSI) we study the performance of a cluster-based cognitivemultihopwireless sensor network with decode-and-forward (DF) partial relay selection over Nakagami-m fading channels The closed-form expressions for the exact outage probability and bit error rate (BER) of the secondary system are derived and validated bysimulations Asymptotic outage analysis in high SNR regime reveals that the diversity order is determined by the minimum fadingseverity parameter of all the secondary transmission links irrespective of theCSI imperfection It is shown that the fading severity ofthe secondary transmission links hasmore influence on the outage performance than that of the interference linksWe also concludethat for secondary nodes whose transmit power is restricted by the interference constraint of the primary user increasing thenumber of relaying hops is an effective way to improve their transmission performance Besides increasing the number of availablerelays in each relay cluster can mitigate the performance degradation caused by CSI imperfection
1 Introduction
Cognitive radio (CR) is a promising technology to improvethe spectrumutilization forwireless systems [1]Theunderlayparadigm in CR has drawn much attention due to itsflexibility [2] By allowing simultaneous transmissions ofsecondary users (SU) and primary users (PU) as long asthe interference caused by SU to PU is below a tolerablethreshold the underlay CR can take full advantage of thespectrum resources However since the transmit power ofSU is strictly limited by the interference power constraintthe secondary transmission often suffers from a short-range drawback To deal with this problem cooperativerelaying techniques [3] are introduced to secondary systemsto extend their coverage area Furthermore two-hop ormultihop sensor (relaying) can help the secondary systemsfor example mobile ad hoc wireless networks and wirelesssensor networks in achieving broader coverage and bettertransmission performance As the combination of cognitive
radio and cooperative communications cognitive dual-hopor multihop networks benefit from both the high spectrumutilization and the cooperative diversity which have beenintensively studied recently [4ndash8]
In [5] the authors investigated the performance ofmultihop cognitive decode-and-forward (DF) networks overRayleigh fading channels In [6] performance for mul-tihop cognitive amplify-and-forward (AF) networks overNakagami-119898 fading was presented Cognitive multihop net-works in the presence of multiple primary receivers withAF and DF protocols were studied in [7 8] respectivelyover Nakagami-119898 fading channels All the aforementionedreferences assumed that only one single relay node is availablein each hop of the secondary transmission When thereare multiple relays available for one-hop transmission relayselection can bring significant performance improvementReference [9] proposed a cluster-based multihop networkwhere relay selection can be applied to multihop trans-mission In [10] performance of a cluster-based multihop
Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2014 Article ID 630823 11 pageshttpdxdoiorg1011552014630823
2 International Journal of Distributed Sensor Networks
sensor network with partial relay selection without spectrumsharing was analyzed Power allocation and relay selectionproblems for cluster-based cognitivemultihop networks werestudied in [11]
In cognitive radio networks channel state information(CSI) plays an important role On one hand to restrict theinterference from SU to PU SU must adjust their transmitpower according to the instantaneous CSI of the interferencelinks between SU and PU On the other hand the relayselection for each hop depends on the instantaneous CSIof the corresponding secondary transmission links Mostexisting works assumed that perfect CSI could be acquired Inpractice however due to channel estimation errors mobilityfeedback delay limited feedback and feedback quantizationthe acquired CSI may sometimes be imperfect In [12 13] theimpact of imperfect CSI of the interference links between SUand PU was studied on the outage performance of cognitivedual-hop networkswithAF andDF protocols respectively In[14 15] the outage performance of cognitive relay networks(CRN) for theNth best relay selection and orthogonal space-time block coding (OSTBC) in Rayleigh channels are consid-ered For multihop relaying the authors in [16] analyzed theperformance ofmultihop cognitiveDFnetworks consideringthe effect of imperfect CSI of the interference links However[12 13 16] all focused on Rayleigh fading channels To thebest of our knowledge there have been no prior works onthe impact of imperfect CSI on cognitive multihop networksin a more general fading environment such as Nakagami-119898 fading environment Motivated by these considerationswe extend our previous research on a cognitive dual-hopnetwork over Rayleigh fading channels in [13] to a cognitivemultihop wireless sensor network over Nakagami-119898 fadingchannels
In this paper we investigate the performance of a cluster-based cognitive multihop wireless sensor network with DFpartial relay selection over Nakagami-119898 fading channelsTheeffect of imperfect CSI of the secondary transmission links istaken into accountWederive the closed-form expressions forthe exact outage probability and bit error rate (BER) and theasymptotic outage probability of the secondary system andvalidate them by simulations The impact of various factors(eg the CSI imperfection the fading severity of all links thenumber of hops and the number of sensorsrelays in clusters)on the performance of the secondary system is analyzed
The remainder of this paper is organized as follows inSection 2 we present the system model for the analysis ofcognitive multihop wireless sensor networks then basedon this model in Section 3 the outage probability andbit-error-rate (BER) are derived over Nakagami-119898 fadingchannels meanwhile the asymptotic outage probability forlarge system SNR is derived to study the diversity order insuch a system in Section 4 simulation results are given andcompared finally we conclude this paper in Section 5
2 System Model
The system model we consider in this paper is illustratedin Figure 1 The secondary system shares the same spectrum
PR
SS SDCluster 1 Cluster 2 Cluster K minus 1
middot middot middot
Interference linksSecondary transmission links
Figure 1 System model
with a primary user in an underlay approach Like moststudies we only consider the existence of the primary receiver(PR) and assume the primary transmitter (PT) is far awayfrom SU so the interference from PT can be ignored Thesecondary system is a cognitive cluster-based multihop sen-sor network whose transmission consists of119870 hops Betweenthe secondary source (SS) and the secondary destination(SD) there are 119870 minus 1 sensor clusters where cluster 119896 (119896 =
1 2 119870 minus 1) contains119873119896 available relay nodes To simplifythe notation we denote SS and SD by cluster 0 and cluster119870respectively So we have119873119870 = 1
Due to the half-duplex mode of the relays the transmis-sion of the 119870 hops happens in 119870 separate time slots Weassume that all channels experience Nakagami-119898 quasistaticfading that is the channel fading coefficients remain constantduring one slot but change independently for every slot Forthe reason that the CSI of the next hops cannot be obtainedin the current hop the relay selection for the current hopmerely depends on the CSI of the current hop To this endpartial relay selection (PRS) can be adopted in the secondarytransmission as a feasible strategy since the relay selectionprocess is based on the channel quality of the current hoponly instead of the end-to-end signal-to-noise ratio (SNR)Besides DF protocol is exploited in the secondary relayswhere the receiver for each hop decodes the signal and thenreencodes it before forwarding it to the next hop
To guarantee PUrsquos communication the maximum toler-able interference power constraint 119876 at PR must be satisfiedIn other words the transmitter in each hop should restrict itstransmit power so that the interference it causes to PR doesnot exceed 119876
In the 119896th hop (119896 = 1 2 119870) the transmitter incluster 119896 minus 1 is denoted by 119905119896 while the119873119896 potential receiversin cluster 119896 (ie the available relays included in cluster 119896)are represented as 119903(119896)
1 119903
(119896)
2 119903
(119896)
119873119896 We denote the channel
fading coefficient of the interference link between transmitter119905119896 and the PR by ℎ119896119901 and the channel fading coefficients ofthe secondary transmission links between transmitter 119905119896 andreceiver 119903(119896)
119894(119894 = 1 2 119873119896) by ℎ119896119894 According to the PRS
protocol the best relay with the maximum received SNR ischosen from the potential receivers 119903(119896)
1 119903
(119896)
2 119903
(119896)
119873119896 as the
transmitter 119905119896+1 in the (119896 + 1)th hop
International Journal of Distributed Sensor Networks 3
For Nakagami-119898 fading the channel power gains 119892119883119884 =
|ℎ119883119884|2(119883 = 119896 119884 = 1 2 119873119896 119901) are independent and non-
identically distributed (INID) random variables followingGamma distribution with fading severity parameters 119898119883119884
(119898119883119884 is positive integer) and mean Ω119883119884 The probabilitydensity function (PDF) and cumulative distributed function(CDF) of 119892119883119884 are given by
119891119892119883119884(119909) =
120573
119898119883119884
119883119884119909119898119883119884minus1
Γ (119898119883119884)
exp (minus120573119883119884119909) (1)
119865119892119883119884(119909) = 1 minus exp (minus120573119883119884119909)
119898119883119884minus1
sum
119899=0
(120573119883119884119909)119899
119899
(2)
respectively where 120573119883119884 = 119898119883119884Ω119883119884 and Γ(sdot) and 120574(sdot sdot)represent the Gamma function [17 (83391)] and the lowerincomplete Gamma function [17 (83501)] We assume thatall relays in the same cluster are relatively centralized so thechannel parameters pertaining to relays in the same clusterare identical that is we have 119898119896119894 = 119898119896 Ω119896119894 = Ω119896 and120573119896119894 = 120573119896 (119894 = 1 2 119873119896) In addition the thermal noiseat each receiver is modeled as independent complex additivewhite Gaussian noise (AWGN) with variance1198730
In this paper we consider the imperfect CSI of thesecondary transmission links Specifically in the 119896th hopthe perfect channel gain of the transmission link betweentransmitter 119905119896 and the potential receiver 119903119894 (119894 = 1 2 119873119896)
and its imperfect counterpart are denoted by 119892119896119894 and 119892119896119894respectively According to [18 equation (9398)] the jointPDF of the perfect channel gain 119892119896119894 and its imperfectcounterpart 119892119896119894 which follows the same distribution as 119892119896119894
where 119868119899(sdot) denotes the 119899th-ordermodified Bessel function ofthe first kind and 120588119896 = Cov(119892119896119894 119892119896119894)radicVar(119892119896119894)Var(119892119896119894) isin
[0 1] represents the correlation coefficient between 119892119896119894 and119892119896119894 120588119896 reflects the degree of CSI imperfection For instance120588119896 = 1 indicates that the CSI is absolutely perfect while 120588119896 = 0represents that the CSI estimation is totally random
3 Exact Performance Analysis
In this section considering the imperfect CSI of the sec-ondary transmission links we derive the exact outage proba-bility and BER of the cognitive multihop sensor network
31 Cumulative Distribution Function of Single-Hop SNRSince the secondary users suffer from low transmit powereach secondary node should try its best to transmit the signalIn the 119896th hop to satisfy the interference power constraint of
PU the transmit power of 119905119896 should be set to 119875119896 = 119876119892119896119901Due to the imperfect CSI of the secondary transmission linksthe imperfect received SNR at receiver 119903(119896)
119894(119894 = 1 2 119873119896)
is calculated as 120574119896119894 = 1198761198921198961198941198730119892119896119901 According to the PRSstrategy the ldquobestrdquo receiver with the maximum imperfectreceived SNR is selected as transmitter 119905119896+1 for the next hopso that the imperfect SNR for the 119896th hop satisfies
120574119896 = max119894=12119873119896
120574119896119894 = max119894=12119873119896
119876119892119896119894
1198730119892119896119901
(4)
The imperfect channel gain between transmitter 119905119896 and theselected receiver 119905119896+1 is given by
where 119892119896 is the actual channel gain between transmitter 119905119896and the selected receiver 119905119896+1 The end-to-end SNR of thesecondary system can be expressed as [10]
1205741198902119890 = min119896=12119870
120574119896 (9)
Next we will first derive the PDF of the actual channelgain for the 119896th hop 119892119896 by the similar approach to [19] From[19 equations (29)-(30)] we have
where 119894 is the imaginary unit and 119869119899(sdot) denotes the 119899th-order Bessel function of the first kind From [17 equation(66434)] (15) can be simplified as
Φ = ℎ120573minusℎminus1
119896(
1
1 minus 120588119896
+ 119895)
minusℎminus119898119896
(
radic120588119896119909
1 minus 120588119896
)
119898119896minus1
times exp(120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
times 119871119898119896minus1
ℎ(minus
120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
(16)
where 119871120572
119899(sdot) represents the associated Laguerre polynomial
By usingRodriguesrsquo formula for119871120572
119899(sdot) [17 equation (89701)]
we can obtain
Φ = ℎ120573119896
minusℎminus1(
1
1 minus 120588119896
+ 119895)
minusℎminus119898119896
(
radic120588119896
1 minus 120588119896
)
119898119896minus1
times 119909(119898119896minus1)2 exp(
120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
times
ℎ
sum
119898=0
(
ℎ + 119898119896 minus 1
ℎ minus 119898)
times
1
119898
(
120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
119898
(17)
By substituting (17) into (14) we get the closed-form expres-sion for the PDF of 119892119896 as follows
119891119892119896(119909)
=
119873119896
Γ (119898119896)
119873119896minus1
sum
119895=0
(
119873119896 minus 1
119895) (minus1)
119895119895
International Journal of Distributed Sensor Networks 5
32 End-to-EndOutage Probability Theoutage probability ofthemultihop secondary systemwhich characterizes the prob-ability that the end-to-end SNR falls below a predeterminedthreshold 120574th is given by
Now we will derive the closed-form expression for119883(119886 119887) By exploiting [17 equation (91311)] we can rewritethe CDF of 120574119896 (22) as follows
[1 + 119895 (1 minus 120588119896)] 120573119896119901120574
)
(29)
Then by substituting (22) into (27) and using the variabletransformation 119909 = (119895 + 1)120573119896120574[1 + 119895(1 minus 120588119896)]120573119896119901120574 we have
119886 [1 + 119895 (1 minus 120588119896)] 120573119896119901120574
(119895 + 1) 120573119896
)
(31)
where 119864() denotes MacRobertrsquos E-Function Since Mac-Robertrsquos E-Function can always be expressed in terms ofMeijerrsquos G-function (which can be computed in MATLAB)we can rewrite (31) as
119886 [1 + 119895 (1 minus 120588119896)] 120573119896119901120574
(119895 + 1) 120573119896
)
(32)
With the help of (32) the single-hop average BER for binarytransmissions (28) can be calculated Then by substitutingit into (25) we can obtain the end-to-end average BER forbinary transmissions
332119872-QAMTransmissions Thesingle-hop averageBERof119872-QAM transmissions is given by [8]
BER119896 =
1
radic119872log2radic119872
log2radic119872
sum
119901=1
V119901
sum
119902=0
120601119902119901radic
120596119902
120587
times int
infin
0
120574minus12
119890minus120596119902120574
119865120574119896(120574) 119889120574
(33)
where V119901 = (1minus2minus119901)radic119872minus1120596119902 = 3(2119902 + 1)
2log2119872(2119872minus2)
and 120601119902119901 = (minus1)lfloor1199022119901minus1
radic119872rfloor(2
119901minus1minus lfloor(1199022
119901minus1radic119872) + (12)rfloor)
From (27) the integral term in (33) can be rewritten as119883(120596119902 12) Thus (33) can be expressed as
BER119896 =
1
radic119872log2radic119872
log2radic119872
sum
119901=1
V119901
sum
119902=0
120601119902119901radic
120596119902
120587
119883(120596119902
1
2
) (34)
With the help of (32) andby substituting (34) into (25) we canget the end-to-end average BER for119872-QAM transmissions
333 119872-PSK Transmissions The BER of 24sdot sdot sdot 119872-PSKtransmissions can be obtained by recursive algorithms basedon the generalized 24-PSK [21] Therefore we only need toderive the BER of 24-PSK transmissions
According to [8] the BER for the most significant bit(MSB) and the least significant bit (LSB) with 24-PSKconstellation through a set of angles (120579 = [1205791 1205792]) are givenby
BER (120574) = 12
erfc (sin 1205792radic120574) (35)
BER (120574) = 12
erfc (cos 1205792radic120574) (36)
respectively
By applying integration by parts we can obtain the single-hop average BER for the MSB with 24-PSK as
BER119896 =
sin 12057922radic120587
int
infin
0
120574minus12
119890minussin21205792120574
119865120574119896(120574) 119889120574
=
sin 12057922radic120587
119883(sin21205792
1
2
)
(37)
The single-hop average BER for the LSB with 24-PSK can beobtained in a similar way
Given the single-hop average BER for the MSB andthe LSB cases the end-to-end average BER for 119872-PSKtransmissions can be written in closed form with the help of(32) and (25)
4 Asymptotic Outage Performance Analysis
In this section to study the diversity performance for thecognitive multihop sensor network we derive the asymptoticoutage probability of the secondary system in high SNRregime
According to the asymptotic behavior of the lower incom-plete Gamma function
By substituting (40) into (24) we can get the asymptoticexpression for the outage probability of the secondary system
Diversity order is an important performance metric forcooperative relay system It is defined as
119889 = minus lim120574rarrinfin
log119875out (120574)log 120574
(41)
From (40) we can see that in high SNR regime
119865120574119896(120574th) prop (
1
120574
)
119898119896
(42)
According to (24) we can conclude that the lowest orderof 1120574 in the outage probability expression is determinedby the minimum among the lowest order of 1120574 in 119865120574119896(120574th)(119896 = 1 2 119870) that is min119896=12119870119898119896 In other words thediversity order of the secondary system is min119896=12119870119898119896This indicates that the diversity order of the secondary systemis merely determined by the fading severity parameters of thesecondary transmission links but not affected by other factorssuch as the fading severity parameters of the interferencelinks and the CSI imperfection
5 Numerical Results
In this section analytical results are presented and validatedby simulations We study the effect of various factors onthe outage and error performance of the cognitive multihopsensor network such as the CSI imperfection the fadingseverity of the interference links and the secondary trans-mission links the number of hops and the number of relays
100
10minus1
10minus2
10minus3
10minus4
10minus5minus5 0 5 10
QN0 (dB)
Out
age p
roba
bilit
y
mk = 1
mk = 3
ExactExactExactExact
AsymptoticAsymptoticAsymptoticAsymptotic
SimulationSimulationSimulationSimulation
mkp = 1 120588k = 1
mkp = 1 120588k = 05
mkp = 3 120588k = 1
mkp = 3 120588k = 05
Figure 2 Exact and asymptotic outage probability of the secondarysystem versus system SNR for various fading severity with 119870 = 3
and119873119896 = 3
in clusters The predetermined SNR threshold for successfuldecoding is set as 120574th = 3 dB
We denote the distance between cluster 119896 minus 1 and cluster119896 by 119889119896 and the distance between cluster 119896 minus 1 and PR by119889119896119901 (119896 = 1 2 119870) To consider the impact of path losswe set Ω119896 = 119889
minus120578
119896and Ω119896119901 = 119889
minus120578
119896119901 where 120578 is the path loss
exponent and is set to 3 in this paperTo study the effect of the number of hops we normalize
the distance between SS and SD Specifically SS and SD arelocated at coordinates (00) and (10) respectivelyWe assumeall relay clusters are uniformly distributed on the straight lineconnecting SS and SD that is cluster 119896 is placed at (119896119870 0)so we have 119889119896 = 1119870 PR is located at (051)
For the implementation of simulations we use MonteCarlo method on MATLAB platform During the simu-lations we have to generate pairs of correlated randomvariables following Nakagami-119898 distribution to representthe perfect and imperfect channel coefficients The detailedprocedure of generating a pair of correlated random variablesfollowing Nakagami-119898 distribution by a modified inversetransform method is referred to in [22] The simulationconfiguration is given in Table 1 The specific values ofparameters are described in the caption of each figure
Figure 2 plots the exact and asymptotic outage probabilitycurves for a cognitive three-hop sensor network with119873119896 = 3
relay nodes in each cluster It shows that the exact analyticalresults match well the simulation results Besides the asymp-totic curves are very close to the exact ones in high SNRregime The slope of the curves which indicates the diversityorder rises with the increase in 119898119896 but remains unchangedwith the change of 119898119896119901 or 120588119896 This reveals that the diversityorder is affected by the fading severity of the secondary
International Journal of Distributed Sensor Networks 9
Table 1 Simulation configuration
Number of hops 119870
Number of relays in cluster 119896 (119896 = 1 2 119870) 119873119896 (119873119870 = 1)Correlation coefficient between 119892119896119894 and 119892119896119894 (119896 = 1 2 119870 119894 = 1 2 119873119896) 120588119896
System SNR119876
1198730
SNR threshold for successful decoding 120574th = 3 dBPath loss exponent 120578 = 3
Distance between cluster 119896 minus 1 and cluster 119896 (119896 = 1 2 119870) 119889119896 =
1
119870
Distance between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119889119896119901 =radic(
119896
119870
minus 05)
2
+ (0 minus 1)2
Fading severity parameter of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) 119898119896
Fading severity parameter of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119898119896119901
Average power of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) Ω119896 = 119889
minus120578
119896
Average power of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 sdot sdot sdot 119870) Ω119896119901 = 119889minus120578
119896119901
10minus33
10minus34
10minus35
10minus36
10minus37
10minus38
10minus39
0 02 04 06 08 1
Out
age p
roba
bilit
y
120588k
Analytical mkp = 1
Analytical mkp = 2
Analytical mkp = 3
Figure 3 Outage probability of the secondary system versus 120588119896 with119898119896 = 3 1198761198730 = 5 dB 119870 = 3 and119873119896 = 3
transmission links irrespective of the fading severity ofthe interference links and the CSI imperfection which isconsistent with our analysis in Section 4 From Figure 2 wecan also observe that the severer the fading gets (ie thesmaller 119898119896 or 119898119896119901 is) the worse the outage performancegets Furthermore the fading severity of the interference linkshas much less impact on the outage performance than thatof the transmission links In particular when 119898119896 is small119898119896119901 almost has no effect on the outage performance of thesecondary system
Figure 3 illustrates the outage probability versus 120588k fora cognitive three-hop sensor network with 119873119896 = 3 relay
nodes in each clusterWe can see that the outage performanceimproves with the increase in 120588119896 Specifically the descent ofthe curves is steep at first but then turns gentle when 120588119896 getsclose to 1 This indicates that the outage performance is quitegood even though the CSI is slightly imperfect for example120588119896 = 08 From Figure 3 we can also observe that as 119898119896119901
gets larger the improvement of the outage performance getssmaller
In Figure 4 the impact of the number of hops thenumber of relays in each cluster and the CSI imperfectionon the outage performance is presented We can observe thatthe outage performance improves with the increase in thenumber of hops When there is only one hop (ie 119870 = 1)120588119896 and 119873119896 have no effect on the outage performance sincethe direct transmission does not involve the relay selectionprocess For the cases of 119870 ge 2 it is indicated that whenthe CSI is perfect (ie 120588119896 = 1) assigning only two relaysto each cluster can achieve the best outage performance andmore relays would not help to achieve better performance Itis also shown that when the CSI estimation is totally random(ie 120588119896 = 0) no matter how many relays each cluster hasthe outage performance is the same with the case of 119873119896 = 1
since the relay selection does not work anymore Howeverwhen the CSI is imperfect but not totally random (ie 0 lt120588119896 lt 1) the outage performance improves with the increasein 119873119896 and gradually approaches the case of perfect CSI Inthis case adding more relays to each cluster can mitigate theperformance degradation caused by CSI imperfection
Figure 5 plots the average end-to-end BER curves of acognitive three-hop sensor network with119873119896 = 3 relay nodesin each cluster We show four different kinds of modulationsthat is orthogonal coherent BFSK (119886 = 12 119887 = 12)orthogonal noncoherent BFSK (119886 = 12 119887 = 1) antipodalcoherent BPSK (119886 = 1 119887 = 12) and antipodal differentiallycoherent BPSK (119886 = 1 119887 = 1) It can be observed that
10 International Journal of Distributed Sensor Networks
100
10minus1
10minus2
10minus3
10minus41 2 3 4 5 6 7
Out
age p
roba
bilit
y
Nk
Analytical 120588k = 1
Analytical 120588k = 05
Analytical 120588k = 0
K = 1
K = 2
K = 3
K = 4
Figure 4 Outage probability of the secondary system versusnumber of relays in each cluster for various number of hops with119898119896 = 119898119896119901 = 3 and 1198761198730 = 5 dB
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6minus5 0 5
Bit e
rror
rate
QN0 (dB)
Coherent BFSK 120588 = 1
Coherent BFSK 120588 = 05
Coherent BPSK 120588 = 1
Coherent BPSK 120588 = 05
Noncoherent BFSK120588 = 1
Noncoherent BFSK120588 = 05
Diff coherent BPSK120588 = 1
Diff coherent BPSK120588 = 05
Figure 5 Average end-to-end BER of the secondary system versussystem SNR for various kinds of modulations with 119870 = 3 119873119896 = 3and119898119896 = 119898119896119901 = 3
the CSI imperfection will increase the bit error rate of thesecondary system almost equally for the four different kindsof modulations
6 Conclusion
In this paper we study the performance of a cluster-basedcognitive multihop wireless sensor network with DF partial
relay selection overNakagami-119898 fading channelsThe imper-fect channel knowledge of the secondary transmission linksis taken into account We derive the closed-form expressionsfor the exact outage probability and BER of the secondarysystem as well as the asymptotic outage probability in highSNR regime We also analyze the influence of various factorsfor example the fading severity parameters the number ofrelaying hops the number of available relays in clustersand the CSI imperfection on the secondary transmissionperformance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Science Foundationof China (NSFC) under Grant 61372114 by the National 973Program of China under Grant 2012CB316005 by the JointFunds of NSFC-Guangdong under Grant U1035001 and byBeijing Higher Education Young Elite Teacher Project (noYETP0434)
References
[1] S Haykin ldquoCognitive radio brain-empowered wireless com-municationsrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 2 pp 201ndash220 2005
[2] M Xia and S Aissa ldquoUnderlay cooperative AF relaying incellular networks performance and challengesrdquo IEEE Commu-nications Magazine vol 51 no 12 pp 170ndash176 2013
[3] J N Laneman D N C Tse and G Wornell ldquoCooperativediversity in wireless networks efficient protocols and outagebehaviorrdquo IEEE Transactions on InformationTheory vol 50 no12 pp 3062ndash3080 2004
[4] Z Yan X Zhang and W Wang ldquoExact outage performance ofcognitive relay networkswithmaximum transmit power limitsrdquoIEEE Communications Letters vol 15 no 12 pp 1317ndash1319 2011
[5] V N Q Bao T T Thanh T D Nguyen and T D VuldquoSpectrum sharing-based multi-hop decode-and-forward relaynetworks under interference constraints performance analysisand relay position optimizationrdquo Journal of Communicationsand Networks vol 15 no 3 Article ID 6559352 pp 266ndash2752013
[6] H Phan H Zepernick and H Tran ldquoImpact of interferencepower constraint on multi-hop cognitive amplify-and-forwardrelay networks overNakagami-m fadingrdquo IETCommunicationsvol 7 no 9 pp 860ndash866 2013
[7] K J Kim T Q Duong T A Tsiftsis and V N Q BaoldquoCognitive multihop networks in spectrum sharing environ-ment with multiple licensed usersrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo13) pp2869ndash2873 Budapest Hungary June 2013
[8] A Hyadi M Benjillali M-S Alouini and D B Costa ldquoPer-formance analysis of underlay cognitive multihop regenerativerelaying systems with multiple primary receiversrdquo IEEE Trans-actions on Wireless Communications vol 12 no 12 pp 6418ndash6429 2013
International Journal of Distributed Sensor Networks 11
[9] A Scaglione D L Goeckel and J N Laneman ldquoCooperativecommunications in mobile ad hoc networksrdquo IEEE SignalProcessing Magazine vol 23 no 5 pp 18ndash29 2006
[10] V N Q Bao and H Y Kong ldquoPerformance analysis ofdecode-and-forward relaying with partial relay selection formultihop transmission over rayleigh fading channelsrdquo Journalof Communications and Networks vol 12 no 5 pp 433ndash4412010
[11] Y Wang Z Feng X Chen R Li and P Zhang ldquoOutageconstrained power allocation and relay selection formulti-HOPcognitive networkrdquo in Proceedings of the 76th IEEE VehicularTechnology Conference (VTC-Fall rsquo12) pp 1ndash5 Quebec CanadaSeptember 2012
[12] J Chen J Si Z Li and H Huang ldquoOn the performance ofspectrum sharing cognitive relay networks with imperfect CSIrdquoIEEE Communications Letters vol 16 no 7 pp 1002ndash1005 2012
[13] X Zhang J Xing Z Yan Y Gao and W Wang ldquoOutageperformance study of cognitive relay networks with imperfectchannel knowledgerdquo IEEE Communications Letters vol 17 no1 pp 27ndash30 2013
[14] X Zhang Z Yan Y Gao andWWang ldquoOn the study of outageperformance for cognitive relay networks (CRN) with the nthbest-relay selection in rayleigh-fading channelsrdquo IEEE WirelessCommunications Letters vol 2 no 1 pp 110ndash113 2013
[15] X Zhang J Xing andWWang ldquoOutage analysis of orthogonalspacendashtime block code transmission in cognitive relay networkswith multiple antennasrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 7 pp 3503ndash3508 2013
[16] V N Q Bao T Q Duong and C Tellambura ldquoOn theperformance of cognitive underlay multihop networks withimperfect channel state informationrdquo IEEE Transactions onCommunications vol 61 no 12 pp 4864ndash4873 2013
[17] I S Gradshteyn IM Ryzhik A Jeffrey andD ZwillingerTableof Integrals Series and Products Academic Press AmsterdamThe Netherlands 7th edition 2007
[18] M K Simon and M-S Alouini Digital Communications overFading Channels John Wiley amp Sons Hoboken NJ USA 2ndedition 2005
[19] D S Michalopoulos H A Suraweera G K Karagiannidis andR Schober ldquoAmplify-and-forward relay selectionwith outdatedchannel estimatesrdquo IEEE Transactions on Communications vol60 no 5 pp 1278ndash1290 2012
[20] M- Alouini and M K Simon ldquoGeneric form for average errorprobability of binary signals over fading channelsrdquo ElectronicsLetters vol 34 no 10 pp 949ndash950 1998
[21] P K Vitthaladevuni and M Alouini ldquoExact BER computationof generalized hierarchical PSK constellationsrdquo IEEE Transac-tions on Communications vol 51 no 12 pp 2030ndash2037 2003
[22] C Tellambura and A D S Jayalath ldquoGeneration of bivariateRayleigh and Nakagami-m fading envelopesrdquo IEEE Communi-cations Letters vol 4 no 5 pp 170ndash172 2000
2 International Journal of Distributed Sensor Networks
sensor network with partial relay selection without spectrumsharing was analyzed Power allocation and relay selectionproblems for cluster-based cognitivemultihop networks werestudied in [11]
In cognitive radio networks channel state information(CSI) plays an important role On one hand to restrict theinterference from SU to PU SU must adjust their transmitpower according to the instantaneous CSI of the interferencelinks between SU and PU On the other hand the relayselection for each hop depends on the instantaneous CSIof the corresponding secondary transmission links Mostexisting works assumed that perfect CSI could be acquired Inpractice however due to channel estimation errors mobilityfeedback delay limited feedback and feedback quantizationthe acquired CSI may sometimes be imperfect In [12 13] theimpact of imperfect CSI of the interference links between SUand PU was studied on the outage performance of cognitivedual-hop networkswithAF andDF protocols respectively In[14 15] the outage performance of cognitive relay networks(CRN) for theNth best relay selection and orthogonal space-time block coding (OSTBC) in Rayleigh channels are consid-ered For multihop relaying the authors in [16] analyzed theperformance ofmultihop cognitiveDFnetworks consideringthe effect of imperfect CSI of the interference links However[12 13 16] all focused on Rayleigh fading channels To thebest of our knowledge there have been no prior works onthe impact of imperfect CSI on cognitive multihop networksin a more general fading environment such as Nakagami-119898 fading environment Motivated by these considerationswe extend our previous research on a cognitive dual-hopnetwork over Rayleigh fading channels in [13] to a cognitivemultihop wireless sensor network over Nakagami-119898 fadingchannels
In this paper we investigate the performance of a cluster-based cognitive multihop wireless sensor network with DFpartial relay selection over Nakagami-119898 fading channelsTheeffect of imperfect CSI of the secondary transmission links istaken into accountWederive the closed-form expressions forthe exact outage probability and bit error rate (BER) and theasymptotic outage probability of the secondary system andvalidate them by simulations The impact of various factors(eg the CSI imperfection the fading severity of all links thenumber of hops and the number of sensorsrelays in clusters)on the performance of the secondary system is analyzed
The remainder of this paper is organized as follows inSection 2 we present the system model for the analysis ofcognitive multihop wireless sensor networks then basedon this model in Section 3 the outage probability andbit-error-rate (BER) are derived over Nakagami-119898 fadingchannels meanwhile the asymptotic outage probability forlarge system SNR is derived to study the diversity order insuch a system in Section 4 simulation results are given andcompared finally we conclude this paper in Section 5
2 System Model
The system model we consider in this paper is illustratedin Figure 1 The secondary system shares the same spectrum
PR
SS SDCluster 1 Cluster 2 Cluster K minus 1
middot middot middot
Interference linksSecondary transmission links
Figure 1 System model
with a primary user in an underlay approach Like moststudies we only consider the existence of the primary receiver(PR) and assume the primary transmitter (PT) is far awayfrom SU so the interference from PT can be ignored Thesecondary system is a cognitive cluster-based multihop sen-sor network whose transmission consists of119870 hops Betweenthe secondary source (SS) and the secondary destination(SD) there are 119870 minus 1 sensor clusters where cluster 119896 (119896 =
1 2 119870 minus 1) contains119873119896 available relay nodes To simplifythe notation we denote SS and SD by cluster 0 and cluster119870respectively So we have119873119870 = 1
Due to the half-duplex mode of the relays the transmis-sion of the 119870 hops happens in 119870 separate time slots Weassume that all channels experience Nakagami-119898 quasistaticfading that is the channel fading coefficients remain constantduring one slot but change independently for every slot Forthe reason that the CSI of the next hops cannot be obtainedin the current hop the relay selection for the current hopmerely depends on the CSI of the current hop To this endpartial relay selection (PRS) can be adopted in the secondarytransmission as a feasible strategy since the relay selectionprocess is based on the channel quality of the current hoponly instead of the end-to-end signal-to-noise ratio (SNR)Besides DF protocol is exploited in the secondary relayswhere the receiver for each hop decodes the signal and thenreencodes it before forwarding it to the next hop
To guarantee PUrsquos communication the maximum toler-able interference power constraint 119876 at PR must be satisfiedIn other words the transmitter in each hop should restrict itstransmit power so that the interference it causes to PR doesnot exceed 119876
In the 119896th hop (119896 = 1 2 119870) the transmitter incluster 119896 minus 1 is denoted by 119905119896 while the119873119896 potential receiversin cluster 119896 (ie the available relays included in cluster 119896)are represented as 119903(119896)
1 119903
(119896)
2 119903
(119896)
119873119896 We denote the channel
fading coefficient of the interference link between transmitter119905119896 and the PR by ℎ119896119901 and the channel fading coefficients ofthe secondary transmission links between transmitter 119905119896 andreceiver 119903(119896)
119894(119894 = 1 2 119873119896) by ℎ119896119894 According to the PRS
protocol the best relay with the maximum received SNR ischosen from the potential receivers 119903(119896)
1 119903
(119896)
2 119903
(119896)
119873119896 as the
transmitter 119905119896+1 in the (119896 + 1)th hop
International Journal of Distributed Sensor Networks 3
For Nakagami-119898 fading the channel power gains 119892119883119884 =
|ℎ119883119884|2(119883 = 119896 119884 = 1 2 119873119896 119901) are independent and non-
identically distributed (INID) random variables followingGamma distribution with fading severity parameters 119898119883119884
(119898119883119884 is positive integer) and mean Ω119883119884 The probabilitydensity function (PDF) and cumulative distributed function(CDF) of 119892119883119884 are given by
119891119892119883119884(119909) =
120573
119898119883119884
119883119884119909119898119883119884minus1
Γ (119898119883119884)
exp (minus120573119883119884119909) (1)
119865119892119883119884(119909) = 1 minus exp (minus120573119883119884119909)
119898119883119884minus1
sum
119899=0
(120573119883119884119909)119899
119899
(2)
respectively where 120573119883119884 = 119898119883119884Ω119883119884 and Γ(sdot) and 120574(sdot sdot)represent the Gamma function [17 (83391)] and the lowerincomplete Gamma function [17 (83501)] We assume thatall relays in the same cluster are relatively centralized so thechannel parameters pertaining to relays in the same clusterare identical that is we have 119898119896119894 = 119898119896 Ω119896119894 = Ω119896 and120573119896119894 = 120573119896 (119894 = 1 2 119873119896) In addition the thermal noiseat each receiver is modeled as independent complex additivewhite Gaussian noise (AWGN) with variance1198730
In this paper we consider the imperfect CSI of thesecondary transmission links Specifically in the 119896th hopthe perfect channel gain of the transmission link betweentransmitter 119905119896 and the potential receiver 119903119894 (119894 = 1 2 119873119896)
and its imperfect counterpart are denoted by 119892119896119894 and 119892119896119894respectively According to [18 equation (9398)] the jointPDF of the perfect channel gain 119892119896119894 and its imperfectcounterpart 119892119896119894 which follows the same distribution as 119892119896119894
where 119868119899(sdot) denotes the 119899th-ordermodified Bessel function ofthe first kind and 120588119896 = Cov(119892119896119894 119892119896119894)radicVar(119892119896119894)Var(119892119896119894) isin
[0 1] represents the correlation coefficient between 119892119896119894 and119892119896119894 120588119896 reflects the degree of CSI imperfection For instance120588119896 = 1 indicates that the CSI is absolutely perfect while 120588119896 = 0represents that the CSI estimation is totally random
3 Exact Performance Analysis
In this section considering the imperfect CSI of the sec-ondary transmission links we derive the exact outage proba-bility and BER of the cognitive multihop sensor network
31 Cumulative Distribution Function of Single-Hop SNRSince the secondary users suffer from low transmit powereach secondary node should try its best to transmit the signalIn the 119896th hop to satisfy the interference power constraint of
PU the transmit power of 119905119896 should be set to 119875119896 = 119876119892119896119901Due to the imperfect CSI of the secondary transmission linksthe imperfect received SNR at receiver 119903(119896)
119894(119894 = 1 2 119873119896)
is calculated as 120574119896119894 = 1198761198921198961198941198730119892119896119901 According to the PRSstrategy the ldquobestrdquo receiver with the maximum imperfectreceived SNR is selected as transmitter 119905119896+1 for the next hopso that the imperfect SNR for the 119896th hop satisfies
120574119896 = max119894=12119873119896
120574119896119894 = max119894=12119873119896
119876119892119896119894
1198730119892119896119901
(4)
The imperfect channel gain between transmitter 119905119896 and theselected receiver 119905119896+1 is given by
where 119892119896 is the actual channel gain between transmitter 119905119896and the selected receiver 119905119896+1 The end-to-end SNR of thesecondary system can be expressed as [10]
1205741198902119890 = min119896=12119870
120574119896 (9)
Next we will first derive the PDF of the actual channelgain for the 119896th hop 119892119896 by the similar approach to [19] From[19 equations (29)-(30)] we have
where 119894 is the imaginary unit and 119869119899(sdot) denotes the 119899th-order Bessel function of the first kind From [17 equation(66434)] (15) can be simplified as
Φ = ℎ120573minusℎminus1
119896(
1
1 minus 120588119896
+ 119895)
minusℎminus119898119896
(
radic120588119896119909
1 minus 120588119896
)
119898119896minus1
times exp(120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
times 119871119898119896minus1
ℎ(minus
120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
(16)
where 119871120572
119899(sdot) represents the associated Laguerre polynomial
By usingRodriguesrsquo formula for119871120572
119899(sdot) [17 equation (89701)]
we can obtain
Φ = ℎ120573119896
minusℎminus1(
1
1 minus 120588119896
+ 119895)
minusℎminus119898119896
(
radic120588119896
1 minus 120588119896
)
119898119896minus1
times 119909(119898119896minus1)2 exp(
120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
times
ℎ
sum
119898=0
(
ℎ + 119898119896 minus 1
ℎ minus 119898)
times
1
119898
(
120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
119898
(17)
By substituting (17) into (14) we get the closed-form expres-sion for the PDF of 119892119896 as follows
119891119892119896(119909)
=
119873119896
Γ (119898119896)
119873119896minus1
sum
119895=0
(
119873119896 minus 1
119895) (minus1)
119895119895
International Journal of Distributed Sensor Networks 5
32 End-to-EndOutage Probability Theoutage probability ofthemultihop secondary systemwhich characterizes the prob-ability that the end-to-end SNR falls below a predeterminedthreshold 120574th is given by
Now we will derive the closed-form expression for119883(119886 119887) By exploiting [17 equation (91311)] we can rewritethe CDF of 120574119896 (22) as follows
[1 + 119895 (1 minus 120588119896)] 120573119896119901120574
)
(29)
Then by substituting (22) into (27) and using the variabletransformation 119909 = (119895 + 1)120573119896120574[1 + 119895(1 minus 120588119896)]120573119896119901120574 we have
119886 [1 + 119895 (1 minus 120588119896)] 120573119896119901120574
(119895 + 1) 120573119896
)
(31)
where 119864() denotes MacRobertrsquos E-Function Since Mac-Robertrsquos E-Function can always be expressed in terms ofMeijerrsquos G-function (which can be computed in MATLAB)we can rewrite (31) as
119886 [1 + 119895 (1 minus 120588119896)] 120573119896119901120574
(119895 + 1) 120573119896
)
(32)
With the help of (32) the single-hop average BER for binarytransmissions (28) can be calculated Then by substitutingit into (25) we can obtain the end-to-end average BER forbinary transmissions
332119872-QAMTransmissions Thesingle-hop averageBERof119872-QAM transmissions is given by [8]
BER119896 =
1
radic119872log2radic119872
log2radic119872
sum
119901=1
V119901
sum
119902=0
120601119902119901radic
120596119902
120587
times int
infin
0
120574minus12
119890minus120596119902120574
119865120574119896(120574) 119889120574
(33)
where V119901 = (1minus2minus119901)radic119872minus1120596119902 = 3(2119902 + 1)
2log2119872(2119872minus2)
and 120601119902119901 = (minus1)lfloor1199022119901minus1
radic119872rfloor(2
119901minus1minus lfloor(1199022
119901minus1radic119872) + (12)rfloor)
From (27) the integral term in (33) can be rewritten as119883(120596119902 12) Thus (33) can be expressed as
BER119896 =
1
radic119872log2radic119872
log2radic119872
sum
119901=1
V119901
sum
119902=0
120601119902119901radic
120596119902
120587
119883(120596119902
1
2
) (34)
With the help of (32) andby substituting (34) into (25) we canget the end-to-end average BER for119872-QAM transmissions
333 119872-PSK Transmissions The BER of 24sdot sdot sdot 119872-PSKtransmissions can be obtained by recursive algorithms basedon the generalized 24-PSK [21] Therefore we only need toderive the BER of 24-PSK transmissions
According to [8] the BER for the most significant bit(MSB) and the least significant bit (LSB) with 24-PSKconstellation through a set of angles (120579 = [1205791 1205792]) are givenby
BER (120574) = 12
erfc (sin 1205792radic120574) (35)
BER (120574) = 12
erfc (cos 1205792radic120574) (36)
respectively
By applying integration by parts we can obtain the single-hop average BER for the MSB with 24-PSK as
BER119896 =
sin 12057922radic120587
int
infin
0
120574minus12
119890minussin21205792120574
119865120574119896(120574) 119889120574
=
sin 12057922radic120587
119883(sin21205792
1
2
)
(37)
The single-hop average BER for the LSB with 24-PSK can beobtained in a similar way
Given the single-hop average BER for the MSB andthe LSB cases the end-to-end average BER for 119872-PSKtransmissions can be written in closed form with the help of(32) and (25)
4 Asymptotic Outage Performance Analysis
In this section to study the diversity performance for thecognitive multihop sensor network we derive the asymptoticoutage probability of the secondary system in high SNRregime
According to the asymptotic behavior of the lower incom-plete Gamma function
By substituting (40) into (24) we can get the asymptoticexpression for the outage probability of the secondary system
Diversity order is an important performance metric forcooperative relay system It is defined as
119889 = minus lim120574rarrinfin
log119875out (120574)log 120574
(41)
From (40) we can see that in high SNR regime
119865120574119896(120574th) prop (
1
120574
)
119898119896
(42)
According to (24) we can conclude that the lowest orderof 1120574 in the outage probability expression is determinedby the minimum among the lowest order of 1120574 in 119865120574119896(120574th)(119896 = 1 2 119870) that is min119896=12119870119898119896 In other words thediversity order of the secondary system is min119896=12119870119898119896This indicates that the diversity order of the secondary systemis merely determined by the fading severity parameters of thesecondary transmission links but not affected by other factorssuch as the fading severity parameters of the interferencelinks and the CSI imperfection
5 Numerical Results
In this section analytical results are presented and validatedby simulations We study the effect of various factors onthe outage and error performance of the cognitive multihopsensor network such as the CSI imperfection the fadingseverity of the interference links and the secondary trans-mission links the number of hops and the number of relays
100
10minus1
10minus2
10minus3
10minus4
10minus5minus5 0 5 10
QN0 (dB)
Out
age p
roba
bilit
y
mk = 1
mk = 3
ExactExactExactExact
AsymptoticAsymptoticAsymptoticAsymptotic
SimulationSimulationSimulationSimulation
mkp = 1 120588k = 1
mkp = 1 120588k = 05
mkp = 3 120588k = 1
mkp = 3 120588k = 05
Figure 2 Exact and asymptotic outage probability of the secondarysystem versus system SNR for various fading severity with 119870 = 3
and119873119896 = 3
in clusters The predetermined SNR threshold for successfuldecoding is set as 120574th = 3 dB
We denote the distance between cluster 119896 minus 1 and cluster119896 by 119889119896 and the distance between cluster 119896 minus 1 and PR by119889119896119901 (119896 = 1 2 119870) To consider the impact of path losswe set Ω119896 = 119889
minus120578
119896and Ω119896119901 = 119889
minus120578
119896119901 where 120578 is the path loss
exponent and is set to 3 in this paperTo study the effect of the number of hops we normalize
the distance between SS and SD Specifically SS and SD arelocated at coordinates (00) and (10) respectivelyWe assumeall relay clusters are uniformly distributed on the straight lineconnecting SS and SD that is cluster 119896 is placed at (119896119870 0)so we have 119889119896 = 1119870 PR is located at (051)
For the implementation of simulations we use MonteCarlo method on MATLAB platform During the simu-lations we have to generate pairs of correlated randomvariables following Nakagami-119898 distribution to representthe perfect and imperfect channel coefficients The detailedprocedure of generating a pair of correlated random variablesfollowing Nakagami-119898 distribution by a modified inversetransform method is referred to in [22] The simulationconfiguration is given in Table 1 The specific values ofparameters are described in the caption of each figure
Figure 2 plots the exact and asymptotic outage probabilitycurves for a cognitive three-hop sensor network with119873119896 = 3
relay nodes in each cluster It shows that the exact analyticalresults match well the simulation results Besides the asymp-totic curves are very close to the exact ones in high SNRregime The slope of the curves which indicates the diversityorder rises with the increase in 119898119896 but remains unchangedwith the change of 119898119896119901 or 120588119896 This reveals that the diversityorder is affected by the fading severity of the secondary
International Journal of Distributed Sensor Networks 9
Table 1 Simulation configuration
Number of hops 119870
Number of relays in cluster 119896 (119896 = 1 2 119870) 119873119896 (119873119870 = 1)Correlation coefficient between 119892119896119894 and 119892119896119894 (119896 = 1 2 119870 119894 = 1 2 119873119896) 120588119896
System SNR119876
1198730
SNR threshold for successful decoding 120574th = 3 dBPath loss exponent 120578 = 3
Distance between cluster 119896 minus 1 and cluster 119896 (119896 = 1 2 119870) 119889119896 =
1
119870
Distance between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119889119896119901 =radic(
119896
119870
minus 05)
2
+ (0 minus 1)2
Fading severity parameter of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) 119898119896
Fading severity parameter of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119898119896119901
Average power of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) Ω119896 = 119889
minus120578
119896
Average power of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 sdot sdot sdot 119870) Ω119896119901 = 119889minus120578
119896119901
10minus33
10minus34
10minus35
10minus36
10minus37
10minus38
10minus39
0 02 04 06 08 1
Out
age p
roba
bilit
y
120588k
Analytical mkp = 1
Analytical mkp = 2
Analytical mkp = 3
Figure 3 Outage probability of the secondary system versus 120588119896 with119898119896 = 3 1198761198730 = 5 dB 119870 = 3 and119873119896 = 3
transmission links irrespective of the fading severity ofthe interference links and the CSI imperfection which isconsistent with our analysis in Section 4 From Figure 2 wecan also observe that the severer the fading gets (ie thesmaller 119898119896 or 119898119896119901 is) the worse the outage performancegets Furthermore the fading severity of the interference linkshas much less impact on the outage performance than thatof the transmission links In particular when 119898119896 is small119898119896119901 almost has no effect on the outage performance of thesecondary system
Figure 3 illustrates the outage probability versus 120588k fora cognitive three-hop sensor network with 119873119896 = 3 relay
nodes in each clusterWe can see that the outage performanceimproves with the increase in 120588119896 Specifically the descent ofthe curves is steep at first but then turns gentle when 120588119896 getsclose to 1 This indicates that the outage performance is quitegood even though the CSI is slightly imperfect for example120588119896 = 08 From Figure 3 we can also observe that as 119898119896119901
gets larger the improvement of the outage performance getssmaller
In Figure 4 the impact of the number of hops thenumber of relays in each cluster and the CSI imperfectionon the outage performance is presented We can observe thatthe outage performance improves with the increase in thenumber of hops When there is only one hop (ie 119870 = 1)120588119896 and 119873119896 have no effect on the outage performance sincethe direct transmission does not involve the relay selectionprocess For the cases of 119870 ge 2 it is indicated that whenthe CSI is perfect (ie 120588119896 = 1) assigning only two relaysto each cluster can achieve the best outage performance andmore relays would not help to achieve better performance Itis also shown that when the CSI estimation is totally random(ie 120588119896 = 0) no matter how many relays each cluster hasthe outage performance is the same with the case of 119873119896 = 1
since the relay selection does not work anymore Howeverwhen the CSI is imperfect but not totally random (ie 0 lt120588119896 lt 1) the outage performance improves with the increasein 119873119896 and gradually approaches the case of perfect CSI Inthis case adding more relays to each cluster can mitigate theperformance degradation caused by CSI imperfection
Figure 5 plots the average end-to-end BER curves of acognitive three-hop sensor network with119873119896 = 3 relay nodesin each cluster We show four different kinds of modulationsthat is orthogonal coherent BFSK (119886 = 12 119887 = 12)orthogonal noncoherent BFSK (119886 = 12 119887 = 1) antipodalcoherent BPSK (119886 = 1 119887 = 12) and antipodal differentiallycoherent BPSK (119886 = 1 119887 = 1) It can be observed that
10 International Journal of Distributed Sensor Networks
100
10minus1
10minus2
10minus3
10minus41 2 3 4 5 6 7
Out
age p
roba
bilit
y
Nk
Analytical 120588k = 1
Analytical 120588k = 05
Analytical 120588k = 0
K = 1
K = 2
K = 3
K = 4
Figure 4 Outage probability of the secondary system versusnumber of relays in each cluster for various number of hops with119898119896 = 119898119896119901 = 3 and 1198761198730 = 5 dB
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6minus5 0 5
Bit e
rror
rate
QN0 (dB)
Coherent BFSK 120588 = 1
Coherent BFSK 120588 = 05
Coherent BPSK 120588 = 1
Coherent BPSK 120588 = 05
Noncoherent BFSK120588 = 1
Noncoherent BFSK120588 = 05
Diff coherent BPSK120588 = 1
Diff coherent BPSK120588 = 05
Figure 5 Average end-to-end BER of the secondary system versussystem SNR for various kinds of modulations with 119870 = 3 119873119896 = 3and119898119896 = 119898119896119901 = 3
the CSI imperfection will increase the bit error rate of thesecondary system almost equally for the four different kindsof modulations
6 Conclusion
In this paper we study the performance of a cluster-basedcognitive multihop wireless sensor network with DF partial
relay selection overNakagami-119898 fading channelsThe imper-fect channel knowledge of the secondary transmission linksis taken into account We derive the closed-form expressionsfor the exact outage probability and BER of the secondarysystem as well as the asymptotic outage probability in highSNR regime We also analyze the influence of various factorsfor example the fading severity parameters the number ofrelaying hops the number of available relays in clustersand the CSI imperfection on the secondary transmissionperformance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Science Foundationof China (NSFC) under Grant 61372114 by the National 973Program of China under Grant 2012CB316005 by the JointFunds of NSFC-Guangdong under Grant U1035001 and byBeijing Higher Education Young Elite Teacher Project (noYETP0434)
References
[1] S Haykin ldquoCognitive radio brain-empowered wireless com-municationsrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 2 pp 201ndash220 2005
[2] M Xia and S Aissa ldquoUnderlay cooperative AF relaying incellular networks performance and challengesrdquo IEEE Commu-nications Magazine vol 51 no 12 pp 170ndash176 2013
[3] J N Laneman D N C Tse and G Wornell ldquoCooperativediversity in wireless networks efficient protocols and outagebehaviorrdquo IEEE Transactions on InformationTheory vol 50 no12 pp 3062ndash3080 2004
[4] Z Yan X Zhang and W Wang ldquoExact outage performance ofcognitive relay networkswithmaximum transmit power limitsrdquoIEEE Communications Letters vol 15 no 12 pp 1317ndash1319 2011
[5] V N Q Bao T T Thanh T D Nguyen and T D VuldquoSpectrum sharing-based multi-hop decode-and-forward relaynetworks under interference constraints performance analysisand relay position optimizationrdquo Journal of Communicationsand Networks vol 15 no 3 Article ID 6559352 pp 266ndash2752013
[6] H Phan H Zepernick and H Tran ldquoImpact of interferencepower constraint on multi-hop cognitive amplify-and-forwardrelay networks overNakagami-m fadingrdquo IETCommunicationsvol 7 no 9 pp 860ndash866 2013
[7] K J Kim T Q Duong T A Tsiftsis and V N Q BaoldquoCognitive multihop networks in spectrum sharing environ-ment with multiple licensed usersrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo13) pp2869ndash2873 Budapest Hungary June 2013
[8] A Hyadi M Benjillali M-S Alouini and D B Costa ldquoPer-formance analysis of underlay cognitive multihop regenerativerelaying systems with multiple primary receiversrdquo IEEE Trans-actions on Wireless Communications vol 12 no 12 pp 6418ndash6429 2013
International Journal of Distributed Sensor Networks 11
[9] A Scaglione D L Goeckel and J N Laneman ldquoCooperativecommunications in mobile ad hoc networksrdquo IEEE SignalProcessing Magazine vol 23 no 5 pp 18ndash29 2006
[10] V N Q Bao and H Y Kong ldquoPerformance analysis ofdecode-and-forward relaying with partial relay selection formultihop transmission over rayleigh fading channelsrdquo Journalof Communications and Networks vol 12 no 5 pp 433ndash4412010
[11] Y Wang Z Feng X Chen R Li and P Zhang ldquoOutageconstrained power allocation and relay selection formulti-HOPcognitive networkrdquo in Proceedings of the 76th IEEE VehicularTechnology Conference (VTC-Fall rsquo12) pp 1ndash5 Quebec CanadaSeptember 2012
[12] J Chen J Si Z Li and H Huang ldquoOn the performance ofspectrum sharing cognitive relay networks with imperfect CSIrdquoIEEE Communications Letters vol 16 no 7 pp 1002ndash1005 2012
[13] X Zhang J Xing Z Yan Y Gao and W Wang ldquoOutageperformance study of cognitive relay networks with imperfectchannel knowledgerdquo IEEE Communications Letters vol 17 no1 pp 27ndash30 2013
[14] X Zhang Z Yan Y Gao andWWang ldquoOn the study of outageperformance for cognitive relay networks (CRN) with the nthbest-relay selection in rayleigh-fading channelsrdquo IEEE WirelessCommunications Letters vol 2 no 1 pp 110ndash113 2013
[15] X Zhang J Xing andWWang ldquoOutage analysis of orthogonalspacendashtime block code transmission in cognitive relay networkswith multiple antennasrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 7 pp 3503ndash3508 2013
[16] V N Q Bao T Q Duong and C Tellambura ldquoOn theperformance of cognitive underlay multihop networks withimperfect channel state informationrdquo IEEE Transactions onCommunications vol 61 no 12 pp 4864ndash4873 2013
[17] I S Gradshteyn IM Ryzhik A Jeffrey andD ZwillingerTableof Integrals Series and Products Academic Press AmsterdamThe Netherlands 7th edition 2007
[18] M K Simon and M-S Alouini Digital Communications overFading Channels John Wiley amp Sons Hoboken NJ USA 2ndedition 2005
[19] D S Michalopoulos H A Suraweera G K Karagiannidis andR Schober ldquoAmplify-and-forward relay selectionwith outdatedchannel estimatesrdquo IEEE Transactions on Communications vol60 no 5 pp 1278ndash1290 2012
[20] M- Alouini and M K Simon ldquoGeneric form for average errorprobability of binary signals over fading channelsrdquo ElectronicsLetters vol 34 no 10 pp 949ndash950 1998
[21] P K Vitthaladevuni and M Alouini ldquoExact BER computationof generalized hierarchical PSK constellationsrdquo IEEE Transac-tions on Communications vol 51 no 12 pp 2030ndash2037 2003
[22] C Tellambura and A D S Jayalath ldquoGeneration of bivariateRayleigh and Nakagami-m fading envelopesrdquo IEEE Communi-cations Letters vol 4 no 5 pp 170ndash172 2000
International Journal of Distributed Sensor Networks 3
For Nakagami-119898 fading the channel power gains 119892119883119884 =
|ℎ119883119884|2(119883 = 119896 119884 = 1 2 119873119896 119901) are independent and non-
identically distributed (INID) random variables followingGamma distribution with fading severity parameters 119898119883119884
(119898119883119884 is positive integer) and mean Ω119883119884 The probabilitydensity function (PDF) and cumulative distributed function(CDF) of 119892119883119884 are given by
119891119892119883119884(119909) =
120573
119898119883119884
119883119884119909119898119883119884minus1
Γ (119898119883119884)
exp (minus120573119883119884119909) (1)
119865119892119883119884(119909) = 1 minus exp (minus120573119883119884119909)
119898119883119884minus1
sum
119899=0
(120573119883119884119909)119899
119899
(2)
respectively where 120573119883119884 = 119898119883119884Ω119883119884 and Γ(sdot) and 120574(sdot sdot)represent the Gamma function [17 (83391)] and the lowerincomplete Gamma function [17 (83501)] We assume thatall relays in the same cluster are relatively centralized so thechannel parameters pertaining to relays in the same clusterare identical that is we have 119898119896119894 = 119898119896 Ω119896119894 = Ω119896 and120573119896119894 = 120573119896 (119894 = 1 2 119873119896) In addition the thermal noiseat each receiver is modeled as independent complex additivewhite Gaussian noise (AWGN) with variance1198730
In this paper we consider the imperfect CSI of thesecondary transmission links Specifically in the 119896th hopthe perfect channel gain of the transmission link betweentransmitter 119905119896 and the potential receiver 119903119894 (119894 = 1 2 119873119896)
and its imperfect counterpart are denoted by 119892119896119894 and 119892119896119894respectively According to [18 equation (9398)] the jointPDF of the perfect channel gain 119892119896119894 and its imperfectcounterpart 119892119896119894 which follows the same distribution as 119892119896119894
where 119868119899(sdot) denotes the 119899th-ordermodified Bessel function ofthe first kind and 120588119896 = Cov(119892119896119894 119892119896119894)radicVar(119892119896119894)Var(119892119896119894) isin
[0 1] represents the correlation coefficient between 119892119896119894 and119892119896119894 120588119896 reflects the degree of CSI imperfection For instance120588119896 = 1 indicates that the CSI is absolutely perfect while 120588119896 = 0represents that the CSI estimation is totally random
3 Exact Performance Analysis
In this section considering the imperfect CSI of the sec-ondary transmission links we derive the exact outage proba-bility and BER of the cognitive multihop sensor network
31 Cumulative Distribution Function of Single-Hop SNRSince the secondary users suffer from low transmit powereach secondary node should try its best to transmit the signalIn the 119896th hop to satisfy the interference power constraint of
PU the transmit power of 119905119896 should be set to 119875119896 = 119876119892119896119901Due to the imperfect CSI of the secondary transmission linksthe imperfect received SNR at receiver 119903(119896)
119894(119894 = 1 2 119873119896)
is calculated as 120574119896119894 = 1198761198921198961198941198730119892119896119901 According to the PRSstrategy the ldquobestrdquo receiver with the maximum imperfectreceived SNR is selected as transmitter 119905119896+1 for the next hopso that the imperfect SNR for the 119896th hop satisfies
120574119896 = max119894=12119873119896
120574119896119894 = max119894=12119873119896
119876119892119896119894
1198730119892119896119901
(4)
The imperfect channel gain between transmitter 119905119896 and theselected receiver 119905119896+1 is given by
where 119892119896 is the actual channel gain between transmitter 119905119896and the selected receiver 119905119896+1 The end-to-end SNR of thesecondary system can be expressed as [10]
1205741198902119890 = min119896=12119870
120574119896 (9)
Next we will first derive the PDF of the actual channelgain for the 119896th hop 119892119896 by the similar approach to [19] From[19 equations (29)-(30)] we have
where 119894 is the imaginary unit and 119869119899(sdot) denotes the 119899th-order Bessel function of the first kind From [17 equation(66434)] (15) can be simplified as
Φ = ℎ120573minusℎminus1
119896(
1
1 minus 120588119896
+ 119895)
minusℎminus119898119896
(
radic120588119896119909
1 minus 120588119896
)
119898119896minus1
times exp(120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
times 119871119898119896minus1
ℎ(minus
120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
(16)
where 119871120572
119899(sdot) represents the associated Laguerre polynomial
By usingRodriguesrsquo formula for119871120572
119899(sdot) [17 equation (89701)]
we can obtain
Φ = ℎ120573119896
minusℎminus1(
1
1 minus 120588119896
+ 119895)
minusℎminus119898119896
(
radic120588119896
1 minus 120588119896
)
119898119896minus1
times 119909(119898119896minus1)2 exp(
120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
times
ℎ
sum
119898=0
(
ℎ + 119898119896 minus 1
ℎ minus 119898)
times
1
119898
(
120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
119898
(17)
By substituting (17) into (14) we get the closed-form expres-sion for the PDF of 119892119896 as follows
119891119892119896(119909)
=
119873119896
Γ (119898119896)
119873119896minus1
sum
119895=0
(
119873119896 minus 1
119895) (minus1)
119895119895
International Journal of Distributed Sensor Networks 5
32 End-to-EndOutage Probability Theoutage probability ofthemultihop secondary systemwhich characterizes the prob-ability that the end-to-end SNR falls below a predeterminedthreshold 120574th is given by
Now we will derive the closed-form expression for119883(119886 119887) By exploiting [17 equation (91311)] we can rewritethe CDF of 120574119896 (22) as follows
[1 + 119895 (1 minus 120588119896)] 120573119896119901120574
)
(29)
Then by substituting (22) into (27) and using the variabletransformation 119909 = (119895 + 1)120573119896120574[1 + 119895(1 minus 120588119896)]120573119896119901120574 we have
119886 [1 + 119895 (1 minus 120588119896)] 120573119896119901120574
(119895 + 1) 120573119896
)
(31)
where 119864() denotes MacRobertrsquos E-Function Since Mac-Robertrsquos E-Function can always be expressed in terms ofMeijerrsquos G-function (which can be computed in MATLAB)we can rewrite (31) as
119886 [1 + 119895 (1 minus 120588119896)] 120573119896119901120574
(119895 + 1) 120573119896
)
(32)
With the help of (32) the single-hop average BER for binarytransmissions (28) can be calculated Then by substitutingit into (25) we can obtain the end-to-end average BER forbinary transmissions
332119872-QAMTransmissions Thesingle-hop averageBERof119872-QAM transmissions is given by [8]
BER119896 =
1
radic119872log2radic119872
log2radic119872
sum
119901=1
V119901
sum
119902=0
120601119902119901radic
120596119902
120587
times int
infin
0
120574minus12
119890minus120596119902120574
119865120574119896(120574) 119889120574
(33)
where V119901 = (1minus2minus119901)radic119872minus1120596119902 = 3(2119902 + 1)
2log2119872(2119872minus2)
and 120601119902119901 = (minus1)lfloor1199022119901minus1
radic119872rfloor(2
119901minus1minus lfloor(1199022
119901minus1radic119872) + (12)rfloor)
From (27) the integral term in (33) can be rewritten as119883(120596119902 12) Thus (33) can be expressed as
BER119896 =
1
radic119872log2radic119872
log2radic119872
sum
119901=1
V119901
sum
119902=0
120601119902119901radic
120596119902
120587
119883(120596119902
1
2
) (34)
With the help of (32) andby substituting (34) into (25) we canget the end-to-end average BER for119872-QAM transmissions
333 119872-PSK Transmissions The BER of 24sdot sdot sdot 119872-PSKtransmissions can be obtained by recursive algorithms basedon the generalized 24-PSK [21] Therefore we only need toderive the BER of 24-PSK transmissions
According to [8] the BER for the most significant bit(MSB) and the least significant bit (LSB) with 24-PSKconstellation through a set of angles (120579 = [1205791 1205792]) are givenby
BER (120574) = 12
erfc (sin 1205792radic120574) (35)
BER (120574) = 12
erfc (cos 1205792radic120574) (36)
respectively
By applying integration by parts we can obtain the single-hop average BER for the MSB with 24-PSK as
BER119896 =
sin 12057922radic120587
int
infin
0
120574minus12
119890minussin21205792120574
119865120574119896(120574) 119889120574
=
sin 12057922radic120587
119883(sin21205792
1
2
)
(37)
The single-hop average BER for the LSB with 24-PSK can beobtained in a similar way
Given the single-hop average BER for the MSB andthe LSB cases the end-to-end average BER for 119872-PSKtransmissions can be written in closed form with the help of(32) and (25)
4 Asymptotic Outage Performance Analysis
In this section to study the diversity performance for thecognitive multihop sensor network we derive the asymptoticoutage probability of the secondary system in high SNRregime
According to the asymptotic behavior of the lower incom-plete Gamma function
By substituting (40) into (24) we can get the asymptoticexpression for the outage probability of the secondary system
Diversity order is an important performance metric forcooperative relay system It is defined as
119889 = minus lim120574rarrinfin
log119875out (120574)log 120574
(41)
From (40) we can see that in high SNR regime
119865120574119896(120574th) prop (
1
120574
)
119898119896
(42)
According to (24) we can conclude that the lowest orderof 1120574 in the outage probability expression is determinedby the minimum among the lowest order of 1120574 in 119865120574119896(120574th)(119896 = 1 2 119870) that is min119896=12119870119898119896 In other words thediversity order of the secondary system is min119896=12119870119898119896This indicates that the diversity order of the secondary systemis merely determined by the fading severity parameters of thesecondary transmission links but not affected by other factorssuch as the fading severity parameters of the interferencelinks and the CSI imperfection
5 Numerical Results
In this section analytical results are presented and validatedby simulations We study the effect of various factors onthe outage and error performance of the cognitive multihopsensor network such as the CSI imperfection the fadingseverity of the interference links and the secondary trans-mission links the number of hops and the number of relays
100
10minus1
10minus2
10minus3
10minus4
10minus5minus5 0 5 10
QN0 (dB)
Out
age p
roba
bilit
y
mk = 1
mk = 3
ExactExactExactExact
AsymptoticAsymptoticAsymptoticAsymptotic
SimulationSimulationSimulationSimulation
mkp = 1 120588k = 1
mkp = 1 120588k = 05
mkp = 3 120588k = 1
mkp = 3 120588k = 05
Figure 2 Exact and asymptotic outage probability of the secondarysystem versus system SNR for various fading severity with 119870 = 3
and119873119896 = 3
in clusters The predetermined SNR threshold for successfuldecoding is set as 120574th = 3 dB
We denote the distance between cluster 119896 minus 1 and cluster119896 by 119889119896 and the distance between cluster 119896 minus 1 and PR by119889119896119901 (119896 = 1 2 119870) To consider the impact of path losswe set Ω119896 = 119889
minus120578
119896and Ω119896119901 = 119889
minus120578
119896119901 where 120578 is the path loss
exponent and is set to 3 in this paperTo study the effect of the number of hops we normalize
the distance between SS and SD Specifically SS and SD arelocated at coordinates (00) and (10) respectivelyWe assumeall relay clusters are uniformly distributed on the straight lineconnecting SS and SD that is cluster 119896 is placed at (119896119870 0)so we have 119889119896 = 1119870 PR is located at (051)
For the implementation of simulations we use MonteCarlo method on MATLAB platform During the simu-lations we have to generate pairs of correlated randomvariables following Nakagami-119898 distribution to representthe perfect and imperfect channel coefficients The detailedprocedure of generating a pair of correlated random variablesfollowing Nakagami-119898 distribution by a modified inversetransform method is referred to in [22] The simulationconfiguration is given in Table 1 The specific values ofparameters are described in the caption of each figure
Figure 2 plots the exact and asymptotic outage probabilitycurves for a cognitive three-hop sensor network with119873119896 = 3
relay nodes in each cluster It shows that the exact analyticalresults match well the simulation results Besides the asymp-totic curves are very close to the exact ones in high SNRregime The slope of the curves which indicates the diversityorder rises with the increase in 119898119896 but remains unchangedwith the change of 119898119896119901 or 120588119896 This reveals that the diversityorder is affected by the fading severity of the secondary
International Journal of Distributed Sensor Networks 9
Table 1 Simulation configuration
Number of hops 119870
Number of relays in cluster 119896 (119896 = 1 2 119870) 119873119896 (119873119870 = 1)Correlation coefficient between 119892119896119894 and 119892119896119894 (119896 = 1 2 119870 119894 = 1 2 119873119896) 120588119896
System SNR119876
1198730
SNR threshold for successful decoding 120574th = 3 dBPath loss exponent 120578 = 3
Distance between cluster 119896 minus 1 and cluster 119896 (119896 = 1 2 119870) 119889119896 =
1
119870
Distance between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119889119896119901 =radic(
119896
119870
minus 05)
2
+ (0 minus 1)2
Fading severity parameter of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) 119898119896
Fading severity parameter of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119898119896119901
Average power of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) Ω119896 = 119889
minus120578
119896
Average power of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 sdot sdot sdot 119870) Ω119896119901 = 119889minus120578
119896119901
10minus33
10minus34
10minus35
10minus36
10minus37
10minus38
10minus39
0 02 04 06 08 1
Out
age p
roba
bilit
y
120588k
Analytical mkp = 1
Analytical mkp = 2
Analytical mkp = 3
Figure 3 Outage probability of the secondary system versus 120588119896 with119898119896 = 3 1198761198730 = 5 dB 119870 = 3 and119873119896 = 3
transmission links irrespective of the fading severity ofthe interference links and the CSI imperfection which isconsistent with our analysis in Section 4 From Figure 2 wecan also observe that the severer the fading gets (ie thesmaller 119898119896 or 119898119896119901 is) the worse the outage performancegets Furthermore the fading severity of the interference linkshas much less impact on the outage performance than thatof the transmission links In particular when 119898119896 is small119898119896119901 almost has no effect on the outage performance of thesecondary system
Figure 3 illustrates the outage probability versus 120588k fora cognitive three-hop sensor network with 119873119896 = 3 relay
nodes in each clusterWe can see that the outage performanceimproves with the increase in 120588119896 Specifically the descent ofthe curves is steep at first but then turns gentle when 120588119896 getsclose to 1 This indicates that the outage performance is quitegood even though the CSI is slightly imperfect for example120588119896 = 08 From Figure 3 we can also observe that as 119898119896119901
gets larger the improvement of the outage performance getssmaller
In Figure 4 the impact of the number of hops thenumber of relays in each cluster and the CSI imperfectionon the outage performance is presented We can observe thatthe outage performance improves with the increase in thenumber of hops When there is only one hop (ie 119870 = 1)120588119896 and 119873119896 have no effect on the outage performance sincethe direct transmission does not involve the relay selectionprocess For the cases of 119870 ge 2 it is indicated that whenthe CSI is perfect (ie 120588119896 = 1) assigning only two relaysto each cluster can achieve the best outage performance andmore relays would not help to achieve better performance Itis also shown that when the CSI estimation is totally random(ie 120588119896 = 0) no matter how many relays each cluster hasthe outage performance is the same with the case of 119873119896 = 1
since the relay selection does not work anymore Howeverwhen the CSI is imperfect but not totally random (ie 0 lt120588119896 lt 1) the outage performance improves with the increasein 119873119896 and gradually approaches the case of perfect CSI Inthis case adding more relays to each cluster can mitigate theperformance degradation caused by CSI imperfection
Figure 5 plots the average end-to-end BER curves of acognitive three-hop sensor network with119873119896 = 3 relay nodesin each cluster We show four different kinds of modulationsthat is orthogonal coherent BFSK (119886 = 12 119887 = 12)orthogonal noncoherent BFSK (119886 = 12 119887 = 1) antipodalcoherent BPSK (119886 = 1 119887 = 12) and antipodal differentiallycoherent BPSK (119886 = 1 119887 = 1) It can be observed that
10 International Journal of Distributed Sensor Networks
100
10minus1
10minus2
10minus3
10minus41 2 3 4 5 6 7
Out
age p
roba
bilit
y
Nk
Analytical 120588k = 1
Analytical 120588k = 05
Analytical 120588k = 0
K = 1
K = 2
K = 3
K = 4
Figure 4 Outage probability of the secondary system versusnumber of relays in each cluster for various number of hops with119898119896 = 119898119896119901 = 3 and 1198761198730 = 5 dB
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6minus5 0 5
Bit e
rror
rate
QN0 (dB)
Coherent BFSK 120588 = 1
Coherent BFSK 120588 = 05
Coherent BPSK 120588 = 1
Coherent BPSK 120588 = 05
Noncoherent BFSK120588 = 1
Noncoherent BFSK120588 = 05
Diff coherent BPSK120588 = 1
Diff coherent BPSK120588 = 05
Figure 5 Average end-to-end BER of the secondary system versussystem SNR for various kinds of modulations with 119870 = 3 119873119896 = 3and119898119896 = 119898119896119901 = 3
the CSI imperfection will increase the bit error rate of thesecondary system almost equally for the four different kindsof modulations
6 Conclusion
In this paper we study the performance of a cluster-basedcognitive multihop wireless sensor network with DF partial
relay selection overNakagami-119898 fading channelsThe imper-fect channel knowledge of the secondary transmission linksis taken into account We derive the closed-form expressionsfor the exact outage probability and BER of the secondarysystem as well as the asymptotic outage probability in highSNR regime We also analyze the influence of various factorsfor example the fading severity parameters the number ofrelaying hops the number of available relays in clustersand the CSI imperfection on the secondary transmissionperformance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Science Foundationof China (NSFC) under Grant 61372114 by the National 973Program of China under Grant 2012CB316005 by the JointFunds of NSFC-Guangdong under Grant U1035001 and byBeijing Higher Education Young Elite Teacher Project (noYETP0434)
References
[1] S Haykin ldquoCognitive radio brain-empowered wireless com-municationsrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 2 pp 201ndash220 2005
[2] M Xia and S Aissa ldquoUnderlay cooperative AF relaying incellular networks performance and challengesrdquo IEEE Commu-nications Magazine vol 51 no 12 pp 170ndash176 2013
[3] J N Laneman D N C Tse and G Wornell ldquoCooperativediversity in wireless networks efficient protocols and outagebehaviorrdquo IEEE Transactions on InformationTheory vol 50 no12 pp 3062ndash3080 2004
[4] Z Yan X Zhang and W Wang ldquoExact outage performance ofcognitive relay networkswithmaximum transmit power limitsrdquoIEEE Communications Letters vol 15 no 12 pp 1317ndash1319 2011
[5] V N Q Bao T T Thanh T D Nguyen and T D VuldquoSpectrum sharing-based multi-hop decode-and-forward relaynetworks under interference constraints performance analysisand relay position optimizationrdquo Journal of Communicationsand Networks vol 15 no 3 Article ID 6559352 pp 266ndash2752013
[6] H Phan H Zepernick and H Tran ldquoImpact of interferencepower constraint on multi-hop cognitive amplify-and-forwardrelay networks overNakagami-m fadingrdquo IETCommunicationsvol 7 no 9 pp 860ndash866 2013
[7] K J Kim T Q Duong T A Tsiftsis and V N Q BaoldquoCognitive multihop networks in spectrum sharing environ-ment with multiple licensed usersrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo13) pp2869ndash2873 Budapest Hungary June 2013
[8] A Hyadi M Benjillali M-S Alouini and D B Costa ldquoPer-formance analysis of underlay cognitive multihop regenerativerelaying systems with multiple primary receiversrdquo IEEE Trans-actions on Wireless Communications vol 12 no 12 pp 6418ndash6429 2013
International Journal of Distributed Sensor Networks 11
[9] A Scaglione D L Goeckel and J N Laneman ldquoCooperativecommunications in mobile ad hoc networksrdquo IEEE SignalProcessing Magazine vol 23 no 5 pp 18ndash29 2006
[10] V N Q Bao and H Y Kong ldquoPerformance analysis ofdecode-and-forward relaying with partial relay selection formultihop transmission over rayleigh fading channelsrdquo Journalof Communications and Networks vol 12 no 5 pp 433ndash4412010
[11] Y Wang Z Feng X Chen R Li and P Zhang ldquoOutageconstrained power allocation and relay selection formulti-HOPcognitive networkrdquo in Proceedings of the 76th IEEE VehicularTechnology Conference (VTC-Fall rsquo12) pp 1ndash5 Quebec CanadaSeptember 2012
[12] J Chen J Si Z Li and H Huang ldquoOn the performance ofspectrum sharing cognitive relay networks with imperfect CSIrdquoIEEE Communications Letters vol 16 no 7 pp 1002ndash1005 2012
[13] X Zhang J Xing Z Yan Y Gao and W Wang ldquoOutageperformance study of cognitive relay networks with imperfectchannel knowledgerdquo IEEE Communications Letters vol 17 no1 pp 27ndash30 2013
[14] X Zhang Z Yan Y Gao andWWang ldquoOn the study of outageperformance for cognitive relay networks (CRN) with the nthbest-relay selection in rayleigh-fading channelsrdquo IEEE WirelessCommunications Letters vol 2 no 1 pp 110ndash113 2013
[15] X Zhang J Xing andWWang ldquoOutage analysis of orthogonalspacendashtime block code transmission in cognitive relay networkswith multiple antennasrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 7 pp 3503ndash3508 2013
[16] V N Q Bao T Q Duong and C Tellambura ldquoOn theperformance of cognitive underlay multihop networks withimperfect channel state informationrdquo IEEE Transactions onCommunications vol 61 no 12 pp 4864ndash4873 2013
[17] I S Gradshteyn IM Ryzhik A Jeffrey andD ZwillingerTableof Integrals Series and Products Academic Press AmsterdamThe Netherlands 7th edition 2007
[18] M K Simon and M-S Alouini Digital Communications overFading Channels John Wiley amp Sons Hoboken NJ USA 2ndedition 2005
[19] D S Michalopoulos H A Suraweera G K Karagiannidis andR Schober ldquoAmplify-and-forward relay selectionwith outdatedchannel estimatesrdquo IEEE Transactions on Communications vol60 no 5 pp 1278ndash1290 2012
[20] M- Alouini and M K Simon ldquoGeneric form for average errorprobability of binary signals over fading channelsrdquo ElectronicsLetters vol 34 no 10 pp 949ndash950 1998
[21] P K Vitthaladevuni and M Alouini ldquoExact BER computationof generalized hierarchical PSK constellationsrdquo IEEE Transac-tions on Communications vol 51 no 12 pp 2030ndash2037 2003
[22] C Tellambura and A D S Jayalath ldquoGeneration of bivariateRayleigh and Nakagami-m fading envelopesrdquo IEEE Communi-cations Letters vol 4 no 5 pp 170ndash172 2000
where 119894 is the imaginary unit and 119869119899(sdot) denotes the 119899th-order Bessel function of the first kind From [17 equation(66434)] (15) can be simplified as
Φ = ℎ120573minusℎminus1
119896(
1
1 minus 120588119896
+ 119895)
minusℎminus119898119896
(
radic120588119896119909
1 minus 120588119896
)
119898119896minus1
times exp(120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
times 119871119898119896minus1
ℎ(minus
120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
(16)
where 119871120572
119899(sdot) represents the associated Laguerre polynomial
By usingRodriguesrsquo formula for119871120572
119899(sdot) [17 equation (89701)]
we can obtain
Φ = ℎ120573119896
minusℎminus1(
1
1 minus 120588119896
+ 119895)
minusℎminus119898119896
(
radic120588119896
1 minus 120588119896
)
119898119896minus1
times 119909(119898119896minus1)2 exp(
120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
times
ℎ
sum
119898=0
(
ℎ + 119898119896 minus 1
ℎ minus 119898)
times
1
119898
(
120573119896120588119896119909
(1 minus 120588119896) [1 + 119895 (1 minus 120588119896)]
)
119898
(17)
By substituting (17) into (14) we get the closed-form expres-sion for the PDF of 119892119896 as follows
119891119892119896(119909)
=
119873119896
Γ (119898119896)
119873119896minus1
sum
119895=0
(
119873119896 minus 1
119895) (minus1)
119895119895
International Journal of Distributed Sensor Networks 5
32 End-to-EndOutage Probability Theoutage probability ofthemultihop secondary systemwhich characterizes the prob-ability that the end-to-end SNR falls below a predeterminedthreshold 120574th is given by
Now we will derive the closed-form expression for119883(119886 119887) By exploiting [17 equation (91311)] we can rewritethe CDF of 120574119896 (22) as follows
[1 + 119895 (1 minus 120588119896)] 120573119896119901120574
)
(29)
Then by substituting (22) into (27) and using the variabletransformation 119909 = (119895 + 1)120573119896120574[1 + 119895(1 minus 120588119896)]120573119896119901120574 we have
119886 [1 + 119895 (1 minus 120588119896)] 120573119896119901120574
(119895 + 1) 120573119896
)
(31)
where 119864() denotes MacRobertrsquos E-Function Since Mac-Robertrsquos E-Function can always be expressed in terms ofMeijerrsquos G-function (which can be computed in MATLAB)we can rewrite (31) as
119886 [1 + 119895 (1 minus 120588119896)] 120573119896119901120574
(119895 + 1) 120573119896
)
(32)
With the help of (32) the single-hop average BER for binarytransmissions (28) can be calculated Then by substitutingit into (25) we can obtain the end-to-end average BER forbinary transmissions
332119872-QAMTransmissions Thesingle-hop averageBERof119872-QAM transmissions is given by [8]
BER119896 =
1
radic119872log2radic119872
log2radic119872
sum
119901=1
V119901
sum
119902=0
120601119902119901radic
120596119902
120587
times int
infin
0
120574minus12
119890minus120596119902120574
119865120574119896(120574) 119889120574
(33)
where V119901 = (1minus2minus119901)radic119872minus1120596119902 = 3(2119902 + 1)
2log2119872(2119872minus2)
and 120601119902119901 = (minus1)lfloor1199022119901minus1
radic119872rfloor(2
119901minus1minus lfloor(1199022
119901minus1radic119872) + (12)rfloor)
From (27) the integral term in (33) can be rewritten as119883(120596119902 12) Thus (33) can be expressed as
BER119896 =
1
radic119872log2radic119872
log2radic119872
sum
119901=1
V119901
sum
119902=0
120601119902119901radic
120596119902
120587
119883(120596119902
1
2
) (34)
With the help of (32) andby substituting (34) into (25) we canget the end-to-end average BER for119872-QAM transmissions
333 119872-PSK Transmissions The BER of 24sdot sdot sdot 119872-PSKtransmissions can be obtained by recursive algorithms basedon the generalized 24-PSK [21] Therefore we only need toderive the BER of 24-PSK transmissions
According to [8] the BER for the most significant bit(MSB) and the least significant bit (LSB) with 24-PSKconstellation through a set of angles (120579 = [1205791 1205792]) are givenby
BER (120574) = 12
erfc (sin 1205792radic120574) (35)
BER (120574) = 12
erfc (cos 1205792radic120574) (36)
respectively
By applying integration by parts we can obtain the single-hop average BER for the MSB with 24-PSK as
BER119896 =
sin 12057922radic120587
int
infin
0
120574minus12
119890minussin21205792120574
119865120574119896(120574) 119889120574
=
sin 12057922radic120587
119883(sin21205792
1
2
)
(37)
The single-hop average BER for the LSB with 24-PSK can beobtained in a similar way
Given the single-hop average BER for the MSB andthe LSB cases the end-to-end average BER for 119872-PSKtransmissions can be written in closed form with the help of(32) and (25)
4 Asymptotic Outage Performance Analysis
In this section to study the diversity performance for thecognitive multihop sensor network we derive the asymptoticoutage probability of the secondary system in high SNRregime
According to the asymptotic behavior of the lower incom-plete Gamma function
By substituting (40) into (24) we can get the asymptoticexpression for the outage probability of the secondary system
Diversity order is an important performance metric forcooperative relay system It is defined as
119889 = minus lim120574rarrinfin
log119875out (120574)log 120574
(41)
From (40) we can see that in high SNR regime
119865120574119896(120574th) prop (
1
120574
)
119898119896
(42)
According to (24) we can conclude that the lowest orderof 1120574 in the outage probability expression is determinedby the minimum among the lowest order of 1120574 in 119865120574119896(120574th)(119896 = 1 2 119870) that is min119896=12119870119898119896 In other words thediversity order of the secondary system is min119896=12119870119898119896This indicates that the diversity order of the secondary systemis merely determined by the fading severity parameters of thesecondary transmission links but not affected by other factorssuch as the fading severity parameters of the interferencelinks and the CSI imperfection
5 Numerical Results
In this section analytical results are presented and validatedby simulations We study the effect of various factors onthe outage and error performance of the cognitive multihopsensor network such as the CSI imperfection the fadingseverity of the interference links and the secondary trans-mission links the number of hops and the number of relays
100
10minus1
10minus2
10minus3
10minus4
10minus5minus5 0 5 10
QN0 (dB)
Out
age p
roba
bilit
y
mk = 1
mk = 3
ExactExactExactExact
AsymptoticAsymptoticAsymptoticAsymptotic
SimulationSimulationSimulationSimulation
mkp = 1 120588k = 1
mkp = 1 120588k = 05
mkp = 3 120588k = 1
mkp = 3 120588k = 05
Figure 2 Exact and asymptotic outage probability of the secondarysystem versus system SNR for various fading severity with 119870 = 3
and119873119896 = 3
in clusters The predetermined SNR threshold for successfuldecoding is set as 120574th = 3 dB
We denote the distance between cluster 119896 minus 1 and cluster119896 by 119889119896 and the distance between cluster 119896 minus 1 and PR by119889119896119901 (119896 = 1 2 119870) To consider the impact of path losswe set Ω119896 = 119889
minus120578
119896and Ω119896119901 = 119889
minus120578
119896119901 where 120578 is the path loss
exponent and is set to 3 in this paperTo study the effect of the number of hops we normalize
the distance between SS and SD Specifically SS and SD arelocated at coordinates (00) and (10) respectivelyWe assumeall relay clusters are uniformly distributed on the straight lineconnecting SS and SD that is cluster 119896 is placed at (119896119870 0)so we have 119889119896 = 1119870 PR is located at (051)
For the implementation of simulations we use MonteCarlo method on MATLAB platform During the simu-lations we have to generate pairs of correlated randomvariables following Nakagami-119898 distribution to representthe perfect and imperfect channel coefficients The detailedprocedure of generating a pair of correlated random variablesfollowing Nakagami-119898 distribution by a modified inversetransform method is referred to in [22] The simulationconfiguration is given in Table 1 The specific values ofparameters are described in the caption of each figure
Figure 2 plots the exact and asymptotic outage probabilitycurves for a cognitive three-hop sensor network with119873119896 = 3
relay nodes in each cluster It shows that the exact analyticalresults match well the simulation results Besides the asymp-totic curves are very close to the exact ones in high SNRregime The slope of the curves which indicates the diversityorder rises with the increase in 119898119896 but remains unchangedwith the change of 119898119896119901 or 120588119896 This reveals that the diversityorder is affected by the fading severity of the secondary
International Journal of Distributed Sensor Networks 9
Table 1 Simulation configuration
Number of hops 119870
Number of relays in cluster 119896 (119896 = 1 2 119870) 119873119896 (119873119870 = 1)Correlation coefficient between 119892119896119894 and 119892119896119894 (119896 = 1 2 119870 119894 = 1 2 119873119896) 120588119896
System SNR119876
1198730
SNR threshold for successful decoding 120574th = 3 dBPath loss exponent 120578 = 3
Distance between cluster 119896 minus 1 and cluster 119896 (119896 = 1 2 119870) 119889119896 =
1
119870
Distance between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119889119896119901 =radic(
119896
119870
minus 05)
2
+ (0 minus 1)2
Fading severity parameter of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) 119898119896
Fading severity parameter of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119898119896119901
Average power of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) Ω119896 = 119889
minus120578
119896
Average power of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 sdot sdot sdot 119870) Ω119896119901 = 119889minus120578
119896119901
10minus33
10minus34
10minus35
10minus36
10minus37
10minus38
10minus39
0 02 04 06 08 1
Out
age p
roba
bilit
y
120588k
Analytical mkp = 1
Analytical mkp = 2
Analytical mkp = 3
Figure 3 Outage probability of the secondary system versus 120588119896 with119898119896 = 3 1198761198730 = 5 dB 119870 = 3 and119873119896 = 3
transmission links irrespective of the fading severity ofthe interference links and the CSI imperfection which isconsistent with our analysis in Section 4 From Figure 2 wecan also observe that the severer the fading gets (ie thesmaller 119898119896 or 119898119896119901 is) the worse the outage performancegets Furthermore the fading severity of the interference linkshas much less impact on the outage performance than thatof the transmission links In particular when 119898119896 is small119898119896119901 almost has no effect on the outage performance of thesecondary system
Figure 3 illustrates the outage probability versus 120588k fora cognitive three-hop sensor network with 119873119896 = 3 relay
nodes in each clusterWe can see that the outage performanceimproves with the increase in 120588119896 Specifically the descent ofthe curves is steep at first but then turns gentle when 120588119896 getsclose to 1 This indicates that the outage performance is quitegood even though the CSI is slightly imperfect for example120588119896 = 08 From Figure 3 we can also observe that as 119898119896119901
gets larger the improvement of the outage performance getssmaller
In Figure 4 the impact of the number of hops thenumber of relays in each cluster and the CSI imperfectionon the outage performance is presented We can observe thatthe outage performance improves with the increase in thenumber of hops When there is only one hop (ie 119870 = 1)120588119896 and 119873119896 have no effect on the outage performance sincethe direct transmission does not involve the relay selectionprocess For the cases of 119870 ge 2 it is indicated that whenthe CSI is perfect (ie 120588119896 = 1) assigning only two relaysto each cluster can achieve the best outage performance andmore relays would not help to achieve better performance Itis also shown that when the CSI estimation is totally random(ie 120588119896 = 0) no matter how many relays each cluster hasthe outage performance is the same with the case of 119873119896 = 1
since the relay selection does not work anymore Howeverwhen the CSI is imperfect but not totally random (ie 0 lt120588119896 lt 1) the outage performance improves with the increasein 119873119896 and gradually approaches the case of perfect CSI Inthis case adding more relays to each cluster can mitigate theperformance degradation caused by CSI imperfection
Figure 5 plots the average end-to-end BER curves of acognitive three-hop sensor network with119873119896 = 3 relay nodesin each cluster We show four different kinds of modulationsthat is orthogonal coherent BFSK (119886 = 12 119887 = 12)orthogonal noncoherent BFSK (119886 = 12 119887 = 1) antipodalcoherent BPSK (119886 = 1 119887 = 12) and antipodal differentiallycoherent BPSK (119886 = 1 119887 = 1) It can be observed that
10 International Journal of Distributed Sensor Networks
100
10minus1
10minus2
10minus3
10minus41 2 3 4 5 6 7
Out
age p
roba
bilit
y
Nk
Analytical 120588k = 1
Analytical 120588k = 05
Analytical 120588k = 0
K = 1
K = 2
K = 3
K = 4
Figure 4 Outage probability of the secondary system versusnumber of relays in each cluster for various number of hops with119898119896 = 119898119896119901 = 3 and 1198761198730 = 5 dB
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6minus5 0 5
Bit e
rror
rate
QN0 (dB)
Coherent BFSK 120588 = 1
Coherent BFSK 120588 = 05
Coherent BPSK 120588 = 1
Coherent BPSK 120588 = 05
Noncoherent BFSK120588 = 1
Noncoherent BFSK120588 = 05
Diff coherent BPSK120588 = 1
Diff coherent BPSK120588 = 05
Figure 5 Average end-to-end BER of the secondary system versussystem SNR for various kinds of modulations with 119870 = 3 119873119896 = 3and119898119896 = 119898119896119901 = 3
the CSI imperfection will increase the bit error rate of thesecondary system almost equally for the four different kindsof modulations
6 Conclusion
In this paper we study the performance of a cluster-basedcognitive multihop wireless sensor network with DF partial
relay selection overNakagami-119898 fading channelsThe imper-fect channel knowledge of the secondary transmission linksis taken into account We derive the closed-form expressionsfor the exact outage probability and BER of the secondarysystem as well as the asymptotic outage probability in highSNR regime We also analyze the influence of various factorsfor example the fading severity parameters the number ofrelaying hops the number of available relays in clustersand the CSI imperfection on the secondary transmissionperformance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Science Foundationof China (NSFC) under Grant 61372114 by the National 973Program of China under Grant 2012CB316005 by the JointFunds of NSFC-Guangdong under Grant U1035001 and byBeijing Higher Education Young Elite Teacher Project (noYETP0434)
References
[1] S Haykin ldquoCognitive radio brain-empowered wireless com-municationsrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 2 pp 201ndash220 2005
[2] M Xia and S Aissa ldquoUnderlay cooperative AF relaying incellular networks performance and challengesrdquo IEEE Commu-nications Magazine vol 51 no 12 pp 170ndash176 2013
[3] J N Laneman D N C Tse and G Wornell ldquoCooperativediversity in wireless networks efficient protocols and outagebehaviorrdquo IEEE Transactions on InformationTheory vol 50 no12 pp 3062ndash3080 2004
[4] Z Yan X Zhang and W Wang ldquoExact outage performance ofcognitive relay networkswithmaximum transmit power limitsrdquoIEEE Communications Letters vol 15 no 12 pp 1317ndash1319 2011
[5] V N Q Bao T T Thanh T D Nguyen and T D VuldquoSpectrum sharing-based multi-hop decode-and-forward relaynetworks under interference constraints performance analysisand relay position optimizationrdquo Journal of Communicationsand Networks vol 15 no 3 Article ID 6559352 pp 266ndash2752013
[6] H Phan H Zepernick and H Tran ldquoImpact of interferencepower constraint on multi-hop cognitive amplify-and-forwardrelay networks overNakagami-m fadingrdquo IETCommunicationsvol 7 no 9 pp 860ndash866 2013
[7] K J Kim T Q Duong T A Tsiftsis and V N Q BaoldquoCognitive multihop networks in spectrum sharing environ-ment with multiple licensed usersrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo13) pp2869ndash2873 Budapest Hungary June 2013
[8] A Hyadi M Benjillali M-S Alouini and D B Costa ldquoPer-formance analysis of underlay cognitive multihop regenerativerelaying systems with multiple primary receiversrdquo IEEE Trans-actions on Wireless Communications vol 12 no 12 pp 6418ndash6429 2013
International Journal of Distributed Sensor Networks 11
[9] A Scaglione D L Goeckel and J N Laneman ldquoCooperativecommunications in mobile ad hoc networksrdquo IEEE SignalProcessing Magazine vol 23 no 5 pp 18ndash29 2006
[10] V N Q Bao and H Y Kong ldquoPerformance analysis ofdecode-and-forward relaying with partial relay selection formultihop transmission over rayleigh fading channelsrdquo Journalof Communications and Networks vol 12 no 5 pp 433ndash4412010
[11] Y Wang Z Feng X Chen R Li and P Zhang ldquoOutageconstrained power allocation and relay selection formulti-HOPcognitive networkrdquo in Proceedings of the 76th IEEE VehicularTechnology Conference (VTC-Fall rsquo12) pp 1ndash5 Quebec CanadaSeptember 2012
[12] J Chen J Si Z Li and H Huang ldquoOn the performance ofspectrum sharing cognitive relay networks with imperfect CSIrdquoIEEE Communications Letters vol 16 no 7 pp 1002ndash1005 2012
[13] X Zhang J Xing Z Yan Y Gao and W Wang ldquoOutageperformance study of cognitive relay networks with imperfectchannel knowledgerdquo IEEE Communications Letters vol 17 no1 pp 27ndash30 2013
[14] X Zhang Z Yan Y Gao andWWang ldquoOn the study of outageperformance for cognitive relay networks (CRN) with the nthbest-relay selection in rayleigh-fading channelsrdquo IEEE WirelessCommunications Letters vol 2 no 1 pp 110ndash113 2013
[15] X Zhang J Xing andWWang ldquoOutage analysis of orthogonalspacendashtime block code transmission in cognitive relay networkswith multiple antennasrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 7 pp 3503ndash3508 2013
[16] V N Q Bao T Q Duong and C Tellambura ldquoOn theperformance of cognitive underlay multihop networks withimperfect channel state informationrdquo IEEE Transactions onCommunications vol 61 no 12 pp 4864ndash4873 2013
[17] I S Gradshteyn IM Ryzhik A Jeffrey andD ZwillingerTableof Integrals Series and Products Academic Press AmsterdamThe Netherlands 7th edition 2007
[18] M K Simon and M-S Alouini Digital Communications overFading Channels John Wiley amp Sons Hoboken NJ USA 2ndedition 2005
[19] D S Michalopoulos H A Suraweera G K Karagiannidis andR Schober ldquoAmplify-and-forward relay selectionwith outdatedchannel estimatesrdquo IEEE Transactions on Communications vol60 no 5 pp 1278ndash1290 2012
[20] M- Alouini and M K Simon ldquoGeneric form for average errorprobability of binary signals over fading channelsrdquo ElectronicsLetters vol 34 no 10 pp 949ndash950 1998
[21] P K Vitthaladevuni and M Alouini ldquoExact BER computationof generalized hierarchical PSK constellationsrdquo IEEE Transac-tions on Communications vol 51 no 12 pp 2030ndash2037 2003
[22] C Tellambura and A D S Jayalath ldquoGeneration of bivariateRayleigh and Nakagami-m fading envelopesrdquo IEEE Communi-cations Letters vol 4 no 5 pp 170ndash172 2000
32 End-to-EndOutage Probability Theoutage probability ofthemultihop secondary systemwhich characterizes the prob-ability that the end-to-end SNR falls below a predeterminedthreshold 120574th is given by
Now we will derive the closed-form expression for119883(119886 119887) By exploiting [17 equation (91311)] we can rewritethe CDF of 120574119896 (22) as follows
[1 + 119895 (1 minus 120588119896)] 120573119896119901120574
)
(29)
Then by substituting (22) into (27) and using the variabletransformation 119909 = (119895 + 1)120573119896120574[1 + 119895(1 minus 120588119896)]120573119896119901120574 we have
119886 [1 + 119895 (1 minus 120588119896)] 120573119896119901120574
(119895 + 1) 120573119896
)
(31)
where 119864() denotes MacRobertrsquos E-Function Since Mac-Robertrsquos E-Function can always be expressed in terms ofMeijerrsquos G-function (which can be computed in MATLAB)we can rewrite (31) as
119886 [1 + 119895 (1 minus 120588119896)] 120573119896119901120574
(119895 + 1) 120573119896
)
(32)
With the help of (32) the single-hop average BER for binarytransmissions (28) can be calculated Then by substitutingit into (25) we can obtain the end-to-end average BER forbinary transmissions
332119872-QAMTransmissions Thesingle-hop averageBERof119872-QAM transmissions is given by [8]
BER119896 =
1
radic119872log2radic119872
log2radic119872
sum
119901=1
V119901
sum
119902=0
120601119902119901radic
120596119902
120587
times int
infin
0
120574minus12
119890minus120596119902120574
119865120574119896(120574) 119889120574
(33)
where V119901 = (1minus2minus119901)radic119872minus1120596119902 = 3(2119902 + 1)
2log2119872(2119872minus2)
and 120601119902119901 = (minus1)lfloor1199022119901minus1
radic119872rfloor(2
119901minus1minus lfloor(1199022
119901minus1radic119872) + (12)rfloor)
From (27) the integral term in (33) can be rewritten as119883(120596119902 12) Thus (33) can be expressed as
BER119896 =
1
radic119872log2radic119872
log2radic119872
sum
119901=1
V119901
sum
119902=0
120601119902119901radic
120596119902
120587
119883(120596119902
1
2
) (34)
With the help of (32) andby substituting (34) into (25) we canget the end-to-end average BER for119872-QAM transmissions
333 119872-PSK Transmissions The BER of 24sdot sdot sdot 119872-PSKtransmissions can be obtained by recursive algorithms basedon the generalized 24-PSK [21] Therefore we only need toderive the BER of 24-PSK transmissions
According to [8] the BER for the most significant bit(MSB) and the least significant bit (LSB) with 24-PSKconstellation through a set of angles (120579 = [1205791 1205792]) are givenby
BER (120574) = 12
erfc (sin 1205792radic120574) (35)
BER (120574) = 12
erfc (cos 1205792radic120574) (36)
respectively
By applying integration by parts we can obtain the single-hop average BER for the MSB with 24-PSK as
BER119896 =
sin 12057922radic120587
int
infin
0
120574minus12
119890minussin21205792120574
119865120574119896(120574) 119889120574
=
sin 12057922radic120587
119883(sin21205792
1
2
)
(37)
The single-hop average BER for the LSB with 24-PSK can beobtained in a similar way
Given the single-hop average BER for the MSB andthe LSB cases the end-to-end average BER for 119872-PSKtransmissions can be written in closed form with the help of(32) and (25)
4 Asymptotic Outage Performance Analysis
In this section to study the diversity performance for thecognitive multihop sensor network we derive the asymptoticoutage probability of the secondary system in high SNRregime
According to the asymptotic behavior of the lower incom-plete Gamma function
By substituting (40) into (24) we can get the asymptoticexpression for the outage probability of the secondary system
Diversity order is an important performance metric forcooperative relay system It is defined as
119889 = minus lim120574rarrinfin
log119875out (120574)log 120574
(41)
From (40) we can see that in high SNR regime
119865120574119896(120574th) prop (
1
120574
)
119898119896
(42)
According to (24) we can conclude that the lowest orderof 1120574 in the outage probability expression is determinedby the minimum among the lowest order of 1120574 in 119865120574119896(120574th)(119896 = 1 2 119870) that is min119896=12119870119898119896 In other words thediversity order of the secondary system is min119896=12119870119898119896This indicates that the diversity order of the secondary systemis merely determined by the fading severity parameters of thesecondary transmission links but not affected by other factorssuch as the fading severity parameters of the interferencelinks and the CSI imperfection
5 Numerical Results
In this section analytical results are presented and validatedby simulations We study the effect of various factors onthe outage and error performance of the cognitive multihopsensor network such as the CSI imperfection the fadingseverity of the interference links and the secondary trans-mission links the number of hops and the number of relays
100
10minus1
10minus2
10minus3
10minus4
10minus5minus5 0 5 10
QN0 (dB)
Out
age p
roba
bilit
y
mk = 1
mk = 3
ExactExactExactExact
AsymptoticAsymptoticAsymptoticAsymptotic
SimulationSimulationSimulationSimulation
mkp = 1 120588k = 1
mkp = 1 120588k = 05
mkp = 3 120588k = 1
mkp = 3 120588k = 05
Figure 2 Exact and asymptotic outage probability of the secondarysystem versus system SNR for various fading severity with 119870 = 3
and119873119896 = 3
in clusters The predetermined SNR threshold for successfuldecoding is set as 120574th = 3 dB
We denote the distance between cluster 119896 minus 1 and cluster119896 by 119889119896 and the distance between cluster 119896 minus 1 and PR by119889119896119901 (119896 = 1 2 119870) To consider the impact of path losswe set Ω119896 = 119889
minus120578
119896and Ω119896119901 = 119889
minus120578
119896119901 where 120578 is the path loss
exponent and is set to 3 in this paperTo study the effect of the number of hops we normalize
the distance between SS and SD Specifically SS and SD arelocated at coordinates (00) and (10) respectivelyWe assumeall relay clusters are uniformly distributed on the straight lineconnecting SS and SD that is cluster 119896 is placed at (119896119870 0)so we have 119889119896 = 1119870 PR is located at (051)
For the implementation of simulations we use MonteCarlo method on MATLAB platform During the simu-lations we have to generate pairs of correlated randomvariables following Nakagami-119898 distribution to representthe perfect and imperfect channel coefficients The detailedprocedure of generating a pair of correlated random variablesfollowing Nakagami-119898 distribution by a modified inversetransform method is referred to in [22] The simulationconfiguration is given in Table 1 The specific values ofparameters are described in the caption of each figure
Figure 2 plots the exact and asymptotic outage probabilitycurves for a cognitive three-hop sensor network with119873119896 = 3
relay nodes in each cluster It shows that the exact analyticalresults match well the simulation results Besides the asymp-totic curves are very close to the exact ones in high SNRregime The slope of the curves which indicates the diversityorder rises with the increase in 119898119896 but remains unchangedwith the change of 119898119896119901 or 120588119896 This reveals that the diversityorder is affected by the fading severity of the secondary
International Journal of Distributed Sensor Networks 9
Table 1 Simulation configuration
Number of hops 119870
Number of relays in cluster 119896 (119896 = 1 2 119870) 119873119896 (119873119870 = 1)Correlation coefficient between 119892119896119894 and 119892119896119894 (119896 = 1 2 119870 119894 = 1 2 119873119896) 120588119896
System SNR119876
1198730
SNR threshold for successful decoding 120574th = 3 dBPath loss exponent 120578 = 3
Distance between cluster 119896 minus 1 and cluster 119896 (119896 = 1 2 119870) 119889119896 =
1
119870
Distance between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119889119896119901 =radic(
119896
119870
minus 05)
2
+ (0 minus 1)2
Fading severity parameter of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) 119898119896
Fading severity parameter of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119898119896119901
Average power of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) Ω119896 = 119889
minus120578
119896
Average power of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 sdot sdot sdot 119870) Ω119896119901 = 119889minus120578
119896119901
10minus33
10minus34
10minus35
10minus36
10minus37
10minus38
10minus39
0 02 04 06 08 1
Out
age p
roba
bilit
y
120588k
Analytical mkp = 1
Analytical mkp = 2
Analytical mkp = 3
Figure 3 Outage probability of the secondary system versus 120588119896 with119898119896 = 3 1198761198730 = 5 dB 119870 = 3 and119873119896 = 3
transmission links irrespective of the fading severity ofthe interference links and the CSI imperfection which isconsistent with our analysis in Section 4 From Figure 2 wecan also observe that the severer the fading gets (ie thesmaller 119898119896 or 119898119896119901 is) the worse the outage performancegets Furthermore the fading severity of the interference linkshas much less impact on the outage performance than thatof the transmission links In particular when 119898119896 is small119898119896119901 almost has no effect on the outage performance of thesecondary system
Figure 3 illustrates the outage probability versus 120588k fora cognitive three-hop sensor network with 119873119896 = 3 relay
nodes in each clusterWe can see that the outage performanceimproves with the increase in 120588119896 Specifically the descent ofthe curves is steep at first but then turns gentle when 120588119896 getsclose to 1 This indicates that the outage performance is quitegood even though the CSI is slightly imperfect for example120588119896 = 08 From Figure 3 we can also observe that as 119898119896119901
gets larger the improvement of the outage performance getssmaller
In Figure 4 the impact of the number of hops thenumber of relays in each cluster and the CSI imperfectionon the outage performance is presented We can observe thatthe outage performance improves with the increase in thenumber of hops When there is only one hop (ie 119870 = 1)120588119896 and 119873119896 have no effect on the outage performance sincethe direct transmission does not involve the relay selectionprocess For the cases of 119870 ge 2 it is indicated that whenthe CSI is perfect (ie 120588119896 = 1) assigning only two relaysto each cluster can achieve the best outage performance andmore relays would not help to achieve better performance Itis also shown that when the CSI estimation is totally random(ie 120588119896 = 0) no matter how many relays each cluster hasthe outage performance is the same with the case of 119873119896 = 1
since the relay selection does not work anymore Howeverwhen the CSI is imperfect but not totally random (ie 0 lt120588119896 lt 1) the outage performance improves with the increasein 119873119896 and gradually approaches the case of perfect CSI Inthis case adding more relays to each cluster can mitigate theperformance degradation caused by CSI imperfection
Figure 5 plots the average end-to-end BER curves of acognitive three-hop sensor network with119873119896 = 3 relay nodesin each cluster We show four different kinds of modulationsthat is orthogonal coherent BFSK (119886 = 12 119887 = 12)orthogonal noncoherent BFSK (119886 = 12 119887 = 1) antipodalcoherent BPSK (119886 = 1 119887 = 12) and antipodal differentiallycoherent BPSK (119886 = 1 119887 = 1) It can be observed that
10 International Journal of Distributed Sensor Networks
100
10minus1
10minus2
10minus3
10minus41 2 3 4 5 6 7
Out
age p
roba
bilit
y
Nk
Analytical 120588k = 1
Analytical 120588k = 05
Analytical 120588k = 0
K = 1
K = 2
K = 3
K = 4
Figure 4 Outage probability of the secondary system versusnumber of relays in each cluster for various number of hops with119898119896 = 119898119896119901 = 3 and 1198761198730 = 5 dB
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6minus5 0 5
Bit e
rror
rate
QN0 (dB)
Coherent BFSK 120588 = 1
Coherent BFSK 120588 = 05
Coherent BPSK 120588 = 1
Coherent BPSK 120588 = 05
Noncoherent BFSK120588 = 1
Noncoherent BFSK120588 = 05
Diff coherent BPSK120588 = 1
Diff coherent BPSK120588 = 05
Figure 5 Average end-to-end BER of the secondary system versussystem SNR for various kinds of modulations with 119870 = 3 119873119896 = 3and119898119896 = 119898119896119901 = 3
the CSI imperfection will increase the bit error rate of thesecondary system almost equally for the four different kindsof modulations
6 Conclusion
In this paper we study the performance of a cluster-basedcognitive multihop wireless sensor network with DF partial
relay selection overNakagami-119898 fading channelsThe imper-fect channel knowledge of the secondary transmission linksis taken into account We derive the closed-form expressionsfor the exact outage probability and BER of the secondarysystem as well as the asymptotic outage probability in highSNR regime We also analyze the influence of various factorsfor example the fading severity parameters the number ofrelaying hops the number of available relays in clustersand the CSI imperfection on the secondary transmissionperformance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Science Foundationof China (NSFC) under Grant 61372114 by the National 973Program of China under Grant 2012CB316005 by the JointFunds of NSFC-Guangdong under Grant U1035001 and byBeijing Higher Education Young Elite Teacher Project (noYETP0434)
References
[1] S Haykin ldquoCognitive radio brain-empowered wireless com-municationsrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 2 pp 201ndash220 2005
[2] M Xia and S Aissa ldquoUnderlay cooperative AF relaying incellular networks performance and challengesrdquo IEEE Commu-nications Magazine vol 51 no 12 pp 170ndash176 2013
[3] J N Laneman D N C Tse and G Wornell ldquoCooperativediversity in wireless networks efficient protocols and outagebehaviorrdquo IEEE Transactions on InformationTheory vol 50 no12 pp 3062ndash3080 2004
[4] Z Yan X Zhang and W Wang ldquoExact outage performance ofcognitive relay networkswithmaximum transmit power limitsrdquoIEEE Communications Letters vol 15 no 12 pp 1317ndash1319 2011
[5] V N Q Bao T T Thanh T D Nguyen and T D VuldquoSpectrum sharing-based multi-hop decode-and-forward relaynetworks under interference constraints performance analysisand relay position optimizationrdquo Journal of Communicationsand Networks vol 15 no 3 Article ID 6559352 pp 266ndash2752013
[6] H Phan H Zepernick and H Tran ldquoImpact of interferencepower constraint on multi-hop cognitive amplify-and-forwardrelay networks overNakagami-m fadingrdquo IETCommunicationsvol 7 no 9 pp 860ndash866 2013
[7] K J Kim T Q Duong T A Tsiftsis and V N Q BaoldquoCognitive multihop networks in spectrum sharing environ-ment with multiple licensed usersrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo13) pp2869ndash2873 Budapest Hungary June 2013
[8] A Hyadi M Benjillali M-S Alouini and D B Costa ldquoPer-formance analysis of underlay cognitive multihop regenerativerelaying systems with multiple primary receiversrdquo IEEE Trans-actions on Wireless Communications vol 12 no 12 pp 6418ndash6429 2013
International Journal of Distributed Sensor Networks 11
[9] A Scaglione D L Goeckel and J N Laneman ldquoCooperativecommunications in mobile ad hoc networksrdquo IEEE SignalProcessing Magazine vol 23 no 5 pp 18ndash29 2006
[10] V N Q Bao and H Y Kong ldquoPerformance analysis ofdecode-and-forward relaying with partial relay selection formultihop transmission over rayleigh fading channelsrdquo Journalof Communications and Networks vol 12 no 5 pp 433ndash4412010
[11] Y Wang Z Feng X Chen R Li and P Zhang ldquoOutageconstrained power allocation and relay selection formulti-HOPcognitive networkrdquo in Proceedings of the 76th IEEE VehicularTechnology Conference (VTC-Fall rsquo12) pp 1ndash5 Quebec CanadaSeptember 2012
[12] J Chen J Si Z Li and H Huang ldquoOn the performance ofspectrum sharing cognitive relay networks with imperfect CSIrdquoIEEE Communications Letters vol 16 no 7 pp 1002ndash1005 2012
[13] X Zhang J Xing Z Yan Y Gao and W Wang ldquoOutageperformance study of cognitive relay networks with imperfectchannel knowledgerdquo IEEE Communications Letters vol 17 no1 pp 27ndash30 2013
[14] X Zhang Z Yan Y Gao andWWang ldquoOn the study of outageperformance for cognitive relay networks (CRN) with the nthbest-relay selection in rayleigh-fading channelsrdquo IEEE WirelessCommunications Letters vol 2 no 1 pp 110ndash113 2013
[15] X Zhang J Xing andWWang ldquoOutage analysis of orthogonalspacendashtime block code transmission in cognitive relay networkswith multiple antennasrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 7 pp 3503ndash3508 2013
[16] V N Q Bao T Q Duong and C Tellambura ldquoOn theperformance of cognitive underlay multihop networks withimperfect channel state informationrdquo IEEE Transactions onCommunications vol 61 no 12 pp 4864ndash4873 2013
[17] I S Gradshteyn IM Ryzhik A Jeffrey andD ZwillingerTableof Integrals Series and Products Academic Press AmsterdamThe Netherlands 7th edition 2007
[18] M K Simon and M-S Alouini Digital Communications overFading Channels John Wiley amp Sons Hoboken NJ USA 2ndedition 2005
[19] D S Michalopoulos H A Suraweera G K Karagiannidis andR Schober ldquoAmplify-and-forward relay selectionwith outdatedchannel estimatesrdquo IEEE Transactions on Communications vol60 no 5 pp 1278ndash1290 2012
[20] M- Alouini and M K Simon ldquoGeneric form for average errorprobability of binary signals over fading channelsrdquo ElectronicsLetters vol 34 no 10 pp 949ndash950 1998
[21] P K Vitthaladevuni and M Alouini ldquoExact BER computationof generalized hierarchical PSK constellationsrdquo IEEE Transac-tions on Communications vol 51 no 12 pp 2030ndash2037 2003
[22] C Tellambura and A D S Jayalath ldquoGeneration of bivariateRayleigh and Nakagami-m fading envelopesrdquo IEEE Communi-cations Letters vol 4 no 5 pp 170ndash172 2000
Now we will derive the closed-form expression for119883(119886 119887) By exploiting [17 equation (91311)] we can rewritethe CDF of 120574119896 (22) as follows
[1 + 119895 (1 minus 120588119896)] 120573119896119901120574
)
(29)
Then by substituting (22) into (27) and using the variabletransformation 119909 = (119895 + 1)120573119896120574[1 + 119895(1 minus 120588119896)]120573119896119901120574 we have
119886 [1 + 119895 (1 minus 120588119896)] 120573119896119901120574
(119895 + 1) 120573119896
)
(31)
where 119864() denotes MacRobertrsquos E-Function Since Mac-Robertrsquos E-Function can always be expressed in terms ofMeijerrsquos G-function (which can be computed in MATLAB)we can rewrite (31) as
119886 [1 + 119895 (1 minus 120588119896)] 120573119896119901120574
(119895 + 1) 120573119896
)
(32)
With the help of (32) the single-hop average BER for binarytransmissions (28) can be calculated Then by substitutingit into (25) we can obtain the end-to-end average BER forbinary transmissions
332119872-QAMTransmissions Thesingle-hop averageBERof119872-QAM transmissions is given by [8]
BER119896 =
1
radic119872log2radic119872
log2radic119872
sum
119901=1
V119901
sum
119902=0
120601119902119901radic
120596119902
120587
times int
infin
0
120574minus12
119890minus120596119902120574
119865120574119896(120574) 119889120574
(33)
where V119901 = (1minus2minus119901)radic119872minus1120596119902 = 3(2119902 + 1)
2log2119872(2119872minus2)
and 120601119902119901 = (minus1)lfloor1199022119901minus1
radic119872rfloor(2
119901minus1minus lfloor(1199022
119901minus1radic119872) + (12)rfloor)
From (27) the integral term in (33) can be rewritten as119883(120596119902 12) Thus (33) can be expressed as
BER119896 =
1
radic119872log2radic119872
log2radic119872
sum
119901=1
V119901
sum
119902=0
120601119902119901radic
120596119902
120587
119883(120596119902
1
2
) (34)
With the help of (32) andby substituting (34) into (25) we canget the end-to-end average BER for119872-QAM transmissions
333 119872-PSK Transmissions The BER of 24sdot sdot sdot 119872-PSKtransmissions can be obtained by recursive algorithms basedon the generalized 24-PSK [21] Therefore we only need toderive the BER of 24-PSK transmissions
According to [8] the BER for the most significant bit(MSB) and the least significant bit (LSB) with 24-PSKconstellation through a set of angles (120579 = [1205791 1205792]) are givenby
BER (120574) = 12
erfc (sin 1205792radic120574) (35)
BER (120574) = 12
erfc (cos 1205792radic120574) (36)
respectively
By applying integration by parts we can obtain the single-hop average BER for the MSB with 24-PSK as
BER119896 =
sin 12057922radic120587
int
infin
0
120574minus12
119890minussin21205792120574
119865120574119896(120574) 119889120574
=
sin 12057922radic120587
119883(sin21205792
1
2
)
(37)
The single-hop average BER for the LSB with 24-PSK can beobtained in a similar way
Given the single-hop average BER for the MSB andthe LSB cases the end-to-end average BER for 119872-PSKtransmissions can be written in closed form with the help of(32) and (25)
4 Asymptotic Outage Performance Analysis
In this section to study the diversity performance for thecognitive multihop sensor network we derive the asymptoticoutage probability of the secondary system in high SNRregime
According to the asymptotic behavior of the lower incom-plete Gamma function
By substituting (40) into (24) we can get the asymptoticexpression for the outage probability of the secondary system
Diversity order is an important performance metric forcooperative relay system It is defined as
119889 = minus lim120574rarrinfin
log119875out (120574)log 120574
(41)
From (40) we can see that in high SNR regime
119865120574119896(120574th) prop (
1
120574
)
119898119896
(42)
According to (24) we can conclude that the lowest orderof 1120574 in the outage probability expression is determinedby the minimum among the lowest order of 1120574 in 119865120574119896(120574th)(119896 = 1 2 119870) that is min119896=12119870119898119896 In other words thediversity order of the secondary system is min119896=12119870119898119896This indicates that the diversity order of the secondary systemis merely determined by the fading severity parameters of thesecondary transmission links but not affected by other factorssuch as the fading severity parameters of the interferencelinks and the CSI imperfection
5 Numerical Results
In this section analytical results are presented and validatedby simulations We study the effect of various factors onthe outage and error performance of the cognitive multihopsensor network such as the CSI imperfection the fadingseverity of the interference links and the secondary trans-mission links the number of hops and the number of relays
100
10minus1
10minus2
10minus3
10minus4
10minus5minus5 0 5 10
QN0 (dB)
Out
age p
roba
bilit
y
mk = 1
mk = 3
ExactExactExactExact
AsymptoticAsymptoticAsymptoticAsymptotic
SimulationSimulationSimulationSimulation
mkp = 1 120588k = 1
mkp = 1 120588k = 05
mkp = 3 120588k = 1
mkp = 3 120588k = 05
Figure 2 Exact and asymptotic outage probability of the secondarysystem versus system SNR for various fading severity with 119870 = 3
and119873119896 = 3
in clusters The predetermined SNR threshold for successfuldecoding is set as 120574th = 3 dB
We denote the distance between cluster 119896 minus 1 and cluster119896 by 119889119896 and the distance between cluster 119896 minus 1 and PR by119889119896119901 (119896 = 1 2 119870) To consider the impact of path losswe set Ω119896 = 119889
minus120578
119896and Ω119896119901 = 119889
minus120578
119896119901 where 120578 is the path loss
exponent and is set to 3 in this paperTo study the effect of the number of hops we normalize
the distance between SS and SD Specifically SS and SD arelocated at coordinates (00) and (10) respectivelyWe assumeall relay clusters are uniformly distributed on the straight lineconnecting SS and SD that is cluster 119896 is placed at (119896119870 0)so we have 119889119896 = 1119870 PR is located at (051)
For the implementation of simulations we use MonteCarlo method on MATLAB platform During the simu-lations we have to generate pairs of correlated randomvariables following Nakagami-119898 distribution to representthe perfect and imperfect channel coefficients The detailedprocedure of generating a pair of correlated random variablesfollowing Nakagami-119898 distribution by a modified inversetransform method is referred to in [22] The simulationconfiguration is given in Table 1 The specific values ofparameters are described in the caption of each figure
Figure 2 plots the exact and asymptotic outage probabilitycurves for a cognitive three-hop sensor network with119873119896 = 3
relay nodes in each cluster It shows that the exact analyticalresults match well the simulation results Besides the asymp-totic curves are very close to the exact ones in high SNRregime The slope of the curves which indicates the diversityorder rises with the increase in 119898119896 but remains unchangedwith the change of 119898119896119901 or 120588119896 This reveals that the diversityorder is affected by the fading severity of the secondary
International Journal of Distributed Sensor Networks 9
Table 1 Simulation configuration
Number of hops 119870
Number of relays in cluster 119896 (119896 = 1 2 119870) 119873119896 (119873119870 = 1)Correlation coefficient between 119892119896119894 and 119892119896119894 (119896 = 1 2 119870 119894 = 1 2 119873119896) 120588119896
System SNR119876
1198730
SNR threshold for successful decoding 120574th = 3 dBPath loss exponent 120578 = 3
Distance between cluster 119896 minus 1 and cluster 119896 (119896 = 1 2 119870) 119889119896 =
1
119870
Distance between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119889119896119901 =radic(
119896
119870
minus 05)
2
+ (0 minus 1)2
Fading severity parameter of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) 119898119896
Fading severity parameter of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119898119896119901
Average power of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) Ω119896 = 119889
minus120578
119896
Average power of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 sdot sdot sdot 119870) Ω119896119901 = 119889minus120578
119896119901
10minus33
10minus34
10minus35
10minus36
10minus37
10minus38
10minus39
0 02 04 06 08 1
Out
age p
roba
bilit
y
120588k
Analytical mkp = 1
Analytical mkp = 2
Analytical mkp = 3
Figure 3 Outage probability of the secondary system versus 120588119896 with119898119896 = 3 1198761198730 = 5 dB 119870 = 3 and119873119896 = 3
transmission links irrespective of the fading severity ofthe interference links and the CSI imperfection which isconsistent with our analysis in Section 4 From Figure 2 wecan also observe that the severer the fading gets (ie thesmaller 119898119896 or 119898119896119901 is) the worse the outage performancegets Furthermore the fading severity of the interference linkshas much less impact on the outage performance than thatof the transmission links In particular when 119898119896 is small119898119896119901 almost has no effect on the outage performance of thesecondary system
Figure 3 illustrates the outage probability versus 120588k fora cognitive three-hop sensor network with 119873119896 = 3 relay
nodes in each clusterWe can see that the outage performanceimproves with the increase in 120588119896 Specifically the descent ofthe curves is steep at first but then turns gentle when 120588119896 getsclose to 1 This indicates that the outage performance is quitegood even though the CSI is slightly imperfect for example120588119896 = 08 From Figure 3 we can also observe that as 119898119896119901
gets larger the improvement of the outage performance getssmaller
In Figure 4 the impact of the number of hops thenumber of relays in each cluster and the CSI imperfectionon the outage performance is presented We can observe thatthe outage performance improves with the increase in thenumber of hops When there is only one hop (ie 119870 = 1)120588119896 and 119873119896 have no effect on the outage performance sincethe direct transmission does not involve the relay selectionprocess For the cases of 119870 ge 2 it is indicated that whenthe CSI is perfect (ie 120588119896 = 1) assigning only two relaysto each cluster can achieve the best outage performance andmore relays would not help to achieve better performance Itis also shown that when the CSI estimation is totally random(ie 120588119896 = 0) no matter how many relays each cluster hasthe outage performance is the same with the case of 119873119896 = 1
since the relay selection does not work anymore Howeverwhen the CSI is imperfect but not totally random (ie 0 lt120588119896 lt 1) the outage performance improves with the increasein 119873119896 and gradually approaches the case of perfect CSI Inthis case adding more relays to each cluster can mitigate theperformance degradation caused by CSI imperfection
Figure 5 plots the average end-to-end BER curves of acognitive three-hop sensor network with119873119896 = 3 relay nodesin each cluster We show four different kinds of modulationsthat is orthogonal coherent BFSK (119886 = 12 119887 = 12)orthogonal noncoherent BFSK (119886 = 12 119887 = 1) antipodalcoherent BPSK (119886 = 1 119887 = 12) and antipodal differentiallycoherent BPSK (119886 = 1 119887 = 1) It can be observed that
10 International Journal of Distributed Sensor Networks
100
10minus1
10minus2
10minus3
10minus41 2 3 4 5 6 7
Out
age p
roba
bilit
y
Nk
Analytical 120588k = 1
Analytical 120588k = 05
Analytical 120588k = 0
K = 1
K = 2
K = 3
K = 4
Figure 4 Outage probability of the secondary system versusnumber of relays in each cluster for various number of hops with119898119896 = 119898119896119901 = 3 and 1198761198730 = 5 dB
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6minus5 0 5
Bit e
rror
rate
QN0 (dB)
Coherent BFSK 120588 = 1
Coherent BFSK 120588 = 05
Coherent BPSK 120588 = 1
Coherent BPSK 120588 = 05
Noncoherent BFSK120588 = 1
Noncoherent BFSK120588 = 05
Diff coherent BPSK120588 = 1
Diff coherent BPSK120588 = 05
Figure 5 Average end-to-end BER of the secondary system versussystem SNR for various kinds of modulations with 119870 = 3 119873119896 = 3and119898119896 = 119898119896119901 = 3
the CSI imperfection will increase the bit error rate of thesecondary system almost equally for the four different kindsof modulations
6 Conclusion
In this paper we study the performance of a cluster-basedcognitive multihop wireless sensor network with DF partial
relay selection overNakagami-119898 fading channelsThe imper-fect channel knowledge of the secondary transmission linksis taken into account We derive the closed-form expressionsfor the exact outage probability and BER of the secondarysystem as well as the asymptotic outage probability in highSNR regime We also analyze the influence of various factorsfor example the fading severity parameters the number ofrelaying hops the number of available relays in clustersand the CSI imperfection on the secondary transmissionperformance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Science Foundationof China (NSFC) under Grant 61372114 by the National 973Program of China under Grant 2012CB316005 by the JointFunds of NSFC-Guangdong under Grant U1035001 and byBeijing Higher Education Young Elite Teacher Project (noYETP0434)
References
[1] S Haykin ldquoCognitive radio brain-empowered wireless com-municationsrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 2 pp 201ndash220 2005
[2] M Xia and S Aissa ldquoUnderlay cooperative AF relaying incellular networks performance and challengesrdquo IEEE Commu-nications Magazine vol 51 no 12 pp 170ndash176 2013
[3] J N Laneman D N C Tse and G Wornell ldquoCooperativediversity in wireless networks efficient protocols and outagebehaviorrdquo IEEE Transactions on InformationTheory vol 50 no12 pp 3062ndash3080 2004
[4] Z Yan X Zhang and W Wang ldquoExact outage performance ofcognitive relay networkswithmaximum transmit power limitsrdquoIEEE Communications Letters vol 15 no 12 pp 1317ndash1319 2011
[5] V N Q Bao T T Thanh T D Nguyen and T D VuldquoSpectrum sharing-based multi-hop decode-and-forward relaynetworks under interference constraints performance analysisand relay position optimizationrdquo Journal of Communicationsand Networks vol 15 no 3 Article ID 6559352 pp 266ndash2752013
[6] H Phan H Zepernick and H Tran ldquoImpact of interferencepower constraint on multi-hop cognitive amplify-and-forwardrelay networks overNakagami-m fadingrdquo IETCommunicationsvol 7 no 9 pp 860ndash866 2013
[7] K J Kim T Q Duong T A Tsiftsis and V N Q BaoldquoCognitive multihop networks in spectrum sharing environ-ment with multiple licensed usersrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo13) pp2869ndash2873 Budapest Hungary June 2013
[8] A Hyadi M Benjillali M-S Alouini and D B Costa ldquoPer-formance analysis of underlay cognitive multihop regenerativerelaying systems with multiple primary receiversrdquo IEEE Trans-actions on Wireless Communications vol 12 no 12 pp 6418ndash6429 2013
International Journal of Distributed Sensor Networks 11
[9] A Scaglione D L Goeckel and J N Laneman ldquoCooperativecommunications in mobile ad hoc networksrdquo IEEE SignalProcessing Magazine vol 23 no 5 pp 18ndash29 2006
[10] V N Q Bao and H Y Kong ldquoPerformance analysis ofdecode-and-forward relaying with partial relay selection formultihop transmission over rayleigh fading channelsrdquo Journalof Communications and Networks vol 12 no 5 pp 433ndash4412010
[11] Y Wang Z Feng X Chen R Li and P Zhang ldquoOutageconstrained power allocation and relay selection formulti-HOPcognitive networkrdquo in Proceedings of the 76th IEEE VehicularTechnology Conference (VTC-Fall rsquo12) pp 1ndash5 Quebec CanadaSeptember 2012
[12] J Chen J Si Z Li and H Huang ldquoOn the performance ofspectrum sharing cognitive relay networks with imperfect CSIrdquoIEEE Communications Letters vol 16 no 7 pp 1002ndash1005 2012
[13] X Zhang J Xing Z Yan Y Gao and W Wang ldquoOutageperformance study of cognitive relay networks with imperfectchannel knowledgerdquo IEEE Communications Letters vol 17 no1 pp 27ndash30 2013
[14] X Zhang Z Yan Y Gao andWWang ldquoOn the study of outageperformance for cognitive relay networks (CRN) with the nthbest-relay selection in rayleigh-fading channelsrdquo IEEE WirelessCommunications Letters vol 2 no 1 pp 110ndash113 2013
[15] X Zhang J Xing andWWang ldquoOutage analysis of orthogonalspacendashtime block code transmission in cognitive relay networkswith multiple antennasrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 7 pp 3503ndash3508 2013
[16] V N Q Bao T Q Duong and C Tellambura ldquoOn theperformance of cognitive underlay multihop networks withimperfect channel state informationrdquo IEEE Transactions onCommunications vol 61 no 12 pp 4864ndash4873 2013
[17] I S Gradshteyn IM Ryzhik A Jeffrey andD ZwillingerTableof Integrals Series and Products Academic Press AmsterdamThe Netherlands 7th edition 2007
[18] M K Simon and M-S Alouini Digital Communications overFading Channels John Wiley amp Sons Hoboken NJ USA 2ndedition 2005
[19] D S Michalopoulos H A Suraweera G K Karagiannidis andR Schober ldquoAmplify-and-forward relay selectionwith outdatedchannel estimatesrdquo IEEE Transactions on Communications vol60 no 5 pp 1278ndash1290 2012
[20] M- Alouini and M K Simon ldquoGeneric form for average errorprobability of binary signals over fading channelsrdquo ElectronicsLetters vol 34 no 10 pp 949ndash950 1998
[21] P K Vitthaladevuni and M Alouini ldquoExact BER computationof generalized hierarchical PSK constellationsrdquo IEEE Transac-tions on Communications vol 51 no 12 pp 2030ndash2037 2003
[22] C Tellambura and A D S Jayalath ldquoGeneration of bivariateRayleigh and Nakagami-m fading envelopesrdquo IEEE Communi-cations Letters vol 4 no 5 pp 170ndash172 2000
119886 [1 + 119895 (1 minus 120588119896)] 120573119896119901120574
(119895 + 1) 120573119896
)
(31)
where 119864() denotes MacRobertrsquos E-Function Since Mac-Robertrsquos E-Function can always be expressed in terms ofMeijerrsquos G-function (which can be computed in MATLAB)we can rewrite (31) as
119886 [1 + 119895 (1 minus 120588119896)] 120573119896119901120574
(119895 + 1) 120573119896
)
(32)
With the help of (32) the single-hop average BER for binarytransmissions (28) can be calculated Then by substitutingit into (25) we can obtain the end-to-end average BER forbinary transmissions
332119872-QAMTransmissions Thesingle-hop averageBERof119872-QAM transmissions is given by [8]
BER119896 =
1
radic119872log2radic119872
log2radic119872
sum
119901=1
V119901
sum
119902=0
120601119902119901radic
120596119902
120587
times int
infin
0
120574minus12
119890minus120596119902120574
119865120574119896(120574) 119889120574
(33)
where V119901 = (1minus2minus119901)radic119872minus1120596119902 = 3(2119902 + 1)
2log2119872(2119872minus2)
and 120601119902119901 = (minus1)lfloor1199022119901minus1
radic119872rfloor(2
119901minus1minus lfloor(1199022
119901minus1radic119872) + (12)rfloor)
From (27) the integral term in (33) can be rewritten as119883(120596119902 12) Thus (33) can be expressed as
BER119896 =
1
radic119872log2radic119872
log2radic119872
sum
119901=1
V119901
sum
119902=0
120601119902119901radic
120596119902
120587
119883(120596119902
1
2
) (34)
With the help of (32) andby substituting (34) into (25) we canget the end-to-end average BER for119872-QAM transmissions
333 119872-PSK Transmissions The BER of 24sdot sdot sdot 119872-PSKtransmissions can be obtained by recursive algorithms basedon the generalized 24-PSK [21] Therefore we only need toderive the BER of 24-PSK transmissions
According to [8] the BER for the most significant bit(MSB) and the least significant bit (LSB) with 24-PSKconstellation through a set of angles (120579 = [1205791 1205792]) are givenby
BER (120574) = 12
erfc (sin 1205792radic120574) (35)
BER (120574) = 12
erfc (cos 1205792radic120574) (36)
respectively
By applying integration by parts we can obtain the single-hop average BER for the MSB with 24-PSK as
BER119896 =
sin 12057922radic120587
int
infin
0
120574minus12
119890minussin21205792120574
119865120574119896(120574) 119889120574
=
sin 12057922radic120587
119883(sin21205792
1
2
)
(37)
The single-hop average BER for the LSB with 24-PSK can beobtained in a similar way
Given the single-hop average BER for the MSB andthe LSB cases the end-to-end average BER for 119872-PSKtransmissions can be written in closed form with the help of(32) and (25)
4 Asymptotic Outage Performance Analysis
In this section to study the diversity performance for thecognitive multihop sensor network we derive the asymptoticoutage probability of the secondary system in high SNRregime
According to the asymptotic behavior of the lower incom-plete Gamma function
By substituting (40) into (24) we can get the asymptoticexpression for the outage probability of the secondary system
Diversity order is an important performance metric forcooperative relay system It is defined as
119889 = minus lim120574rarrinfin
log119875out (120574)log 120574
(41)
From (40) we can see that in high SNR regime
119865120574119896(120574th) prop (
1
120574
)
119898119896
(42)
According to (24) we can conclude that the lowest orderof 1120574 in the outage probability expression is determinedby the minimum among the lowest order of 1120574 in 119865120574119896(120574th)(119896 = 1 2 119870) that is min119896=12119870119898119896 In other words thediversity order of the secondary system is min119896=12119870119898119896This indicates that the diversity order of the secondary systemis merely determined by the fading severity parameters of thesecondary transmission links but not affected by other factorssuch as the fading severity parameters of the interferencelinks and the CSI imperfection
5 Numerical Results
In this section analytical results are presented and validatedby simulations We study the effect of various factors onthe outage and error performance of the cognitive multihopsensor network such as the CSI imperfection the fadingseverity of the interference links and the secondary trans-mission links the number of hops and the number of relays
100
10minus1
10minus2
10minus3
10minus4
10minus5minus5 0 5 10
QN0 (dB)
Out
age p
roba
bilit
y
mk = 1
mk = 3
ExactExactExactExact
AsymptoticAsymptoticAsymptoticAsymptotic
SimulationSimulationSimulationSimulation
mkp = 1 120588k = 1
mkp = 1 120588k = 05
mkp = 3 120588k = 1
mkp = 3 120588k = 05
Figure 2 Exact and asymptotic outage probability of the secondarysystem versus system SNR for various fading severity with 119870 = 3
and119873119896 = 3
in clusters The predetermined SNR threshold for successfuldecoding is set as 120574th = 3 dB
We denote the distance between cluster 119896 minus 1 and cluster119896 by 119889119896 and the distance between cluster 119896 minus 1 and PR by119889119896119901 (119896 = 1 2 119870) To consider the impact of path losswe set Ω119896 = 119889
minus120578
119896and Ω119896119901 = 119889
minus120578
119896119901 where 120578 is the path loss
exponent and is set to 3 in this paperTo study the effect of the number of hops we normalize
the distance between SS and SD Specifically SS and SD arelocated at coordinates (00) and (10) respectivelyWe assumeall relay clusters are uniformly distributed on the straight lineconnecting SS and SD that is cluster 119896 is placed at (119896119870 0)so we have 119889119896 = 1119870 PR is located at (051)
For the implementation of simulations we use MonteCarlo method on MATLAB platform During the simu-lations we have to generate pairs of correlated randomvariables following Nakagami-119898 distribution to representthe perfect and imperfect channel coefficients The detailedprocedure of generating a pair of correlated random variablesfollowing Nakagami-119898 distribution by a modified inversetransform method is referred to in [22] The simulationconfiguration is given in Table 1 The specific values ofparameters are described in the caption of each figure
Figure 2 plots the exact and asymptotic outage probabilitycurves for a cognitive three-hop sensor network with119873119896 = 3
relay nodes in each cluster It shows that the exact analyticalresults match well the simulation results Besides the asymp-totic curves are very close to the exact ones in high SNRregime The slope of the curves which indicates the diversityorder rises with the increase in 119898119896 but remains unchangedwith the change of 119898119896119901 or 120588119896 This reveals that the diversityorder is affected by the fading severity of the secondary
International Journal of Distributed Sensor Networks 9
Table 1 Simulation configuration
Number of hops 119870
Number of relays in cluster 119896 (119896 = 1 2 119870) 119873119896 (119873119870 = 1)Correlation coefficient between 119892119896119894 and 119892119896119894 (119896 = 1 2 119870 119894 = 1 2 119873119896) 120588119896
System SNR119876
1198730
SNR threshold for successful decoding 120574th = 3 dBPath loss exponent 120578 = 3
Distance between cluster 119896 minus 1 and cluster 119896 (119896 = 1 2 119870) 119889119896 =
1
119870
Distance between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119889119896119901 =radic(
119896
119870
minus 05)
2
+ (0 minus 1)2
Fading severity parameter of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) 119898119896
Fading severity parameter of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119898119896119901
Average power of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) Ω119896 = 119889
minus120578
119896
Average power of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 sdot sdot sdot 119870) Ω119896119901 = 119889minus120578
119896119901
10minus33
10minus34
10minus35
10minus36
10minus37
10minus38
10minus39
0 02 04 06 08 1
Out
age p
roba
bilit
y
120588k
Analytical mkp = 1
Analytical mkp = 2
Analytical mkp = 3
Figure 3 Outage probability of the secondary system versus 120588119896 with119898119896 = 3 1198761198730 = 5 dB 119870 = 3 and119873119896 = 3
transmission links irrespective of the fading severity ofthe interference links and the CSI imperfection which isconsistent with our analysis in Section 4 From Figure 2 wecan also observe that the severer the fading gets (ie thesmaller 119898119896 or 119898119896119901 is) the worse the outage performancegets Furthermore the fading severity of the interference linkshas much less impact on the outage performance than thatof the transmission links In particular when 119898119896 is small119898119896119901 almost has no effect on the outage performance of thesecondary system
Figure 3 illustrates the outage probability versus 120588k fora cognitive three-hop sensor network with 119873119896 = 3 relay
nodes in each clusterWe can see that the outage performanceimproves with the increase in 120588119896 Specifically the descent ofthe curves is steep at first but then turns gentle when 120588119896 getsclose to 1 This indicates that the outage performance is quitegood even though the CSI is slightly imperfect for example120588119896 = 08 From Figure 3 we can also observe that as 119898119896119901
gets larger the improvement of the outage performance getssmaller
In Figure 4 the impact of the number of hops thenumber of relays in each cluster and the CSI imperfectionon the outage performance is presented We can observe thatthe outage performance improves with the increase in thenumber of hops When there is only one hop (ie 119870 = 1)120588119896 and 119873119896 have no effect on the outage performance sincethe direct transmission does not involve the relay selectionprocess For the cases of 119870 ge 2 it is indicated that whenthe CSI is perfect (ie 120588119896 = 1) assigning only two relaysto each cluster can achieve the best outage performance andmore relays would not help to achieve better performance Itis also shown that when the CSI estimation is totally random(ie 120588119896 = 0) no matter how many relays each cluster hasthe outage performance is the same with the case of 119873119896 = 1
since the relay selection does not work anymore Howeverwhen the CSI is imperfect but not totally random (ie 0 lt120588119896 lt 1) the outage performance improves with the increasein 119873119896 and gradually approaches the case of perfect CSI Inthis case adding more relays to each cluster can mitigate theperformance degradation caused by CSI imperfection
Figure 5 plots the average end-to-end BER curves of acognitive three-hop sensor network with119873119896 = 3 relay nodesin each cluster We show four different kinds of modulationsthat is orthogonal coherent BFSK (119886 = 12 119887 = 12)orthogonal noncoherent BFSK (119886 = 12 119887 = 1) antipodalcoherent BPSK (119886 = 1 119887 = 12) and antipodal differentiallycoherent BPSK (119886 = 1 119887 = 1) It can be observed that
10 International Journal of Distributed Sensor Networks
100
10minus1
10minus2
10minus3
10minus41 2 3 4 5 6 7
Out
age p
roba
bilit
y
Nk
Analytical 120588k = 1
Analytical 120588k = 05
Analytical 120588k = 0
K = 1
K = 2
K = 3
K = 4
Figure 4 Outage probability of the secondary system versusnumber of relays in each cluster for various number of hops with119898119896 = 119898119896119901 = 3 and 1198761198730 = 5 dB
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6minus5 0 5
Bit e
rror
rate
QN0 (dB)
Coherent BFSK 120588 = 1
Coherent BFSK 120588 = 05
Coherent BPSK 120588 = 1
Coherent BPSK 120588 = 05
Noncoherent BFSK120588 = 1
Noncoherent BFSK120588 = 05
Diff coherent BPSK120588 = 1
Diff coherent BPSK120588 = 05
Figure 5 Average end-to-end BER of the secondary system versussystem SNR for various kinds of modulations with 119870 = 3 119873119896 = 3and119898119896 = 119898119896119901 = 3
the CSI imperfection will increase the bit error rate of thesecondary system almost equally for the four different kindsof modulations
6 Conclusion
In this paper we study the performance of a cluster-basedcognitive multihop wireless sensor network with DF partial
relay selection overNakagami-119898 fading channelsThe imper-fect channel knowledge of the secondary transmission linksis taken into account We derive the closed-form expressionsfor the exact outage probability and BER of the secondarysystem as well as the asymptotic outage probability in highSNR regime We also analyze the influence of various factorsfor example the fading severity parameters the number ofrelaying hops the number of available relays in clustersand the CSI imperfection on the secondary transmissionperformance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Science Foundationof China (NSFC) under Grant 61372114 by the National 973Program of China under Grant 2012CB316005 by the JointFunds of NSFC-Guangdong under Grant U1035001 and byBeijing Higher Education Young Elite Teacher Project (noYETP0434)
References
[1] S Haykin ldquoCognitive radio brain-empowered wireless com-municationsrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 2 pp 201ndash220 2005
[2] M Xia and S Aissa ldquoUnderlay cooperative AF relaying incellular networks performance and challengesrdquo IEEE Commu-nications Magazine vol 51 no 12 pp 170ndash176 2013
[3] J N Laneman D N C Tse and G Wornell ldquoCooperativediversity in wireless networks efficient protocols and outagebehaviorrdquo IEEE Transactions on InformationTheory vol 50 no12 pp 3062ndash3080 2004
[4] Z Yan X Zhang and W Wang ldquoExact outage performance ofcognitive relay networkswithmaximum transmit power limitsrdquoIEEE Communications Letters vol 15 no 12 pp 1317ndash1319 2011
[5] V N Q Bao T T Thanh T D Nguyen and T D VuldquoSpectrum sharing-based multi-hop decode-and-forward relaynetworks under interference constraints performance analysisand relay position optimizationrdquo Journal of Communicationsand Networks vol 15 no 3 Article ID 6559352 pp 266ndash2752013
[6] H Phan H Zepernick and H Tran ldquoImpact of interferencepower constraint on multi-hop cognitive amplify-and-forwardrelay networks overNakagami-m fadingrdquo IETCommunicationsvol 7 no 9 pp 860ndash866 2013
[7] K J Kim T Q Duong T A Tsiftsis and V N Q BaoldquoCognitive multihop networks in spectrum sharing environ-ment with multiple licensed usersrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo13) pp2869ndash2873 Budapest Hungary June 2013
[8] A Hyadi M Benjillali M-S Alouini and D B Costa ldquoPer-formance analysis of underlay cognitive multihop regenerativerelaying systems with multiple primary receiversrdquo IEEE Trans-actions on Wireless Communications vol 12 no 12 pp 6418ndash6429 2013
International Journal of Distributed Sensor Networks 11
[9] A Scaglione D L Goeckel and J N Laneman ldquoCooperativecommunications in mobile ad hoc networksrdquo IEEE SignalProcessing Magazine vol 23 no 5 pp 18ndash29 2006
[10] V N Q Bao and H Y Kong ldquoPerformance analysis ofdecode-and-forward relaying with partial relay selection formultihop transmission over rayleigh fading channelsrdquo Journalof Communications and Networks vol 12 no 5 pp 433ndash4412010
[11] Y Wang Z Feng X Chen R Li and P Zhang ldquoOutageconstrained power allocation and relay selection formulti-HOPcognitive networkrdquo in Proceedings of the 76th IEEE VehicularTechnology Conference (VTC-Fall rsquo12) pp 1ndash5 Quebec CanadaSeptember 2012
[12] J Chen J Si Z Li and H Huang ldquoOn the performance ofspectrum sharing cognitive relay networks with imperfect CSIrdquoIEEE Communications Letters vol 16 no 7 pp 1002ndash1005 2012
[13] X Zhang J Xing Z Yan Y Gao and W Wang ldquoOutageperformance study of cognitive relay networks with imperfectchannel knowledgerdquo IEEE Communications Letters vol 17 no1 pp 27ndash30 2013
[14] X Zhang Z Yan Y Gao andWWang ldquoOn the study of outageperformance for cognitive relay networks (CRN) with the nthbest-relay selection in rayleigh-fading channelsrdquo IEEE WirelessCommunications Letters vol 2 no 1 pp 110ndash113 2013
[15] X Zhang J Xing andWWang ldquoOutage analysis of orthogonalspacendashtime block code transmission in cognitive relay networkswith multiple antennasrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 7 pp 3503ndash3508 2013
[16] V N Q Bao T Q Duong and C Tellambura ldquoOn theperformance of cognitive underlay multihop networks withimperfect channel state informationrdquo IEEE Transactions onCommunications vol 61 no 12 pp 4864ndash4873 2013
[17] I S Gradshteyn IM Ryzhik A Jeffrey andD ZwillingerTableof Integrals Series and Products Academic Press AmsterdamThe Netherlands 7th edition 2007
[18] M K Simon and M-S Alouini Digital Communications overFading Channels John Wiley amp Sons Hoboken NJ USA 2ndedition 2005
[19] D S Michalopoulos H A Suraweera G K Karagiannidis andR Schober ldquoAmplify-and-forward relay selectionwith outdatedchannel estimatesrdquo IEEE Transactions on Communications vol60 no 5 pp 1278ndash1290 2012
[20] M- Alouini and M K Simon ldquoGeneric form for average errorprobability of binary signals over fading channelsrdquo ElectronicsLetters vol 34 no 10 pp 949ndash950 1998
[21] P K Vitthaladevuni and M Alouini ldquoExact BER computationof generalized hierarchical PSK constellationsrdquo IEEE Transac-tions on Communications vol 51 no 12 pp 2030ndash2037 2003
[22] C Tellambura and A D S Jayalath ldquoGeneration of bivariateRayleigh and Nakagami-m fading envelopesrdquo IEEE Communi-cations Letters vol 4 no 5 pp 170ndash172 2000
By substituting (40) into (24) we can get the asymptoticexpression for the outage probability of the secondary system
Diversity order is an important performance metric forcooperative relay system It is defined as
119889 = minus lim120574rarrinfin
log119875out (120574)log 120574
(41)
From (40) we can see that in high SNR regime
119865120574119896(120574th) prop (
1
120574
)
119898119896
(42)
According to (24) we can conclude that the lowest orderof 1120574 in the outage probability expression is determinedby the minimum among the lowest order of 1120574 in 119865120574119896(120574th)(119896 = 1 2 119870) that is min119896=12119870119898119896 In other words thediversity order of the secondary system is min119896=12119870119898119896This indicates that the diversity order of the secondary systemis merely determined by the fading severity parameters of thesecondary transmission links but not affected by other factorssuch as the fading severity parameters of the interferencelinks and the CSI imperfection
5 Numerical Results
In this section analytical results are presented and validatedby simulations We study the effect of various factors onthe outage and error performance of the cognitive multihopsensor network such as the CSI imperfection the fadingseverity of the interference links and the secondary trans-mission links the number of hops and the number of relays
100
10minus1
10minus2
10minus3
10minus4
10minus5minus5 0 5 10
QN0 (dB)
Out
age p
roba
bilit
y
mk = 1
mk = 3
ExactExactExactExact
AsymptoticAsymptoticAsymptoticAsymptotic
SimulationSimulationSimulationSimulation
mkp = 1 120588k = 1
mkp = 1 120588k = 05
mkp = 3 120588k = 1
mkp = 3 120588k = 05
Figure 2 Exact and asymptotic outage probability of the secondarysystem versus system SNR for various fading severity with 119870 = 3
and119873119896 = 3
in clusters The predetermined SNR threshold for successfuldecoding is set as 120574th = 3 dB
We denote the distance between cluster 119896 minus 1 and cluster119896 by 119889119896 and the distance between cluster 119896 minus 1 and PR by119889119896119901 (119896 = 1 2 119870) To consider the impact of path losswe set Ω119896 = 119889
minus120578
119896and Ω119896119901 = 119889
minus120578
119896119901 where 120578 is the path loss
exponent and is set to 3 in this paperTo study the effect of the number of hops we normalize
the distance between SS and SD Specifically SS and SD arelocated at coordinates (00) and (10) respectivelyWe assumeall relay clusters are uniformly distributed on the straight lineconnecting SS and SD that is cluster 119896 is placed at (119896119870 0)so we have 119889119896 = 1119870 PR is located at (051)
For the implementation of simulations we use MonteCarlo method on MATLAB platform During the simu-lations we have to generate pairs of correlated randomvariables following Nakagami-119898 distribution to representthe perfect and imperfect channel coefficients The detailedprocedure of generating a pair of correlated random variablesfollowing Nakagami-119898 distribution by a modified inversetransform method is referred to in [22] The simulationconfiguration is given in Table 1 The specific values ofparameters are described in the caption of each figure
Figure 2 plots the exact and asymptotic outage probabilitycurves for a cognitive three-hop sensor network with119873119896 = 3
relay nodes in each cluster It shows that the exact analyticalresults match well the simulation results Besides the asymp-totic curves are very close to the exact ones in high SNRregime The slope of the curves which indicates the diversityorder rises with the increase in 119898119896 but remains unchangedwith the change of 119898119896119901 or 120588119896 This reveals that the diversityorder is affected by the fading severity of the secondary
International Journal of Distributed Sensor Networks 9
Table 1 Simulation configuration
Number of hops 119870
Number of relays in cluster 119896 (119896 = 1 2 119870) 119873119896 (119873119870 = 1)Correlation coefficient between 119892119896119894 and 119892119896119894 (119896 = 1 2 119870 119894 = 1 2 119873119896) 120588119896
System SNR119876
1198730
SNR threshold for successful decoding 120574th = 3 dBPath loss exponent 120578 = 3
Distance between cluster 119896 minus 1 and cluster 119896 (119896 = 1 2 119870) 119889119896 =
1
119870
Distance between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119889119896119901 =radic(
119896
119870
minus 05)
2
+ (0 minus 1)2
Fading severity parameter of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) 119898119896
Fading severity parameter of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119898119896119901
Average power of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) Ω119896 = 119889
minus120578
119896
Average power of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 sdot sdot sdot 119870) Ω119896119901 = 119889minus120578
119896119901
10minus33
10minus34
10minus35
10minus36
10minus37
10minus38
10minus39
0 02 04 06 08 1
Out
age p
roba
bilit
y
120588k
Analytical mkp = 1
Analytical mkp = 2
Analytical mkp = 3
Figure 3 Outage probability of the secondary system versus 120588119896 with119898119896 = 3 1198761198730 = 5 dB 119870 = 3 and119873119896 = 3
transmission links irrespective of the fading severity ofthe interference links and the CSI imperfection which isconsistent with our analysis in Section 4 From Figure 2 wecan also observe that the severer the fading gets (ie thesmaller 119898119896 or 119898119896119901 is) the worse the outage performancegets Furthermore the fading severity of the interference linkshas much less impact on the outage performance than thatof the transmission links In particular when 119898119896 is small119898119896119901 almost has no effect on the outage performance of thesecondary system
Figure 3 illustrates the outage probability versus 120588k fora cognitive three-hop sensor network with 119873119896 = 3 relay
nodes in each clusterWe can see that the outage performanceimproves with the increase in 120588119896 Specifically the descent ofthe curves is steep at first but then turns gentle when 120588119896 getsclose to 1 This indicates that the outage performance is quitegood even though the CSI is slightly imperfect for example120588119896 = 08 From Figure 3 we can also observe that as 119898119896119901
gets larger the improvement of the outage performance getssmaller
In Figure 4 the impact of the number of hops thenumber of relays in each cluster and the CSI imperfectionon the outage performance is presented We can observe thatthe outage performance improves with the increase in thenumber of hops When there is only one hop (ie 119870 = 1)120588119896 and 119873119896 have no effect on the outage performance sincethe direct transmission does not involve the relay selectionprocess For the cases of 119870 ge 2 it is indicated that whenthe CSI is perfect (ie 120588119896 = 1) assigning only two relaysto each cluster can achieve the best outage performance andmore relays would not help to achieve better performance Itis also shown that when the CSI estimation is totally random(ie 120588119896 = 0) no matter how many relays each cluster hasthe outage performance is the same with the case of 119873119896 = 1
since the relay selection does not work anymore Howeverwhen the CSI is imperfect but not totally random (ie 0 lt120588119896 lt 1) the outage performance improves with the increasein 119873119896 and gradually approaches the case of perfect CSI Inthis case adding more relays to each cluster can mitigate theperformance degradation caused by CSI imperfection
Figure 5 plots the average end-to-end BER curves of acognitive three-hop sensor network with119873119896 = 3 relay nodesin each cluster We show four different kinds of modulationsthat is orthogonal coherent BFSK (119886 = 12 119887 = 12)orthogonal noncoherent BFSK (119886 = 12 119887 = 1) antipodalcoherent BPSK (119886 = 1 119887 = 12) and antipodal differentiallycoherent BPSK (119886 = 1 119887 = 1) It can be observed that
10 International Journal of Distributed Sensor Networks
100
10minus1
10minus2
10minus3
10minus41 2 3 4 5 6 7
Out
age p
roba
bilit
y
Nk
Analytical 120588k = 1
Analytical 120588k = 05
Analytical 120588k = 0
K = 1
K = 2
K = 3
K = 4
Figure 4 Outage probability of the secondary system versusnumber of relays in each cluster for various number of hops with119898119896 = 119898119896119901 = 3 and 1198761198730 = 5 dB
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6minus5 0 5
Bit e
rror
rate
QN0 (dB)
Coherent BFSK 120588 = 1
Coherent BFSK 120588 = 05
Coherent BPSK 120588 = 1
Coherent BPSK 120588 = 05
Noncoherent BFSK120588 = 1
Noncoherent BFSK120588 = 05
Diff coherent BPSK120588 = 1
Diff coherent BPSK120588 = 05
Figure 5 Average end-to-end BER of the secondary system versussystem SNR for various kinds of modulations with 119870 = 3 119873119896 = 3and119898119896 = 119898119896119901 = 3
the CSI imperfection will increase the bit error rate of thesecondary system almost equally for the four different kindsof modulations
6 Conclusion
In this paper we study the performance of a cluster-basedcognitive multihop wireless sensor network with DF partial
relay selection overNakagami-119898 fading channelsThe imper-fect channel knowledge of the secondary transmission linksis taken into account We derive the closed-form expressionsfor the exact outage probability and BER of the secondarysystem as well as the asymptotic outage probability in highSNR regime We also analyze the influence of various factorsfor example the fading severity parameters the number ofrelaying hops the number of available relays in clustersand the CSI imperfection on the secondary transmissionperformance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Science Foundationof China (NSFC) under Grant 61372114 by the National 973Program of China under Grant 2012CB316005 by the JointFunds of NSFC-Guangdong under Grant U1035001 and byBeijing Higher Education Young Elite Teacher Project (noYETP0434)
References
[1] S Haykin ldquoCognitive radio brain-empowered wireless com-municationsrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 2 pp 201ndash220 2005
[2] M Xia and S Aissa ldquoUnderlay cooperative AF relaying incellular networks performance and challengesrdquo IEEE Commu-nications Magazine vol 51 no 12 pp 170ndash176 2013
[3] J N Laneman D N C Tse and G Wornell ldquoCooperativediversity in wireless networks efficient protocols and outagebehaviorrdquo IEEE Transactions on InformationTheory vol 50 no12 pp 3062ndash3080 2004
[4] Z Yan X Zhang and W Wang ldquoExact outage performance ofcognitive relay networkswithmaximum transmit power limitsrdquoIEEE Communications Letters vol 15 no 12 pp 1317ndash1319 2011
[5] V N Q Bao T T Thanh T D Nguyen and T D VuldquoSpectrum sharing-based multi-hop decode-and-forward relaynetworks under interference constraints performance analysisand relay position optimizationrdquo Journal of Communicationsand Networks vol 15 no 3 Article ID 6559352 pp 266ndash2752013
[6] H Phan H Zepernick and H Tran ldquoImpact of interferencepower constraint on multi-hop cognitive amplify-and-forwardrelay networks overNakagami-m fadingrdquo IETCommunicationsvol 7 no 9 pp 860ndash866 2013
[7] K J Kim T Q Duong T A Tsiftsis and V N Q BaoldquoCognitive multihop networks in spectrum sharing environ-ment with multiple licensed usersrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo13) pp2869ndash2873 Budapest Hungary June 2013
[8] A Hyadi M Benjillali M-S Alouini and D B Costa ldquoPer-formance analysis of underlay cognitive multihop regenerativerelaying systems with multiple primary receiversrdquo IEEE Trans-actions on Wireless Communications vol 12 no 12 pp 6418ndash6429 2013
International Journal of Distributed Sensor Networks 11
[9] A Scaglione D L Goeckel and J N Laneman ldquoCooperativecommunications in mobile ad hoc networksrdquo IEEE SignalProcessing Magazine vol 23 no 5 pp 18ndash29 2006
[10] V N Q Bao and H Y Kong ldquoPerformance analysis ofdecode-and-forward relaying with partial relay selection formultihop transmission over rayleigh fading channelsrdquo Journalof Communications and Networks vol 12 no 5 pp 433ndash4412010
[11] Y Wang Z Feng X Chen R Li and P Zhang ldquoOutageconstrained power allocation and relay selection formulti-HOPcognitive networkrdquo in Proceedings of the 76th IEEE VehicularTechnology Conference (VTC-Fall rsquo12) pp 1ndash5 Quebec CanadaSeptember 2012
[12] J Chen J Si Z Li and H Huang ldquoOn the performance ofspectrum sharing cognitive relay networks with imperfect CSIrdquoIEEE Communications Letters vol 16 no 7 pp 1002ndash1005 2012
[13] X Zhang J Xing Z Yan Y Gao and W Wang ldquoOutageperformance study of cognitive relay networks with imperfectchannel knowledgerdquo IEEE Communications Letters vol 17 no1 pp 27ndash30 2013
[14] X Zhang Z Yan Y Gao andWWang ldquoOn the study of outageperformance for cognitive relay networks (CRN) with the nthbest-relay selection in rayleigh-fading channelsrdquo IEEE WirelessCommunications Letters vol 2 no 1 pp 110ndash113 2013
[15] X Zhang J Xing andWWang ldquoOutage analysis of orthogonalspacendashtime block code transmission in cognitive relay networkswith multiple antennasrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 7 pp 3503ndash3508 2013
[16] V N Q Bao T Q Duong and C Tellambura ldquoOn theperformance of cognitive underlay multihop networks withimperfect channel state informationrdquo IEEE Transactions onCommunications vol 61 no 12 pp 4864ndash4873 2013
[17] I S Gradshteyn IM Ryzhik A Jeffrey andD ZwillingerTableof Integrals Series and Products Academic Press AmsterdamThe Netherlands 7th edition 2007
[18] M K Simon and M-S Alouini Digital Communications overFading Channels John Wiley amp Sons Hoboken NJ USA 2ndedition 2005
[19] D S Michalopoulos H A Suraweera G K Karagiannidis andR Schober ldquoAmplify-and-forward relay selectionwith outdatedchannel estimatesrdquo IEEE Transactions on Communications vol60 no 5 pp 1278ndash1290 2012
[20] M- Alouini and M K Simon ldquoGeneric form for average errorprobability of binary signals over fading channelsrdquo ElectronicsLetters vol 34 no 10 pp 949ndash950 1998
[21] P K Vitthaladevuni and M Alouini ldquoExact BER computationof generalized hierarchical PSK constellationsrdquo IEEE Transac-tions on Communications vol 51 no 12 pp 2030ndash2037 2003
[22] C Tellambura and A D S Jayalath ldquoGeneration of bivariateRayleigh and Nakagami-m fading envelopesrdquo IEEE Communi-cations Letters vol 4 no 5 pp 170ndash172 2000
International Journal of Distributed Sensor Networks 9
Table 1 Simulation configuration
Number of hops 119870
Number of relays in cluster 119896 (119896 = 1 2 119870) 119873119896 (119873119870 = 1)Correlation coefficient between 119892119896119894 and 119892119896119894 (119896 = 1 2 119870 119894 = 1 2 119873119896) 120588119896
System SNR119876
1198730
SNR threshold for successful decoding 120574th = 3 dBPath loss exponent 120578 = 3
Distance between cluster 119896 minus 1 and cluster 119896 (119896 = 1 2 119870) 119889119896 =
1
119870
Distance between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119889119896119901 =radic(
119896
119870
minus 05)
2
+ (0 minus 1)2
Fading severity parameter of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) 119898119896
Fading severity parameter of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 119870) 119898119896119901
Average power of the secondary transmission link between cluster 119896 minus 1 and cluster119896 (119896 = 1 2 119870) Ω119896 = 119889
minus120578
119896
Average power of the interference link between cluster 119896 minus 1 and PR (119896 = 1 2 sdot sdot sdot 119870) Ω119896119901 = 119889minus120578
119896119901
10minus33
10minus34
10minus35
10minus36
10minus37
10minus38
10minus39
0 02 04 06 08 1
Out
age p
roba
bilit
y
120588k
Analytical mkp = 1
Analytical mkp = 2
Analytical mkp = 3
Figure 3 Outage probability of the secondary system versus 120588119896 with119898119896 = 3 1198761198730 = 5 dB 119870 = 3 and119873119896 = 3
transmission links irrespective of the fading severity ofthe interference links and the CSI imperfection which isconsistent with our analysis in Section 4 From Figure 2 wecan also observe that the severer the fading gets (ie thesmaller 119898119896 or 119898119896119901 is) the worse the outage performancegets Furthermore the fading severity of the interference linkshas much less impact on the outage performance than thatof the transmission links In particular when 119898119896 is small119898119896119901 almost has no effect on the outage performance of thesecondary system
Figure 3 illustrates the outage probability versus 120588k fora cognitive three-hop sensor network with 119873119896 = 3 relay
nodes in each clusterWe can see that the outage performanceimproves with the increase in 120588119896 Specifically the descent ofthe curves is steep at first but then turns gentle when 120588119896 getsclose to 1 This indicates that the outage performance is quitegood even though the CSI is slightly imperfect for example120588119896 = 08 From Figure 3 we can also observe that as 119898119896119901
gets larger the improvement of the outage performance getssmaller
In Figure 4 the impact of the number of hops thenumber of relays in each cluster and the CSI imperfectionon the outage performance is presented We can observe thatthe outage performance improves with the increase in thenumber of hops When there is only one hop (ie 119870 = 1)120588119896 and 119873119896 have no effect on the outage performance sincethe direct transmission does not involve the relay selectionprocess For the cases of 119870 ge 2 it is indicated that whenthe CSI is perfect (ie 120588119896 = 1) assigning only two relaysto each cluster can achieve the best outage performance andmore relays would not help to achieve better performance Itis also shown that when the CSI estimation is totally random(ie 120588119896 = 0) no matter how many relays each cluster hasthe outage performance is the same with the case of 119873119896 = 1
since the relay selection does not work anymore Howeverwhen the CSI is imperfect but not totally random (ie 0 lt120588119896 lt 1) the outage performance improves with the increasein 119873119896 and gradually approaches the case of perfect CSI Inthis case adding more relays to each cluster can mitigate theperformance degradation caused by CSI imperfection
Figure 5 plots the average end-to-end BER curves of acognitive three-hop sensor network with119873119896 = 3 relay nodesin each cluster We show four different kinds of modulationsthat is orthogonal coherent BFSK (119886 = 12 119887 = 12)orthogonal noncoherent BFSK (119886 = 12 119887 = 1) antipodalcoherent BPSK (119886 = 1 119887 = 12) and antipodal differentiallycoherent BPSK (119886 = 1 119887 = 1) It can be observed that
10 International Journal of Distributed Sensor Networks
100
10minus1
10minus2
10minus3
10minus41 2 3 4 5 6 7
Out
age p
roba
bilit
y
Nk
Analytical 120588k = 1
Analytical 120588k = 05
Analytical 120588k = 0
K = 1
K = 2
K = 3
K = 4
Figure 4 Outage probability of the secondary system versusnumber of relays in each cluster for various number of hops with119898119896 = 119898119896119901 = 3 and 1198761198730 = 5 dB
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6minus5 0 5
Bit e
rror
rate
QN0 (dB)
Coherent BFSK 120588 = 1
Coherent BFSK 120588 = 05
Coherent BPSK 120588 = 1
Coherent BPSK 120588 = 05
Noncoherent BFSK120588 = 1
Noncoherent BFSK120588 = 05
Diff coherent BPSK120588 = 1
Diff coherent BPSK120588 = 05
Figure 5 Average end-to-end BER of the secondary system versussystem SNR for various kinds of modulations with 119870 = 3 119873119896 = 3and119898119896 = 119898119896119901 = 3
the CSI imperfection will increase the bit error rate of thesecondary system almost equally for the four different kindsof modulations
6 Conclusion
In this paper we study the performance of a cluster-basedcognitive multihop wireless sensor network with DF partial
relay selection overNakagami-119898 fading channelsThe imper-fect channel knowledge of the secondary transmission linksis taken into account We derive the closed-form expressionsfor the exact outage probability and BER of the secondarysystem as well as the asymptotic outage probability in highSNR regime We also analyze the influence of various factorsfor example the fading severity parameters the number ofrelaying hops the number of available relays in clustersand the CSI imperfection on the secondary transmissionperformance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Science Foundationof China (NSFC) under Grant 61372114 by the National 973Program of China under Grant 2012CB316005 by the JointFunds of NSFC-Guangdong under Grant U1035001 and byBeijing Higher Education Young Elite Teacher Project (noYETP0434)
References
[1] S Haykin ldquoCognitive radio brain-empowered wireless com-municationsrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 2 pp 201ndash220 2005
[2] M Xia and S Aissa ldquoUnderlay cooperative AF relaying incellular networks performance and challengesrdquo IEEE Commu-nications Magazine vol 51 no 12 pp 170ndash176 2013
[3] J N Laneman D N C Tse and G Wornell ldquoCooperativediversity in wireless networks efficient protocols and outagebehaviorrdquo IEEE Transactions on InformationTheory vol 50 no12 pp 3062ndash3080 2004
[4] Z Yan X Zhang and W Wang ldquoExact outage performance ofcognitive relay networkswithmaximum transmit power limitsrdquoIEEE Communications Letters vol 15 no 12 pp 1317ndash1319 2011
[5] V N Q Bao T T Thanh T D Nguyen and T D VuldquoSpectrum sharing-based multi-hop decode-and-forward relaynetworks under interference constraints performance analysisand relay position optimizationrdquo Journal of Communicationsand Networks vol 15 no 3 Article ID 6559352 pp 266ndash2752013
[6] H Phan H Zepernick and H Tran ldquoImpact of interferencepower constraint on multi-hop cognitive amplify-and-forwardrelay networks overNakagami-m fadingrdquo IETCommunicationsvol 7 no 9 pp 860ndash866 2013
[7] K J Kim T Q Duong T A Tsiftsis and V N Q BaoldquoCognitive multihop networks in spectrum sharing environ-ment with multiple licensed usersrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo13) pp2869ndash2873 Budapest Hungary June 2013
[8] A Hyadi M Benjillali M-S Alouini and D B Costa ldquoPer-formance analysis of underlay cognitive multihop regenerativerelaying systems with multiple primary receiversrdquo IEEE Trans-actions on Wireless Communications vol 12 no 12 pp 6418ndash6429 2013
International Journal of Distributed Sensor Networks 11
[9] A Scaglione D L Goeckel and J N Laneman ldquoCooperativecommunications in mobile ad hoc networksrdquo IEEE SignalProcessing Magazine vol 23 no 5 pp 18ndash29 2006
[10] V N Q Bao and H Y Kong ldquoPerformance analysis ofdecode-and-forward relaying with partial relay selection formultihop transmission over rayleigh fading channelsrdquo Journalof Communications and Networks vol 12 no 5 pp 433ndash4412010
[11] Y Wang Z Feng X Chen R Li and P Zhang ldquoOutageconstrained power allocation and relay selection formulti-HOPcognitive networkrdquo in Proceedings of the 76th IEEE VehicularTechnology Conference (VTC-Fall rsquo12) pp 1ndash5 Quebec CanadaSeptember 2012
[12] J Chen J Si Z Li and H Huang ldquoOn the performance ofspectrum sharing cognitive relay networks with imperfect CSIrdquoIEEE Communications Letters vol 16 no 7 pp 1002ndash1005 2012
[13] X Zhang J Xing Z Yan Y Gao and W Wang ldquoOutageperformance study of cognitive relay networks with imperfectchannel knowledgerdquo IEEE Communications Letters vol 17 no1 pp 27ndash30 2013
[14] X Zhang Z Yan Y Gao andWWang ldquoOn the study of outageperformance for cognitive relay networks (CRN) with the nthbest-relay selection in rayleigh-fading channelsrdquo IEEE WirelessCommunications Letters vol 2 no 1 pp 110ndash113 2013
[15] X Zhang J Xing andWWang ldquoOutage analysis of orthogonalspacendashtime block code transmission in cognitive relay networkswith multiple antennasrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 7 pp 3503ndash3508 2013
[16] V N Q Bao T Q Duong and C Tellambura ldquoOn theperformance of cognitive underlay multihop networks withimperfect channel state informationrdquo IEEE Transactions onCommunications vol 61 no 12 pp 4864ndash4873 2013
[17] I S Gradshteyn IM Ryzhik A Jeffrey andD ZwillingerTableof Integrals Series and Products Academic Press AmsterdamThe Netherlands 7th edition 2007
[18] M K Simon and M-S Alouini Digital Communications overFading Channels John Wiley amp Sons Hoboken NJ USA 2ndedition 2005
[19] D S Michalopoulos H A Suraweera G K Karagiannidis andR Schober ldquoAmplify-and-forward relay selectionwith outdatedchannel estimatesrdquo IEEE Transactions on Communications vol60 no 5 pp 1278ndash1290 2012
[20] M- Alouini and M K Simon ldquoGeneric form for average errorprobability of binary signals over fading channelsrdquo ElectronicsLetters vol 34 no 10 pp 949ndash950 1998
[21] P K Vitthaladevuni and M Alouini ldquoExact BER computationof generalized hierarchical PSK constellationsrdquo IEEE Transac-tions on Communications vol 51 no 12 pp 2030ndash2037 2003
[22] C Tellambura and A D S Jayalath ldquoGeneration of bivariateRayleigh and Nakagami-m fading envelopesrdquo IEEE Communi-cations Letters vol 4 no 5 pp 170ndash172 2000
10 International Journal of Distributed Sensor Networks
100
10minus1
10minus2
10minus3
10minus41 2 3 4 5 6 7
Out
age p
roba
bilit
y
Nk
Analytical 120588k = 1
Analytical 120588k = 05
Analytical 120588k = 0
K = 1
K = 2
K = 3
K = 4
Figure 4 Outage probability of the secondary system versusnumber of relays in each cluster for various number of hops with119898119896 = 119898119896119901 = 3 and 1198761198730 = 5 dB
10minus1
10minus2
10minus3
10minus4
10minus5
10minus6minus5 0 5
Bit e
rror
rate
QN0 (dB)
Coherent BFSK 120588 = 1
Coherent BFSK 120588 = 05
Coherent BPSK 120588 = 1
Coherent BPSK 120588 = 05
Noncoherent BFSK120588 = 1
Noncoherent BFSK120588 = 05
Diff coherent BPSK120588 = 1
Diff coherent BPSK120588 = 05
Figure 5 Average end-to-end BER of the secondary system versussystem SNR for various kinds of modulations with 119870 = 3 119873119896 = 3and119898119896 = 119898119896119901 = 3
the CSI imperfection will increase the bit error rate of thesecondary system almost equally for the four different kindsof modulations
6 Conclusion
In this paper we study the performance of a cluster-basedcognitive multihop wireless sensor network with DF partial
relay selection overNakagami-119898 fading channelsThe imper-fect channel knowledge of the secondary transmission linksis taken into account We derive the closed-form expressionsfor the exact outage probability and BER of the secondarysystem as well as the asymptotic outage probability in highSNR regime We also analyze the influence of various factorsfor example the fading severity parameters the number ofrelaying hops the number of available relays in clustersand the CSI imperfection on the secondary transmissionperformance
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by the National Science Foundationof China (NSFC) under Grant 61372114 by the National 973Program of China under Grant 2012CB316005 by the JointFunds of NSFC-Guangdong under Grant U1035001 and byBeijing Higher Education Young Elite Teacher Project (noYETP0434)
References
[1] S Haykin ldquoCognitive radio brain-empowered wireless com-municationsrdquo IEEE Journal on Selected Areas in Communica-tions vol 23 no 2 pp 201ndash220 2005
[2] M Xia and S Aissa ldquoUnderlay cooperative AF relaying incellular networks performance and challengesrdquo IEEE Commu-nications Magazine vol 51 no 12 pp 170ndash176 2013
[3] J N Laneman D N C Tse and G Wornell ldquoCooperativediversity in wireless networks efficient protocols and outagebehaviorrdquo IEEE Transactions on InformationTheory vol 50 no12 pp 3062ndash3080 2004
[4] Z Yan X Zhang and W Wang ldquoExact outage performance ofcognitive relay networkswithmaximum transmit power limitsrdquoIEEE Communications Letters vol 15 no 12 pp 1317ndash1319 2011
[5] V N Q Bao T T Thanh T D Nguyen and T D VuldquoSpectrum sharing-based multi-hop decode-and-forward relaynetworks under interference constraints performance analysisand relay position optimizationrdquo Journal of Communicationsand Networks vol 15 no 3 Article ID 6559352 pp 266ndash2752013
[6] H Phan H Zepernick and H Tran ldquoImpact of interferencepower constraint on multi-hop cognitive amplify-and-forwardrelay networks overNakagami-m fadingrdquo IETCommunicationsvol 7 no 9 pp 860ndash866 2013
[7] K J Kim T Q Duong T A Tsiftsis and V N Q BaoldquoCognitive multihop networks in spectrum sharing environ-ment with multiple licensed usersrdquo in Proceedings of the IEEEInternational Conference on Communications (ICC rsquo13) pp2869ndash2873 Budapest Hungary June 2013
[8] A Hyadi M Benjillali M-S Alouini and D B Costa ldquoPer-formance analysis of underlay cognitive multihop regenerativerelaying systems with multiple primary receiversrdquo IEEE Trans-actions on Wireless Communications vol 12 no 12 pp 6418ndash6429 2013
International Journal of Distributed Sensor Networks 11
[9] A Scaglione D L Goeckel and J N Laneman ldquoCooperativecommunications in mobile ad hoc networksrdquo IEEE SignalProcessing Magazine vol 23 no 5 pp 18ndash29 2006
[10] V N Q Bao and H Y Kong ldquoPerformance analysis ofdecode-and-forward relaying with partial relay selection formultihop transmission over rayleigh fading channelsrdquo Journalof Communications and Networks vol 12 no 5 pp 433ndash4412010
[11] Y Wang Z Feng X Chen R Li and P Zhang ldquoOutageconstrained power allocation and relay selection formulti-HOPcognitive networkrdquo in Proceedings of the 76th IEEE VehicularTechnology Conference (VTC-Fall rsquo12) pp 1ndash5 Quebec CanadaSeptember 2012
[12] J Chen J Si Z Li and H Huang ldquoOn the performance ofspectrum sharing cognitive relay networks with imperfect CSIrdquoIEEE Communications Letters vol 16 no 7 pp 1002ndash1005 2012
[13] X Zhang J Xing Z Yan Y Gao and W Wang ldquoOutageperformance study of cognitive relay networks with imperfectchannel knowledgerdquo IEEE Communications Letters vol 17 no1 pp 27ndash30 2013
[14] X Zhang Z Yan Y Gao andWWang ldquoOn the study of outageperformance for cognitive relay networks (CRN) with the nthbest-relay selection in rayleigh-fading channelsrdquo IEEE WirelessCommunications Letters vol 2 no 1 pp 110ndash113 2013
[15] X Zhang J Xing andWWang ldquoOutage analysis of orthogonalspacendashtime block code transmission in cognitive relay networkswith multiple antennasrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 7 pp 3503ndash3508 2013
[16] V N Q Bao T Q Duong and C Tellambura ldquoOn theperformance of cognitive underlay multihop networks withimperfect channel state informationrdquo IEEE Transactions onCommunications vol 61 no 12 pp 4864ndash4873 2013
[17] I S Gradshteyn IM Ryzhik A Jeffrey andD ZwillingerTableof Integrals Series and Products Academic Press AmsterdamThe Netherlands 7th edition 2007
[18] M K Simon and M-S Alouini Digital Communications overFading Channels John Wiley amp Sons Hoboken NJ USA 2ndedition 2005
[19] D S Michalopoulos H A Suraweera G K Karagiannidis andR Schober ldquoAmplify-and-forward relay selectionwith outdatedchannel estimatesrdquo IEEE Transactions on Communications vol60 no 5 pp 1278ndash1290 2012
[20] M- Alouini and M K Simon ldquoGeneric form for average errorprobability of binary signals over fading channelsrdquo ElectronicsLetters vol 34 no 10 pp 949ndash950 1998
[21] P K Vitthaladevuni and M Alouini ldquoExact BER computationof generalized hierarchical PSK constellationsrdquo IEEE Transac-tions on Communications vol 51 no 12 pp 2030ndash2037 2003
[22] C Tellambura and A D S Jayalath ldquoGeneration of bivariateRayleigh and Nakagami-m fading envelopesrdquo IEEE Communi-cations Letters vol 4 no 5 pp 170ndash172 2000
International Journal of Distributed Sensor Networks 11
[9] A Scaglione D L Goeckel and J N Laneman ldquoCooperativecommunications in mobile ad hoc networksrdquo IEEE SignalProcessing Magazine vol 23 no 5 pp 18ndash29 2006
[10] V N Q Bao and H Y Kong ldquoPerformance analysis ofdecode-and-forward relaying with partial relay selection formultihop transmission over rayleigh fading channelsrdquo Journalof Communications and Networks vol 12 no 5 pp 433ndash4412010
[11] Y Wang Z Feng X Chen R Li and P Zhang ldquoOutageconstrained power allocation and relay selection formulti-HOPcognitive networkrdquo in Proceedings of the 76th IEEE VehicularTechnology Conference (VTC-Fall rsquo12) pp 1ndash5 Quebec CanadaSeptember 2012
[12] J Chen J Si Z Li and H Huang ldquoOn the performance ofspectrum sharing cognitive relay networks with imperfect CSIrdquoIEEE Communications Letters vol 16 no 7 pp 1002ndash1005 2012
[13] X Zhang J Xing Z Yan Y Gao and W Wang ldquoOutageperformance study of cognitive relay networks with imperfectchannel knowledgerdquo IEEE Communications Letters vol 17 no1 pp 27ndash30 2013
[14] X Zhang Z Yan Y Gao andWWang ldquoOn the study of outageperformance for cognitive relay networks (CRN) with the nthbest-relay selection in rayleigh-fading channelsrdquo IEEE WirelessCommunications Letters vol 2 no 1 pp 110ndash113 2013
[15] X Zhang J Xing andWWang ldquoOutage analysis of orthogonalspacendashtime block code transmission in cognitive relay networkswith multiple antennasrdquo IEEE Transactions on Vehicular Tech-nology vol 62 no 7 pp 3503ndash3508 2013
[16] V N Q Bao T Q Duong and C Tellambura ldquoOn theperformance of cognitive underlay multihop networks withimperfect channel state informationrdquo IEEE Transactions onCommunications vol 61 no 12 pp 4864ndash4873 2013
[17] I S Gradshteyn IM Ryzhik A Jeffrey andD ZwillingerTableof Integrals Series and Products Academic Press AmsterdamThe Netherlands 7th edition 2007
[18] M K Simon and M-S Alouini Digital Communications overFading Channels John Wiley amp Sons Hoboken NJ USA 2ndedition 2005
[19] D S Michalopoulos H A Suraweera G K Karagiannidis andR Schober ldquoAmplify-and-forward relay selectionwith outdatedchannel estimatesrdquo IEEE Transactions on Communications vol60 no 5 pp 1278ndash1290 2012
[20] M- Alouini and M K Simon ldquoGeneric form for average errorprobability of binary signals over fading channelsrdquo ElectronicsLetters vol 34 no 10 pp 949ndash950 1998
[21] P K Vitthaladevuni and M Alouini ldquoExact BER computationof generalized hierarchical PSK constellationsrdquo IEEE Transac-tions on Communications vol 51 no 12 pp 2030ndash2037 2003
[22] C Tellambura and A D S Jayalath ldquoGeneration of bivariateRayleigh and Nakagami-m fading envelopesrdquo IEEE Communi-cations Letters vol 4 no 5 pp 170ndash172 2000
