Performance Analysis of Partial Relay Selectionwith Multi-antenna Destination Cooperation

(Invited Paper)Vo Nguyen Quoc Bao, Dang Hoai Bac, Le Quoc Cuong, Le Quang Phu, and Tran Dinh Thuan

Telecommunications DepartmentPosts and Telecommunications Institute of Technology, Vietnam

Email: baovnq@ptithcm.edu.vn

Abstract—Selection diversity combining employed in conjunc-tion with partial relay selection is proposed to reduce thecomplexity at the destination installed multi-antennae. The per-formance measure of the system including outage probability,probability of signal-to-noise ratio (SNR) gain, ergodic capacity,symbol error rate and bit error rate over Rayleigh fadingchannels are derived. Numerical results show that the diversitygain of the proposed protocol is comparable to that of the partialrelay selection networks with maximal-ratio-combiner-equippeddestination while offering lower implementation complexity.

Index Terms—Selection Combining, Rayleigh fading channels,Partial relay selection, Cooperative communication, MIMO.

I. INTRODUCTION

Recent studies have shown that spatial diversity gain canbe achieved in a distributed manner through the concept ofcooperative communication [1]. The basic premise behindcooperative diversity is to improve the direct communicationbetween the source and the destination by making use the helpof geographically distributed wireless relays. In such networks,relay selection technique is pivotal to the operation of co-operative network for achieving spatial diversity. Dependingon the strategy to select the best relay, there are two typesof relay selection protocols classied in the literature: fullrelay selection [2] and partial relay selection [3]. Althoughlimited by its achievable diversity gain [3], [4], partial relayselection is a promising relay selection technique for resource-constrained networks since it reduces the need of perfecttime synchronization and the centralized processing approach,desired by the full relay selection approach, thereby prolongingthe network lifetime [3].

Thus far, the performance of partial relay selection networkshas been investigated corresponding to various system andchannel models, e.g., see [5]–[9]. In particular, the perfor-mance of dual hop cooperative networks using semi-blink(xed gain) relays and partial relay selection was providedin [5], [6] for Rayleigh fading channels. For Ricean fadingchannels, the work in [7] was devoted to offer asymptoticexpressions of the average symbol error rate of M -ary phase-shift keying (M -PSK) and M -ary quadrature amplitude (M -QAM) signals with amplify-and-forward (AF) protocol. Tak-ing into account the impact of outdated channel state infor-mation due to feedback delay, Suraweera et. al. has derivedthe outage probability and the average bit error rate of partial

relay selection networks [8]. Very recently, as a promisingsolution to improve diversity performance, the partial relayselection system with a multi-antenna sited destination hasbeen proposed in [9], although, its performance evaluation wasrestricted to outage probability and achievable diversity order.Beside, the proposed scheme in [9] has some limitations:

• The system complexity arises from employing a separateradio frequency (RF) chain for every employed antennaresulting in a signicant increase in the implementationcost manner as well as power consumption.

• The performance of maximal ratio combiner (MRC) atthe destination is sensitive to channel-estimation errors,which becomes more signicant when the instantaneoussignal-to-noise ratio is low.

In this paper, motivated by all of the above, we, for therst time, propose dual-hop partial relay selection networkswith selection combining (SC) at the destination. The benetof using these two low complexity techniques, i.e., partialrelay selection and selection combining, is clear, particularlyin practical cost-stringent applications on power- and/or size-limited terminals, such as wireless sensor or ad hoc networks.To set up an analytical framework for the proposed system,we rst derive the probability density function (PDF) of theend-to-end signal-to-noise ratio (SNR) in Rayleigh fadingchannels. By expressing the PDF in a mathematically tractableform, we obtain the system performance measure includingoutage probability (OP), probability of SNR gain, averagecapacity, symbol error rate (SER) and bit error rate (BER).Some selected numerical results are presented to verify the an-alytical ones showing that the analysis results are in excellentagreement with the simulated one in all range of operatingSNRs and the combining technique as well as number ofantennae at the destination will not contribute and thus will notlead to any effect on the system performance if the bottle neckof the network is the rst hop. Otherwise, the performance lossdue to the use of selection combining is not substantial.

The rest of this paper is organized as follows. In Sect. II,the model of a partial relay selection network with selectioncombining is briey introduced. In Sect. III, we rst derivethe PDF of the end-to-end SNR, then provide some systemperformance metrics. In Sect. IV, we present our analyticaland simulation results. Finally, in Section V, we conclude with

978-1-4577-1268-5/11/$26.00 ©2011 IEEE ICTC 2011101

a brief summary of our work.

II. SYSTEM MODEL

Fig. 1. Partial relay selection networks with selection diversity.

We consider a wireless dual-hop network consisting of a sin-gle antenna source (S), N single antenna relays (R), and an M -antenna installed destination. We also consider the case wherethe network is located in highly shadowed areas resulting in nodirect communication between the source and the destination.Without the direct link, the communication between the sourceand the destination takes place two consecutive time slots viathe help of the relays. In particular, in the rst time slot, thesource transmits its signals which are received by all relaysdue to the broadcast nature of wireless channels. In the secondtime slot, according to the partial relay selection scheme [3],only the relay providing highest SNR of the link from thesource, denoted by Rb, will forward the received signal towardthe destination using either AF or decode-and-forward (DF)protocol. In the former case, the relay is thought of as a simpleanalog repeater, which amplies the received signal and thenforwards it to the destination without demodulation while inthe latter case, the relay fully decodes the received signaland forwards the re-encoded version to the destination. Inan attempt at simplifying the complexity, selection combining(SC) is used at the destination instead of MRC, i.e. it choosesthe link with the highest SNR and then performs detectionbased on the signal from the selected link.

Let us denote the channel gains of the links from thesource to relay i and from the best relay to antenna j ofthe destination as hSRi

and hRbj where i = 1, . . . , N andj = 1, . . . ,M . Due to Rayleigh fading, the respective channelpowers, represented by |hSRi

|2 and |hRbj |2, are exponentially

distributed with expected values λ1 and λ2, respectively. Forboth AF and DF schemes employed at relays, the end-to-end instantaneous SNR at the destination, γΣ, can be tightlyapproximated in the high SNR regime as follows [10], [11]:

γΣ ≈ min {β1, β2} , (1)

where β1 = maxi=1,...,N P1|hSRi|2/N0 and β2 =

maxj=1,...,M P2|hRbj |2/N0 are the overall instantaneous SNR

of the rst and second hop, respectively where Pk denotesthe transmit power of hop k with k ∈ {1, 2} and N0 is

the power of the additive white Gaussian noise (AWGN) ateach receiver. Correspondingly, the per-hop average SNRs aregiven by γ̄1 = E{β1} = P1λ1 and γ̄2 = E{β2} = P2λ2

where E{.} is the expectation operator. If all the links areindependently faded, thanks to the binomial expansion [12,eq. (7-14)], the PDF of βk is given by

fβk(γ) =

Z

γ̄ke− γ

γ̄k

�1−e

− γ

γ̄k

�Z−1

=Z�

i=1

(−1)i−1

�Z

i

�i

γ̄ke− iγ

γ̄k , (2)

where Z ∈ {N,M}. The corresponding cumulative distribu-tion function (CDF) of βk is

Fβk(γ) =

� γ

0

fβk(γ)dγ

=

Z�i=1

(−1)i−1

�Z

i

��1− e

− iγ

γ̄k

�. (3)

Under the assumption that the hops are subject to indepen-dent fading, order statistics gives the CDF of γΣ as [12, eq.(6-81)]

fγΣ(γ) = fβ1

(γ) + fβ2(γ)− fβ1

(γ)Fβ2(γ)− fβ2

(γ)Fβ1(γ)

= fβ1(γ) [1− Fβ2

(γ)] + fβ2(γ) [1− Fβ1

(γ)] . (4)

With the help of [13, eq. (1.111)], the CDF of γk is re-written as

Fβk(γ) =

Z�i=1

(−1)i−1

�Z

i

�−

Z�i=1

(−1)i−1

�Z

i

�e− iγ

γ̄k

= 1−Z�

i=1

(−1)i−1

�Z

i

�e− iγ

γ̄k . (5)

Substituting (2) and (5) into (4) and after some manipula-tions, the PDF of γΣ can be determined as follows:

fγΣ(γ) = [1− Fβ2

(γ)] fβ1(γ) + [1− Fβ1

(γ)] fβ2(γ)

=

⎡⎣

M�j=1

(−1)j−1

�M

j

�e−

jγ

γ̄2

⎤⎦�

N�i=1

(−1)i−1

�N

i

�i

γ̄1e−

iγ

γ̄1

�

+

�N�i=1

(−1)i−1

�N

i

�e−

iγ

γ̄1

�⎡⎣

M�j=1

(−1)j−1

�M

j

�j

γ̄2e−

jγ

γ̄2

⎤⎦

=N�i=1

M�j=1

(−1)i−1

(−1)j−1

�N

i

��M

j

�i

γ̄1e−�

γ

γ̄1+ γ

γ̄2

�γ

+N�i=1

M�j=1

(−1)i−1

(−1)j−1

�N

i

��M

j

�j

γ̄2e−�

γ

γ̄1+ γ

γ̄2

�γ

=

N�i=1

M�j=1

κχe−γχ, (6)

where κ = (−1)i+j−2�Ni

��Mj

�and χ = i/γ̄1 + j/γ̄2.

102

III. PERFORMANCE ANALYSIS

A. Outage Probability

The rst step in evaluating the system performance is tocompute the outage probability, dened as the probability that1/2 log2(1+γΣ) is less than the end-to-end spectral efciency,R, in bps/Hz. Note that the ratio 1/2 is included to reect thatthe source-to-destination information transmission via relaystakes place in two time slots. Mathematically, the systemoutage probability is given by

Po =

� 22R−1

0

fγΣ(γ)dγ

=

N�i=1

M�j=1

κ�1− e−χ(22R−1)

�. (7)

In the limiting case of interest, (7) can be further simplied inthe high SNR regime. Specically, when the rst hop is theweaker hop, e.g. N < M , by applying ex ≈ 1 + x for smallx and making use of [13, eq. (0.154.2)], we obtain

Po =N�i=1

(−1)i−1

�N

i

�i

γ̄1

�22R − 1

�. (8)

From (8), we can conclude that the outage probability does notdepend neither on the combining technique nor on the numberof antennae (M ) used at the destination if the rst hop is theworst link.

B. Probability of SNR gain

To quantify the benet offered by the proposed system,in this subsection, we study the probability of SNR gaindened in [14]. Such a SNR gain offers us an explicit view ofthe advantage achieved by the proposed protocol over directtransmission. The probability of SNR gain is given by

ΩΔ=Pr

�γΣγ0

>μ

�

= 1−� ∞

0

Pr {γΣ ≤ μγ0|γ0 = γ} fγ0(γ)dγ, (9)

where μ is the pre-determined SNR gain that we wish toobtain, γ0 denotes the instantaneous received SNR at thedestination in a single-hop system. Under Rayleigh fading, thePDF of γ0 is given by fγ0

(γ) = 1γ̄0e−

γ

γ̄0 where γ̄0 = E{γ0}.Substituting (6) into (9), we can obtain the probability of theSNR gain for the proposed system as

Ω = 1−� ∞

0

FγΣ(μγ)fγ0

(γ)dγ

=N�i=1

M�j=1

κ

1 + μγ̄0χ. (10)

C. Ergodic CapacityIn information-theoretical sense, the ergodic capacity refers

to the expected value of the instantaneous maximum mutualinformation between the source and the destination, which canbe mathematically formulated as

CΔ= EγΣ

�1

2log2 (1 + γΣ)

�

=1

2 ln 2

∞�

0

ln (1 + γ)

N�i=1

M�j=1

κχe−γχdγ

=1

2 ln 2

N�i=1

M�j=1

κχ

∞�

0

ln (1 + γ) e−γχdγ. (11)

The integral dened in (11) is evaluated using partial integra-tion, namely

C =1

2 ln 2

N�i=1

M�j=1

κχ

⎡⎢⎢⎢⎣ln (1+γ)

χe−γχ

����+∞

0� �� �=0

+1

χ

� ∞

0

e−γχ

1 + γdγ

⎤⎥⎥⎥⎦

=1

2 ln 2

N�i=1

M�j=1

κ

� ∞

0

e−γχ

1 + γdγ. (12)

Making the change of variables u = 1 + γ, after somemanipulation, we arrive at the equivalent compact result as

C =1

2 ln 2

N�i=1

M�j=1

κeχ∞�

1

e−uχ

udu. (13)

Changing variable again, t = χu ⇒ dt = χdu, yields

C =1

2 ln 2

N�i=1

M�j=1

κeχ∞�

χ

e−t

tdt

=1

2 ln 2

N�i=1

M�j=1

κeχΓ(0, χ), (14)

where Γ(a, x) =+∞�x

e−tta−1dt denotes the incomplete

Gamma function [15, p. 160, eq. (6.5.3)].

D. Symbol Error RateUsing the well-known Moment-Generating Function (MGF)

approach [16], the system SER for M -PSK over Rayleighfading channels is given

Ps =1

π

π−π/M�

0

MγΣ

�−gMPSKlog2M

sin2θ

�, (15)

where gMPSK = sin2(π/M) and MγΣ(s) is the MGF of γΣ

dened as

MγΣ(γ) =

∞�

0

fγΣ(γ)esγdγ =

N�i=1

M�j=1

κ�1− sχ−1.

�−1 (16)

103

Ps =N∑i=1

M∑j=1

κ

π

π−π/M∫

0

sin2θ

sin2θ + ζ

=N∑i=1

M∑j=1

{κ

(M − 1

M

)[1−

√ζ

1 + ζ

(M

(M − 1)π

)[π

2+ tan−1

(√ζ

1 + ζcot

( π

M

))]]}(17)

Substituting (16) into (15), we obtain the closed-form expres-sion for the SER with the help of [16, p. 142, eq. (5.79)] shownat the top of the next page where ζ = χ−1gMPSKlog2M .

E. Bit Error RateHaving the PDF in the desired form, we are now in a

position to derive the average BER for square M -QAM(M = 4m,m = 1, 2, . . .). Implicitly assuming that the Graycode is used, the average BER is given by [17]:

Pb =

∞∫

0

log2

√M∑

l=1

υl∑n=0

φlnerfc (

√ωnγ) fγΣ

(γ)dγ, (18)

where υl = (1 − 2−l)√M − 1, φl

n =

(−1)

⌊n2l−1√

M

⌋ (2l−1 −

⌊n2l−1√M

+ 12

⌋)/√M log2

√M and

ωn = (2n+1)23log2M

2M−2 . Furthermore, we dene �.� and erfc(.)as the oor and complementary error function, respectively.Substituting (6) into (18) and integrating with respect to γ,we obtain the closed-form expression for the BER as

Pb =

log2

√M∑

l=1

υl∑n=0

N∑i=1

M∑j=1

φlnκ

∞∫

0

erfc (√ωnγ)χe

−γχdγ

=

log2

√M∑

l=1

υl∑n=0

N∑i=1

M∑j=1

φlnκ

(1−

√ωnχ−1

1 + ωnχ−1

).(19)

IV. NUMERICAL RESULTS

In this section, we validate the analytical results by compar-ing with Monte-Carlo simulation. In the following numericalexamples, we consider both AF and DF protocol and the uni-form power allocation is used, i.e., ξ = P1/(P2 +P2) = 0.5.

Figure 2 plots the outage probability versus average SNRsfor R = 1 bps/Hz. The gure demonstrates that the asymptoticOP curves match well the simulation results for both thecases of AF and DF. We further observe that a signicantOP improvement is attained for the cases of N = 2,M = 2and N = 2,M = 3 as compared with the other cases.Also, the OP curses conrm that, under the same channelconditions, the performance of the system employing MRCreceiver is not always better than that of the equivalent systemusing SC. In particular, the proposed systems are shown toapproach the system employing MRC if the second hop isstronger than the rst hop, namely N < M , otherwise, theperformance loss due to the use of SC is not signicant. Moreimportantly, it should be noted that in the case of N < M ,

0 5 10 15 20 25 30 35 4010−4

10−3

10−2

10−1

100

Average SNR per bit [dB]O

utag

e Pr

obab

ility

N=2, M=1N=1, M=2N=2, M=2N=2, M=3N=2, M=3 MRC [9]N=2, M=3 Approx. (Eq. 8)Simulation (AF)Simulation (DF)

Fig. 2. Outage probability, R = 1, λ1 = 1, λ2 = 2.

the performance of the proposed system does not dependneither on the combining technique used nor on the number ofantennae. It can be explained by using the fact that in dual-hopnetworks, the overall system performance is always dominatedby the weakest hop.

0 5 10 15 20 25 30 35 40

10−4

10−3

10−2

10−1

100

Average SNR per Bit [dB]

Aer

age

SER

& B

ER

QPSK64−QAM1/SNR1/SNR2

Simulation (AF)Simulation (DF)

N = 2, M = 1

N = 2, M = 2

λ1 = 1, λ2 = 2

Fig. 3. Effective modulation schemes on the system SER and BER.

In Fig. 3, the average SER of QPSK and the average BERof 64-QAM are illustrated. We see that the simulation resultsare well-superimposed on the analytical curves in all rangeof operating SNRs and the modulation schemes used do notchange the diversity order of the system. In order to showthe diversity order explicitly, as references, we include thecurves having diversity order of 1 and 2 obtained by plotting

104

1/SNR and 1/SNR2. By comparing the slopes of the curves,the partial relay selection network with selection combiningindeed achieves the diversity order min(M,N), as expected.

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

Average SNR per bit [dB]

Ergo

dic

Cap

acity

N=1, M=1N=2, M=1N=1, M=2N=2, M=2Simulation (AF)Simulation (DF)

λ1 = 2, λ2 = 1

Fig. 4. Ergodic capacity over Rayleigh fading channels.

−10 −5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

μ [dB]

Prob

abili

ty o

f SN

R G

ain

( Ω)

N=1, M=1N=2, M=1N=1, M=2N=2, M=2Simulation (AF)Simulation (DF)

λ1 = 1, λ2 = 2

Fig. 5. Probability SNR gain over direct transmission, Eb/N0 = 20 dB.

The behavior of the proposed protocol can be further ascer-tained by referring to Fig. 4 and 5 where the average capacityand the probability of SNR gain are shown, respectively, for 4network congurations. In particular, the network with N = 2and M = 2 provides the best performance while that of N = 1and M = 1 gives the worst performance. In comparing thecurves, it can be clearly seen that the average capacity and theprobability of SNR gain can be improved by increasing eitherthe number of relays or the number of antennae installed atthe destination.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−5

10−4

10−3

10−2

10−1

ζ

Bit

Erro

r Pro

babi

lity

N = 2, M = 1N = 2, M = 2N = 2, M = 3

Eb/N0 = 25 dB

Eb/N0 = 15 dB

Fig. 6. Effects of power allocation, λ1 = λ2 = 1.

As a nal remark, regarding Fig. 6, one can conclude aftercomparing the curves that the performance of the proposedprotocol signicantly depend on the power allocation betweenthe source and the relays. To get better performance, unlikethe conventional partial relay selection network (N = 2,M =1) where more power should be allocated to the source, theproposed protocol obtains the best performance since the linkqualities are balanced, namely lower power should be put inthe stronger link and vice verse.

V. CONCLUSION

In this paper, we have proposed the partial relay selectionnetworks with selection diversity at the destination and studiedits performance by applying the well-known relationship ofSNRs in dual-hop networks. Performance results show thatthe proposed protocol still obtain the same diversity of itscounterpart and strikes a good compromise between perfor-mance and complexity. From the implementation stand point,the simplicity of the protocol promises low implementationcost and lends itself to a feasible solution for sensor/ad hocwireless networks.

ACKNOWLEDGMENT

This research was supported by the Vietnam’s Na-tional Foundation for Science and Technology Development(NAFOSTED) (No. 102.99-2010.10).

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