4226 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, … · 4226 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008 Threshold Selection for SNR-based Selective

4226 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 11, NOVEMBER 2008

Threshold Selection for SNR-based SelectiveDigital Relaying in Cooperative Wireless Networks

Furuzan Atay Onat, Abdulkareem Adinoyi, Yijia Fan, Halim Yanikomeroglu, John S. Thompson,and Ian D. Marsland

Abstract—This paper studies selective relaying schemes basedon signal-to-noise-ratio (SNR) to minimize the end-to-end (e2e)bit error rate (BER) in cooperative digital relaying systems usingBPSK modulation. In the SNR-based selective relaying, the relayeither retransmits or remains silent depending on the SNRs ofthe source-relay, relay-destination, and source-destination links.Different models assuming the availability of different sets ofinstantaneous and average SNR information at the relay arestudied. For each model, the optimal strategy to minimize the e2eBER is a different threshold rule on the source-relay SNR, if thelink SNRs are uncorrelated in time and space. Approximationsfor the optimal threshold values that minimize the e2e BER andthe resulting performance are derived analytically for BPSKmodulation. Using the derived threshold the e2e BER can bereduced significantly compared to simple digital relaying. Bystudying the performance under different models, it is shownthat knowledge of the instantaneous source-destination SNR atthe relay can be exploited. The gain from this knowledge is higherwhen the average source-destination SNR is large. However,knowledge of the instantaneous relay-destination SNR at therelay does not change performance significantly.

Index Terms—Multihop communication, cooperative diversity,threshold based digital relaying, selective digital relaying, SNRbased selective relaying.

I. INTRODUCTION

IN wireless networks, multihop relaying improves averagelink SNRs by replacing longer hops with multiple shorter

hops. Relay transmissions are also used to induce coopera-tive diversity, which can increase system reliability withoutrelying on multiple antennas. Several relaying schemes torealize cooperative diversity have been proposed in [1] and[2]. Cooperative relaying schemes are classified as digital oranalog depending on the level of signal processing performed

Manuscript received April 4, 2007; revised September 19, 2007, November26, 2007, and January 19, 2008; accepted March 14, 2008. The associateeditor coordinating the review of this paper and approving it for publicationwas A. Stefanov. A short version of this paper appeared in the proceedings ofthe IEEE Wireless Communications and Networking Conference (WCNC),Hong Kong, 2007. This work was supported by Wireless Tech. Labs, NortelNetworks. John Thompson acknowledges the support of the Scottish FundingCouncil for the Joint Research Institute with the Heriot-Watt University whichis a part of the Edinburgh Research Partnership.

F. A. Onat, A. Adinoyi, H. Yanikomeroglu, and I. D. Marsland are with theBroadband Communications and Wireless Systems (BCWS) Centre, Dept. ofSystems and Computer Engineering, Carleton University, Ottawa, ON, Canada(e-mail: {furuzan, adinoyi, halim, ianm}@sce.carleton.ca).

Y. Fan is with the Department of Electrical Engineering, Princeton Univer-sity, Princeton, NJ, USA (e-mail: [email protected]).

J. S. Thompson is with the Institute for Digital Communications, JointResearch Institute for Signal and Image Processing, School of Engineer-ing and Electronics, University of Edinburgh, Edinburgh, UK (e-mail:[email protected]).

Digital Object Identifier 10.1109/T-WC.2008.070359

at the relay. In digital relaying the relay detects and thenretransmits the detected signal, whereas in analog relaying therelay amplifies and retransmits the received signal. This workfocuses on digital cooperative relaying.

In digital cooperative relaying, if the relay detection is cor-rect, the destination receives the signal through two branches(from the source and the relay) thereby achieves diversity bycombining them. However, if the relay has a detection error,the effective SNR at the destination after combining is signifi-cantly reduced. This phenomenon is called error propagation.The e2e performance of simple digital relaying, where therelay always retransmits, is limited by error propagation.

The relay can forward the data selectively in order to reducethe probability of error propagation. One measure that can beused for forwarding decisions is the link SNR. If the receivedSNR at the relay is low, the data is likely to have errors andhence the relay discards the data. In many wireless appli-cations, relaying schemes might incorporate channel codingtechniques. In this case, other measures of reliability that areextracted from the received signal at the relay can be used inconjunction with SNR [3].

If the reliability information is extracted from the receiveddata, the relay is required to perform channel estimation,demodulation, and then error detection for each data blockbefore making a forwarding decision. These operations causeadditional delay and extra power consumption even if therelay eventually decides not to transmit. In cellular systems,the amount of power consumed by the terminals in receivemode is less significant compared to that in transmit mode.However, these two power levels are comparable in low powerdevices such as battery powered sensor nodes [4]. In SNR-based selective relaying, the relaying decisions are simplerand remain the same for a time duration in the scale of thechannel coherence time in the network. Thus, when the source-relay SNR is low, the relay can be put into sleep mode. Moreimportantly, sensor networks can adopt uncoded transmissionor avoid decoding at the relay due to resource constraints [5],[6]. Hence, in networks that include nodes with a wide rangeof computation and communication capabilities, SNR-basedrelaying can be desirable in order not to isolate the nodeswith scarce power and limited computational capability. SNR-based selective relaying is especially suited for applicationswhere either uncoded transmission is used, or the relayingand channel coding are required to be transparent to eachother, or the delay and the power consumption incurred forextracting the reliability information from the received dataare significant.

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ONAT et al.: THRESHOLD SELECTION FOR SNR-BASED SELECTIVE DIGITAL RELAYING IN COOPERATIVE WIRELESS NETWORKS 4227

In this paper we address the design of SNR-based relayingpolicies for cooperative two-hop networks employing uncodedBPSK signaling. These polices minimize the e2e BER andlead to threshold rules for the source-relay link. The relaytransmits only if the source-relay SNR is above this threshold.The choice of the threshold has considerable impact on thee2e performance of the cooperative diversity schemes. Forinstance, consider a relay detection threshold value of zero.This protocol is akin to simple digital relaying and its diversityorder is equal to one [7]. On the other hand, for a very highthreshold setting, the system degenerates to one path channel,which is the source-destination channel and dual diversity isnot realized.

We formulate the selection of the optimal threshold asa simple decision problem from the relay’s point of view.Four models that differ in the amount of SNR informationavailable at the relay are considered. In the first model,Model 1, the relay makes decisions based on the instantaneoussource-relay SNR, the average relay-destination SNR, and theaverage source-destination SNR. Model 2 assumes that theinstantaneous SNR of source-relay and relay-destination linksare available to the relay while Model 3 assumes that theinstantaneous SNR of the source-relay and source-destinationlinks are available to the relay. Finally, Model 4 assumesthat the relay knows the instantaneous SNRs of all threelinks. Expressions for the optimal threshold values1 and theminimum e2e BER are derived for Rayleigh fading and BPSKmodulation.

For all the models considered, although the e2e BERsdepend on the average SNR of all three links, the optimalthreshold values that minimize the e2e BER are functionsof the relay-destination and source-destination link SNRsonly. For all the models, it is shown that using the derivedthreshold values results in significant improvement of the e2eBER compared to simple digital relaying. By analyzing theperformance under four different models, we observe thathaving the instantaneous source-destination SNR informationfor relaying decisions reduces the e2e BER. The gain fromthis information is higher when the source-destination linkis stronger. However, the gain from instantaneous relay-destination SNR is negligible in most cases. We also comparethe performance of the SNR based selective relaying to a per-formance upper bound that assumes perfect error detection foreach symbol, which is not necessarily achievable in practice.The gap between the performance of selective relaying andthis upper bound suggests that hybrid schemes that incorporateselective relaying at the relay and smarter detection methodsat the destination could provide for further improved e2e BERperformance.

The rest of this paper is organized as follows: In Section II,related literature is discussed. The system model is presentedin Section III and the optimal threshold and the e2e BER forselective relaying schemes are analyzed in Section IV. In Sec-tion V, performance benchmarks are described and numericalexamples on the e2e BER performance are presented. Thepaper concludes with a summary of our findings.

1For the first three models we derive approximations to the optimalthresholds and for Model 4 we derive an exact expression for the optimalthreshold.

II. RELATED WORK

The trade-off between creating the required diversitybranches to the destination and minimizing the risk of errorpropagation has motivated research on SNR-based thresholdrelaying [1], [8]–[10]. Some studies considered a system withideal coding, where no error occurs at the relay as long assource-relay SNR is larger than a target SNR which dependson a specified target rate [1], [11]. This assumption impliesthat the SNR threshold for relaying must be equal to the targetSNR. Herhold et al. studied SNR-based threshold relaying foran uncoded system [8]. In this work, the authors formulatethe power allocation and threshold selection jointly. Theynumerically obtain power allocation fraction and thresholdpairs that minimize the e2e BER for a given modulationscheme used by the source and the relay. Based on thesenumerical results, they also provide empirical rules to ap-proximate the optimal parameters. In [9], the performance ofthreshold relaying in a multi-antenna multi-relay architectureis studied. It is shown that threshold relaying is essentialin uncoded systems when the relay has a small number ofreceive antennas. In [8], the threshold – if used jointly with theoptimal power fraction – is a function of the average SNRs ofthe source-relay, relay-destination and source-destination linkswhile in [9] the threshold depends on the average SNR of thesource-relay link only. Our analytical formulation shows thatfor arbitrary network configurations and given fixed transmitpowers used by the source and the relay, the optimal thresholdis independent of the average source-relay SNR.

In [10], the authors derive the BER of threshold-basedrelaying for an arbitrary threshold value and obtain the optimalthreshold and power allocation by minimizing the BER numer-ically. However, their assumption that the channel coefficientsare real Gaussian random variables does not apply to practicalwireless scenarios.

Lin et al. [12] study the relay selection problem in thecontext of coded cooperation. The authors derive the criteriafor selecting one of the available relays. If none of the relaysis selected, the system reduces to direct transmission from thesource. The relay selection is valid for a long period of time,which is longer than the duration of small scale fading. Hence,the relay selection is made based on the average link SNRs.

The idea of selective relaying, or on-off relaying, can begeneralized to the adaptation of relay transmit power. In [13]and [14], the authors consider a scheme to control the relaypower adaptively based on the link SNRs in order to mitigateerror propagation. They propose a scaling factor for relaypower that is based on the source-relay and relay-destinationSNRs. In these papers, it is reported that if the relay powercan be controlled continuously, the proposed scaling factorachieves full diversity. It is also claimed that the proposedscheme has no diversity gain if the relay can only performon-off power control, i.e., selective relaying. The numericalresults of our present paper hints that the latter conclusion isdue to the particular choice of the scaling factor rather thanthe limitation of on-off power control; we observe that, if it isdone intelligently, on-off power control can increase the slopeof the e2e BER curve, (i.e., the diversity order) compared tosimple digital relaying. In [15], it is shown that the SNR-based



selective relaying achieves dual diversity.An alternative approach to mitigate error propagation is to

design the destination receiver by taking error propagationinto account. In [16], cooperative demodulation techniquesfor a two-hop parallel relaying system are considered. In thissystem, the relays always retransmit, which would result in adiversity order of 1 under simple maximal ratio combining(MRC) at the destination. The authors propose maximum-likelihood combining and demodulation at the destinationassuming that the destination knows the average bit errorprobability at each relay during the first hop. They deriveML receivers and piecewise linear approximations to MLreceivers for different relaying schemes. They show that indigital relaying systems these receivers can achieve a diversityorder of (M + 1)/2 ≤ d ≤ M/2 + 1 for M even andd = (M + 1)/2 for M odd, where M − 1 is the numberof relays. In addition, it is shown that with a single relaydiversity order of 2 can be achieved.

Wang et al. [17] propose a novel combining scheme that canbe employed at the destination for digital parallel relaying.This scheme, which is called Cooperative-MRC (C-MRC),exploits the instantaneous BER of source-relay links at thedestination. The C-MRC can achieve full diversity in uncodeddigital relaying systems. However, it requires the relays to sendtheir instantaneous BER values to the destination.

The models used by [16] and [17] both place the computingburden on the destination while keeping the relays relativelysimple. In our model, however, the relay implicitly participatesin combining the two branches; the relay assigns weightzero to the relay-destination signal by remaining silent. Then,the destination performs MRC. Avoiding transmissions frombranches that make little contribution to the post-processingSNR can reduce interference in the network. Furthermore,in threshold relaying the instantaneous source-relay SNR isexploited at the relay while C-MRC needs the instantaneoussource-relay SNR at the destination, which requires additionalsignaling.

III. SYSTEM MODEL

The network model is shown in Fig. 1. It includes a sourcenode S, a destination node D, and a relay node R that assiststhe communication between S and D. For clarity of exposi-tion, it is assumed that all the links use BPSK modulation.Appendix C provides a sketch for the extension of some ofthe analysis to MPSK. S and R work in time division modein accordance with the half-duplex constraint. This constraintprohibits most practical relay terminals from transmitting andreceiving simultaneously on the same channel. The protocolhas two phases: In phase 1, S transmits and R and D listen. Inphase 2, R detects the signal and either retransmits, in whichcase S is silent, or declares that it will remain silent and Sstarts phase 1 with the next data. If R retransmits in phase 2,D combines the signals received in phase 1 and phase 2 usingMRC and performs detection based on the combined signal.

Let the signal received at the destination from the sourcebe denoted by ysd.

ysd = αsd

√Eb,s xs + nsd, (1)

Fig. 1. The system model.

where xs ∈ {+1,−1}, Eb,s is the energy per bit spent bythe source, αsd is the fading coefficient and nsd is a complexGaussian random variable with zero mean and a variance ofN0/2. Similarly, the signal received at the relay is equal toysr = αsr

√Eb,s xs + nsr. If the relay transmits, the received

signal at the destination as a result of this transmission is givenby

yrd = αrd

√Eb,r xr + nrd, (2)

where xr ∈ {+1,−1} is the symbol sent by the relay based onits detection of xs and Eb,r is the energy per bit spent by therelay. The noise components nsr, nrd, and nsd are assumedto be i.i.d. random variables. The instantaneous link SNRs areequal to γsr = |αsr|2Eb,s/N0, γrd = |αrd|2Eb,r/N0, andγsd = |αsd|2Eb,s/N0. All the links are assumed to exhibit flatfading with Rayleigh envelope distribution. However, some ofthe analysis in the paper is general and not limited to Rayleighdistribution. We assume that both Eb,s and Eb,r are fixed,predetermined values. Hence, the instantaneous link SNRs canbe expressed as γij = σ2

ij X2ij , where X2

ij is an exponentialrandom variable and σ2

ij is the average SNR. All X2ij’s are

independently and identically distributed with unit mean. Thepdf of γij is then given by pγij (γij) = (1/σ2

ij) exp(−γij/σ2ij)

for γij ≥ 0. The average SNR σ2ij , incorporates the energy

per bit spent by node i and the path loss between node iand node j. Hence, the average SNR of S-R, R-D, and S-Dlinks, denoted by σ2

sr , σ2rd, and σ2

sd, respectively, are knownparameters that are not necessarily identical but constant forat least the duration of the two phases.

The channel states remain constant during phase 1 andphase 2. The two phases constitute one block. We assumethat the channel states are either independent from block toblock or their correlation is not exploited. We assume that theCSI is available at the receiver side for all three links and thesignal is demodulated coherently. We consider various modelswith different levels of adaptation in relaying decisions. Inthese models, the relay makes use of either the mean or theinstantaneous SNR for each link. In Model j, the relay usesthe set of parameters denoted by Sj , where j = 1, 2, 3, 4, tomake relaying decisions. The following sets are considered:

S1 = {γsr, σ2rd, σ

2sd}, S2 = {γsr, γrd, σ

2sd},

S3 = {γsr, σ2rd, γsd}, and S4 = {γsr, γrd, γsd}.

How well a relaying configuration can adapt to varyingchannel conditions depends on the information used by the



relay. In general, the average SNR values change muchmore slowly than the instantaneous values. Although a moreadaptive scheme is expected to perform better, a system usingaverage channel characteristics is easier to implement since itrequires less frequent updates to resource allocations. Anotherchallenge is to acquire the necessary channel state information(CSI) at the relay. Since the relay is the receiver in the S-Rlink, it can estimate γsr and additional CSI overhead of Model1 is minimal. Model 2 requires the relay to make decisionsbased on the instantaneous SNR of its forward channel γrd.Thus, a feedback channel from D to R might be necessary.Similarly, Model 3 requires γsd, which can be estimated inthe first phase at D and can be sent to R through the samefeedback channel. Model 4 has the highest complexity sinceit requires that both γrd and γsd are sent to R by D. Theanalysis in this paper focuses on the best possible performanceunder the different models. Therefore, we assume that the CSIrequired by each model is available at the relay.

In the rest of the paper we use the following definitionsand notation. The error events in the S-R and S-D links aredenoted by Esr and Esd, respectively. The event that an erroroccurs after the destination combines the source signal and theincorrectly regenerated relay signal is referred to as error prop-agation and is denoted by Eprop. We use the term cooperativeerror for the event that an error occurs after the destinationcombines the source signal and the correctly regenerated relaysignal. The cooperative error event is denoted by Ecoop. TheBERs for BPSK in point-to-point links conditioned on theinstantaneous link SNR and average link SNR are denoted byBERawgn(γij) and BERray(σ2

ij), respectively, and are givenby [18, pp. 817-818]

P(Eij |γij) = BERawgn(γij) = Q(√

2γij) and

P(Eij |σ2ij) = BERray(σ2

ij) =12

(1 −

√σ2

ij

1 + σ2ij

), (3)

where the Q function is defined as Q(x) =∫∞

x1√2π

e−z2/2dz.The function used to calculate the optimal threshold value

for Model j is denoted by fj ; the policy used by the relay tomake forwarding decisions is denoted by π; and the e2e biterror probability calculated at the relay based on the link SNRobservations Sj when the relay follows policy π is denotedby P{Ee2e|Sj , π(Sj)}. The average e2e BER of the optimalrelaying under Model j is denoted by BER(j)

e2e.

IV. ANALYSIS OF THE SNR-BASED SELECTIVE RELAYING

There are two actions that can be taken by the relay node:a0, which represents remaining silent and a1, which representsdetecting and retransmitting the source signal. In this work,we focus on analyzing the potential of selective relaying toprevent error propagation and to decrease e2e BER. The relaymakes decisions to minimize the expected e2e error probabilitywith given SNR observations.2

2If the relay retransmits in phase 2, the overall transmission uses morebandwidth and more power compared to direct transmission. To keep theanalysis tractable these factors are not taken into account in relaying decisions.However, any selective relaying scheme compares favorably to simple relayingin terms multiplexing loss and total average power.

Then, the relaying policy that minimizes the e2e BER isgiven by

π∗(Sj) = arg minai∈{a0, a1}

P{Ee2e|Sj , ai},

which can be expressed as

P{Ee2e|Sj , a0}a1

a0≷P{Ee2e|Sj , a1}. (4)

This policy is optimal for minimizing the e2e BER for mem-oryless fading channels. The result could be different if linkSNRs were correlated in time and the previous instantaneousSNR values were fed back to the relay.

If the relay does not forward the signal received in the firsthop, the e2e bit error probability for the block depends onlyon the S-D channel: P{Ee2e|Sj , a0} = P{Esd|Sj}. If the relaydoes forward, we can express the e2e bit error probability as

P{Ee2e|Sj , a1}=P{Esr|Sj}P{Eprop|Sj}+(1 − P{Esr|Sj}) P{Ecoop|Sj}. (5)

By substituting (5) into (4), we obtain

P{Esr|Sj}a0

a1≷ P{Esd|Sj} − P{Ecoop|Sj}

P{Eprop|Sj} − P{Ecoop|Sj} . (6)

The derivation up to this point is not specific to Rayleighchannels and is valid under any SNR distribution.

A. Probability of Cooperative Error

Since the destination employs MRC, the SNR after com-bining the two signals is the sum of the SNRs of the S-D andthe R-D channels. If the relay has S4 = {γsr, γrd, γsd}, theprobability of cooperative error calculated at the relay is equalto

P{Ecoop|S4}=P{Ecoop|γrd, γsd} = BERawgn(γrd + γsd)

=Q(√

2(γrd + γsd))

. (7)

The cooperative error probability given S3 = {γsr, σ2rd, γsd},

is equal to

P{Ecoop|S3} = P{Ecoop|σ2rd, γsd}

= Eγrd

[Q(√

2(γsd + γrd))]

(8)

=∫ ∞

0

1σ2

rd

e−γrd/σ2rdQ

(√2(γrd + γsd)

)dγrd (9)

= eγsd/σ2rd

∫ ∞

γsd

1σ2

rd

e−t/σ2rdQ

(√2t)

dt

= eγsd/σ2rdh(γsd, σ2

rd), (10)

where we use change of variables to obtain (10) from (9) anddefine h(., .) as h(x, y) =

∫∞x

1yQ(

√2t)e−t/ydt. This func-

tion can be calculated in terms of Q function (See Appendix Afor the derivation.) :

h(x, y)=e−x/yQ(√

2x) −√

y

1 + yQ

(√2x

(1 +

1y

)).

(11)

Similarly, the cooperative error for S2 = {γsr, γrd, σ2sd}

is equal to P{Ecoop|S2} = Eγsd

[Q(√

2(γsd + γrd))]

. Since



this expression is the same as (8) with γrd and γsd exchanged,P{Ecoop|S2} is given by

P{Ecoop|S2} = P{Ecoop|γrd, σ2sd}

= Eγsd

[Q(√

2(γsd + γrd))]

= eγrd/σ2sd h(γrd, σ

2sd).(12)

If the relay utilizes only S1 = {γsr, σ2rd, σ

2sd} to make

decisions, then the probability of cooperative error is equalto the BER of a 2-branch MRC receiver in Rayleigh fading,which is given as [18, pp. 846-847]

P{Ecoop|S1} = P{Ecoop|σ2rd, σ

2sd}

= Eγsd,γrd

[Q(√

2(γsd + γrd))]

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

12

(1 −

√σ2

rd

1+σ2rd

)2(1 + 1

2

√σ2

rd

1+σ2rd

), σ2

rd = σ2sd;

12

[1 − 1

σ2sd−σ2

rd

(σ2

sd

√σ2

sd

1+σ2sd

− σ2rd

√σ2

rd

1+σ2rd

)]σ2

rd �= σ2sd.

B. Approximate Expression for the Probability of Error Prop-agation

Without loss of generality, we assume that the source sendsthe symbol xs = +1 and the relay sends the symbol xr =−1. The error occurs if the destination decides that −1 wassent by the source. The decision variable after the destinationcombines the received signals (given in (1) and (2)) usingMRC is given by:

y=α∗

sd

√Eb,s

N0ysd +

α∗rd

√Eb,r

N0yrd

=( |αsd|2Eb,s

N0− |αrd|2Eb,r

N0

)+

α∗sd

√Eb,s

N0nsd

+α∗

rd

√Eb,r

N0nrd

=(γsd − γrd) + n, (13)

where n is the effective noise. The mean and the varianceof n are equal to E[n] = 0 and E[|n|2] = 1

2 (γsd + γrd).The decision rule at the destination is to declare +1 ify ≥ 0. Then, the probability of error propagation underS4 = {γsr, γrd, γsd} is equal to

P{Eprop|S4}=P{Eprop|γrd, γsd} = P {y < 0|γrd, γsd}=P {n > (γsd − γrd)|γrd, γsd}

=Q

(γsd − γrd√

(γsd + γrd)/2

). (14)

The probability of error propagation under S3 ={γsr, σ

2rd, γsd} can be found by averaging (14) with

respect to γrd

P{Eprop|S3}=P{Eprop|σ2rd, γsd} = Eγrd

[P{Eprop|γsd, γrd}]

=∫ ∞

0

Q

(γsd − γrd√

(γsd + γrd)/2

)1

σ2rd

e−γrd/σ2rddγrd. (15)

Similarly

P{Eprop|S2}=P{Eprop|γrd, σ2sd}

=∫ ∞

0

Q

(γsd − γrd√

(γsd + γrd)/2

)1

σ2sd

e−γsd/σ2sddγsd, (16)

and

P{Eprop|S1} = P{Eprop|σ2rd, σ

2sd}

=∫ ∞

0

∫ ∞

0

Q

(γsd − γrd√

(γsd + γrd)/2

)1

σ2sdσ2

rd

e−γsd/σ2sd

×e−γrd/σ2rddγsddγrd.

(17)

Due to the complexity of the exact expressions given in(15)-(17), we provide approximate expressions for calculatingthe probability of error propagation for these models. Equation(13) shows that, if relay forwards an incorrect signal, this hasa strong impact on the decision variable y. For instance, forγrd ≈ γsd, the post-combining SNR is close to zero even ifboth γrd and γsd are large. Assuming that the incorrect relaysignal - not the noise term - is the dominant factor that causesthe decision variable y to be negative, we approximate theprobability of error by the probability of {γsd − γrd < 0}.

For S3, using the fact that γrd is an exponential randomvariable with mean σ2

rd, we obtain the approximate probabilityof error as

P{Eprop|S3}≈P{γsd − γrd < 0|σ2rd, γsd}

=∫ ∞

γsd

1σ2

rd

e−γrd/σ2rd dγrd = e−γsd/σ2

rd . (18)

Similarly for S2

P{Eprop|S2}≈P{γsd − γrd < 0|γrd, σ2sd}

=∫ γrd

0

1σ2

sd

e−γsd/σ2sd dγsd = 1 − e−γrd/σ2

sd . (19)

For S1, since γsd and γrd are independent, we obtain

P{Eprop|S1} ≈ P{γsd − γrd < 0|σ2rd, σ

2sd}

=∫ ∞

0

∫ γrd

0

1σ2

sdσ2rd

e−γsd/σ2sde−γrd/σ2

rddγsd dγrd

=σ2

rd

σ2sd + σ2

rd

.

(20)

To check the accuracy of these approximations at practicalSNR values, we compare them with the exact values obtainedthrough the numerical integration of (15)-(17). Fig.s 2-4 showthat all three approximations are reasonably accurate for alarge range of SNR values.

C. Optimal Threshold Functions and Average e2e BER forSNR-based Selective Relaying

In this section, the optimal decision rule given in (6) isevaluated for all the models using the probability of er-ror propagation and cooperative error expressions derived inSection IV-A and Section IV-B. All the rules simplify to athreshold on the instantaneous SNR of the S-R link.



0 2 4 6 8 10 12 14 1610−6

10−5

10−4

10−3

10−2

10−1

100

γsd (dB)

P{εprop

|S3}

σ2rd = 5 dB

σ2rd = 10 dB

σ2rd = 20 dB

Fig. 2. Comparison of P{Eprop|S3} values obtained from the approximationin (18) and from the numerical integration of (15) as a function of γsd fordifferent σ2

rd values. Exact values are shown in solid lines and approximatevalues are shown in dashed lines.

0 5 10 15 20 25 3010−3

10−2

10−1

100

γrd (dB)

P{εprop

|S2}

σ2sd = 5 dB

σ2sd= 10 dB

σ2sd= 20 dB

Fig. 3. Comparison of P{Eprop|S2} values obtained from the approximationin (19) and from the numerical integration of (16) as a function of γrd fordifferent σ2

sd values. Exact values are shown in solid lines and approximatevalues are shown in dashed lines.

1) Relaying based on Model 1: From (6) we obtain therelaying policy for Model 1 asP{Esr|γsr}

a0

a1≷δ1(σ2

rd, σ2sd), where δ1 is defined as

δ1(σ2rd, σ

2sd) =

P{Esd|S1} − P{Ecoop|S1}P{Eprop|S1} − P{Ecoop|S1}

≈1

σ2sd−σ2

rd

(σ2

sd

√σ2

sd

1+σ2sd

− σ2rd

√σ2

rd

1+σ2rd

)−√

σ2sd

1+σ2sd

2σ2rd

σ2rd

+σ2sd

−[1 − 1

σ2sd

−σ2rd

(σ2

sd

√σ2

sd

1+σ2sd

− σ2rd

√σ2

rd

1+σ2rd

)] ,

(21)

0 5 10 15 20 25 3010−3

10−2

10−1

100

σ2rd (dB)

P{εprop

|S1}

σ2sd = 5 dB

σ2sd = 10 dB

σ2sd = 20 dB

Fig. 4. Comparison of P{Eprop|S1} values obtained from the approximationin (20) and from the numerical integration of (17) as a function of σ2

rd fordifferent σ2

sd values. Exact values are shown in solid lines and approximatevalues are shown in dashed lines.

where (3), (13) and (20) have been used to arrive at (21).If δ1(σ2

rd, σ2sd) > 1/2, the relay should always transmit

since P{Esr|γsr} is always less than 1/2. On the other hand,if δ1(σ2

rd, σ2sd) ≤ 1/2, the relaying policy can be further

simplified to

γsr

a1

a0≷f1(σ2

rd, σ2sd), (22)

where

f1(σ2rd, σ

2sd) =

{12

(Q−1(δ1(σ2

rd, σ2sd))

)2, δ1(σ2

rd, σ2sd) ≤ 1/2;

0, otherwise,

and Q−1(z) denotes the inverse of the Q function, which isdefined for 0 ≤ z ≤ 1.

The average e2e BER for Model 1 is derived in (26) (SeeAppendix B for all average e2e BER derivations.):

BER(1)e2e(σ

2sr , σ

2rd, σ

2sd)

= P{Esd|σ2sd}(1 − exp(−f1(σ2

rd, σ2sd)/σ2

sr))

+P{Ecoop|σ2rd, σ

2sd} exp(−f1(σ2

rd, σ2sd)/σ2

sr)+(P{Eprop|σ2

rd, σ2sd} − P{Ecoop|σ2

rd, σ2sd})h

(f1(σ2

rd, σ2sd), σ

2sr

).

(23)

An approximate closed-form expression forBER(1)

e2e(σ2sr , σ

2rd, σ

2sd) can be found by substituting (11),

(13), and (20) into (23).2) Relaying based on Model 2: The optimal decision rule

for the case of S2 is equal toP{Esr|γsr}

a0

a1≷δ2(γrd, σ

2sd), where δ2 is found as

δ2(γrd, σ2sd)=

P{Esd|σ2sd} − P{Ecoop|γrd, σ

2sd}

P{Eprop|γrd, σ2sd} − P{Ecoop|γrd, σ2

sd}

≈12

(1 −

√σ2

sd

1+σ2sd

)− eγrd/σ2

sdh(γrd, σ2sd)

1 − e−γrd/σ2sd − eγrd/σ2

sdh(γrd, σ2sd)



by using (12) and (19). This rule can be expressed as

γsr

a1

a0≷f2(γrd, σ

2sd),

where

f2(γrd, σ2sd) =

{12

(Q−1(δ2(γrd, σ

2sd))

)2, δ2(γrd, σ

2sd) ≤ 1/2;

0, otherwise.

The average e2e BER for Model 2 is given by (28):

BER(2)e2e(σ

2sr , σ

2rd, σ

2sd)

≈∫ ∞

0

[12

(1 −

√σ2

sd

1 + σ2sd

)(1 − exp(−f2(γrd, σ

2sd)/σ2

sr))

+((1 − e−γrd/σ2

sd) − eγrd/σ2sdh(γrd, σ

2sd))

×h(f2(γrd, σ2sd), σ2

sr)

+(1 − e−γrd/σ2sd) exp(−f2(γrd, σ

2sd)/σ2

sr)]

1σ2

rd

eγrd/σ2rddγrd.

Since the integrals to calculate the average e2e BERs of Model2, Model 3, and Model 4 are intractable analytically, we usenumerical integration to evaluate them in Section V.

3) Relaying based on Model 3: For Model 3 the optimaldecision rule is given by P{Esr|γsr}

a0

a1≷δ3(σ2

rd, γsd), where δ3

is equal to

δ3(σ2rd, γsd)=

P{Esd|γsd} − P{Ecoop|σ2rd, γsd}

P{Eprop|σ2rd, γsd} − P{Ecoop|σ2

rd, γsd}

≈ Q(√

2γsd) − eγsd/σ2rd h(γsd, σ

2rd)

e−γsd/σ2rd − eγsd/σ2

rd h(γsd, σ2rd)

.

This rule is equivalent to

γsr

a1

a0≷f3(σ2

rd, γsd),

where

f3(σ2rd, γsd) =

{12

(Q−1(δ3(σ2

rd, γsd)))2

, δ3(σ2rd, γsd) ≤ 1/2;

0, otherwise.

The average e2e BER is given by (29):

BER(3)e2e(σ

2sr, σ

2rd, σ

2sd)

≈∫ ∞

0

[Q(√

2γsd)(1 − exp(−f3(σ2rd, γsd)/σ2

sr))

+(e−γsd/σ2

rd − eγsd/σ2rdh(γsd, σ

2rd))

h(f3(σ2rd, γsd), σ2

sr)

+eγsd/σ2rdh(γsd, σ2

rd) exp(−f3(σ2rd, γsd)/σ2

sr)]

× 1σ2

sd

e−γsd/σ2sd dγsd.

4) Relaying based on Model 4: The optimal decision rulein the case of Model 4 is P{Esr|γsr}

a0

a1≷δ4(γrd, γsd), where δ4

is equal to

δ4(γrd, γsd)=P{Esd|γsd} − P{Ecoop|γrd, γsd}

P{Eprop|γrd, γsd} − P{Ecoop|γrd, γsd}

=Q(

√2γsd) − Q

(√2(γsd + γrd)

)Q

(γsd−γrd√

(γsd+γrd)/2

)− Q

(√2(γsd + γrd)

) ,

and this rule can be expressed as γsr

a1

a0≷f4(γrd, γsd), where

f4(γrd, γsd) ={

12

(Q−1(δ4(γrd, γsd))

)2, δ4(γrd, γsd) ≤ 1/2;

0, otherwise.

The average e2e BER is derived in (30) and is equal to:

BER(4)e2e(σ

2sr , σ

2rd, σ

2sd)

=∫ ∞

0

∫ ∞

0

[Q(√

2γsd)(1 − exp(−f4(γrd, γsd)/σ2sr))

+

(Q

(γsd − γrd√

(γsd + γrd)/2

)− Q(

√γrd + γsd)

)

×h(f4(γrd, γsd), σ2sr)

+Q(√

γrd + γsd) exp(−f4(γrd, γsd)/σ2sr)]

×e−γsd/σ2sde−γrd/σ2

rd

σ2sdσ

2rd

dγrd dγsd.

V. RESULTS

In this section, we first describe two benchmark schemes:simple digital relaying and genie-aided digital relaying. Wethen present numerical examples comparing the e2e BERof SNR-based selective relaying under the different modelspresented in this paper to these benchmark schemes. All theresults are obtained from the analytical formulae derived in thepaper. We resort to numerical integration where it is required.

A. Benchmark Schemes

The descriptions and e2e BERs of the benchmark schemesare given below.

1) Genie-aided digital relaying: Genie-aided digital relay-ing is a protocol designed under the hypothetical assumptionthat the relay has perfect error detection for each symbol.In phase 2, the relay retransmits only those symbols receivedcorrectly in phase 1. Since retransmitting a correctly detectedsymbol decreases e2e BER while transmitting an incorrectlydetected symbol increases it, genie-aided protocol constitutesa performance upper bound for any selective digital relayingscheme. The e2e BER of genie-aided digital relaying is equalto

BERgeniee2e (σ2

sr , σ2rd, σ

2sd)=P{Esr|σ2

sr}P{Esd|σ2sd}

+(1 − P{Esr|σ2sr})P{Ecoop|σ2

rd, σ2sd},

which can be calculated by using (3) and (12).2) Simple digital relaying: In simple digital relaying, the

relay always transmits in phase 2. The e2e BER of simpledigital relaying is equal to:

BERsimplee2e (σ2

sr , σ2rd, σ

2sd)=P{Esr|σ2

sr}P{Eprop|σ2rd, σ

2sd}

+(1 − P{Esr|σ2sr})P{Ecoop|σ2

rd, σ2sd},

which can be calculated by using (3), (12), and (19).



B. Numerical Results

In Fig. 5, we fix σ2rd = 15 dB and σ2

sd = 0 dB and plot thee2e BER as a function of σ2

sr . In this case, f1 (the optimalthreshold for Model 1) remains fixed as seen in (23). Thethreshold is very low (f1 = 0.545), which can be attributed tothe poor quality of the direct link. We observe that when theS-R link is favorable, selective relaying schemes have a smallSNR gain (only 1 to 2 dB) compared to simple relaying.

Fig. 6(a) shows the BER performance at σ2sr = 15 dB and

σ2sd = 5 dB as a function of σ2

rd. For simple digital relaying asσ2

rd increases, on one hand the probability of error propagationincreases, on the other hand the probability of cooperativeerror decreases. In Fig. 6(a), the decrease in the probabilityof cooperative error is the dominant factor. In Fig. 7(a), weplot the e2e BER at σ2

sr = 15 dB and σ2sd = 15 dB as a

function of σ2rd. We observe that in Fig. 7(a) the e2e BER of

simple digital relaying increases as the R-D channel becomesstronger. This is because in this case the increase in errorpropagation dominates over the decrease in the cooperativeerror. In Fig.s 6(a) and 7(a) for large σ2

rd, P{Eprop|σ2rd, σ

2sd} ≈

1 and P{Ecoop|σ2rd, σ

2sd} ≈ 0. Thus, the performance of

simple digital relaying is limited by the S-R link and canbe approximated as BER(simple)

e2e ≈ BERray(σ2sr) for large

σ2rd. Similarly, BER(genie)

e2e ≈ BERray(σ2sr) × BERray(σ2

sd)for large σ2

rd. Model 1 has a significant performance gain oversimple digital relaying in both Fig. 6(a) and Fig. 7(a), since,as shown in Fig. 6(b) and Fig. 7(b), it adaptively increasesthreshold f1 as σ2

rd increases.Finally, we study a scenario where all the average link SNRs

are varied simultaneously. In this scenario, the S-R and R-Dlinks have the same average SNR, while the S-D link hasa lower average SNR, which is a typical scenario when Ris located around the midpoint of S and D. Specifically, weassume σ2

sr = σ2rd = 16 σ2

sd = σ2. In Fig. 8(a), we plot e2eBER as a function of σ2. It is observed that simple digitalrelaying and direct transmission (i.e., no relay) have the sameslope that is equal to 1, while the rest of relaying schemeshave a common slope larger than 1, indicating cooperativediversity gains. The asymptotic diversity gains achieved bySNR-based selective relaying is studied in [15]. Fig. 8(b)depicts the behavior of the optimal threshold for Model 1.It is observed that the threshold must be increased as the linkSNRs increases.

In all the numerical results, we observe that the performanceof SNR-based selective relaying under Model 2 is very closeto Model 1 and the performance under Model 4 is very closeto Model 3. These observations show that the benefit fromexploiting γrd at the relay is marginal. However, there is again both from Model 1 to Model 3 and from Model 2 toModel 4. Hence, it is useful to make use of γsd in relayingdecisions. The gain from adapting according to γsd increasesas the average SNR σ2

sd increases.Although SNR-based selection relaying improves the e2e

BER compared to simple digital relaying, it still has a signif-icant performance gap compared to genie-aided digital relay-ing. Therefore, there might be room for improvement throughhybrid methods combining SNR-based selection relaying withother methods proposed in the literature such as power control

0 5 10 15 2010−4

10−3

10−2

10−1

100

σ2sr (dB)

BER

simple DRModel 1Model 2Model 3Model 4genie −aided DRno relay

Fig. 5. The e2e BER for different relaying schemes as a function of σ2sr

for σ2rd = 15 dB, σ2

sd = 0 dB.

0 5 10 15 2010−4

10−3

10−2

10−1

100

σ2rd (dB)

BER

0 5 10 15 200.5

1

1.5

2

2.5

3

3.5

σ2rd (dB)

f 1 (t

hres

hold

for M

odel

1)


(a) (b)

Fig. 6. The e2e BER for different relaying schemes and the threshold forModel 1 (obtained from (23)) as a function of σ2

rd for σ2sr = 15 dB, σ2

sd =5 dB.

at the relay and better detection methods at the destination.

VI. CONCLUSIONS

In this paper, we proposed and analyzed SNR-based selec-tive relaying schemes to minimize the end-to-end bit error ratein cooperative digital relaying systems. We considered variousmodels for the knowledge of the relay on the link SNRs inthe network. For all the models, the optimal threshold for thesource-relay SNR below which the relay must remain silentdepends on the SNRs (average or instantaneous) of the relay-destination and source-destination links. In contrast to the as-sumption in the literature, the optimal threshold is independentof the average source-relay SNR. For BPSK modulation, wederived exact expressions and in some cases approximationsfor these optimal thresholds and their corresponding averageBER. The average BER of the SNR-based selective relayingwith these thresholds is compared to the performance ofsimple digital relaying. Studying the performance of differentmodels, it has been observed that the instantaneous source-destination SNR information can be exploited while makingrelaying decisions. However, the benefit from knowledge ofthe instantaneous relay-destination SNR is marginal.



0 5 10 15 2010−4

10−3

10−2

10−1

100

σ2rd (dB)

BER

0 5 10 15 200

5

10

15

20

25

σ2rd (dB)

f 1 (t

hres

hold

for M

odel

1)


(a) (b)

Fig. 7. The e2e BER for different relaying schemes and the threshold forModel 1 (obtained from (23)) as a function of σ2

rd for σ2sr = 15 dB, σ2

sd =15 dB.

0 10 20 300

0.5

1

1.5

2

2.5

3

3.5

σ2 (dB)

f 1 (t

hres

hold

for M

odel

1)

0 5 10 15 20 25 3010−6

10−4

10−2

100

σ2 (dB)

BER


(a) (b)

Fig. 8. The e2e BER for different relaying schemes and the threshold forModel 1 (obtained from (23)) as a function of σ2, where σ2

sr = σ2, σ2rd =

σ2 and σ2sd = (1/16) σ2.

APPENDIX ADERIVATION OF h(x, y) GIVEN IN (11)

Using integration by parts, the function h(x, y), which isdefined as h(x, y) =

∫∞x

1y e−t/yQ(

√2t)dt, can be expressed

as

h(x, y) = Q(√

2t)(−e−t/y)]∞

x

−∫ ∞

x

(−e−t/y)ddt

Q(√

2t) dt.

From the definition of Q function (Q(t) =∫∞

t1√2π

e−z2/2dz),

we have ddtQ(

√at) = − 1

2√

2π

√at e−at/2. Hence,

h(x, y)=Q(√

2x)e−x/y

−∫ ∞

x

1√4π

1√texp (−t(1 + 1/y)) dt.

Rewriting the integral term in terms of the new integrationvariable u =

√2t(1 + 1/y), we obtain

h(x, y)=Q(√

2x)e−x/y

−√

y

1 + y

∫ ∞√

2x(1+1/y)

1√2π

e−u2/2du

=Q(√

2x)e−x/y −√

y

1 + yQ

(√2x

(1 +

1y

)),

which is the same as the expression given in (11). In [8], theauthors used a similar derivation to obtain Eqn. (15) of theirpaper.

APPENDIX BAVERAGE E2E BER CALCULATION

Conditioned on γsr, the e2e BER for Model 1 is equal to

BER(1)e2e(γsr, σ

2rd, σ

2sd) = P{Esd|σ2

sd}, γsr < f1(σ2rd, σ

2sd);

and

BER(1)e2e(γsr , σ

2rd, σ

2sd) = P{Esr|γsr}

(P{Eprop|σ2

rd, σ2sd}

−P{Ecoop|σ2rd, σ

2sd})

+ P{Ecoop|σ2rd, σ

2sd}, otherwise.

The average e2e BER can be obtained by averaging theconditional BER over γsr:

BER(1)e2e(σ

2sr , σ

2rd, σ

2sd) = Eγsr

[BER(1)

e2e(γsr , σ2rd, σ

2sd)]

= P{Esd|σ2sd}∫ f1(σ2

rd,σ2sd)

0

pγsr(γsr)dγsr

+((P{Eprop|σ2


rd, σ2sd})

×∫ ∞

f1(σ2rd,σ2

sd)

P{Esr|γsr}pγsr(γsr)dγsr

+P{Ecoop|σ2rd, σ

2sd}∫ ∞

f1(σ2rd,σ2

sd)

pγsr(γsr)dγsr (24)


rd, σ2sd)/σ2

sr))

+P{Ecoop|σ2rd, σ

2sd} exp(−f1(σ2

rd, σ2sd)/σ2

sr)+(P{Eprop|σ2


rd, σ2sd})

×∫ ∞

f1(σ2rd,σ2

sd)

Q(√

2γsr)pγsr(γsr)dγsr (25)


rd, σ2sd)/σ2

sr))

+P{Ecoop|σ2rd, σ

2sd} exp(−f1(σ2

rd, σ2sd)/σ2

sr)+(P{Eprop|σ2


rd, σ2sd})

×h(f1(σ2

rd, σ2sd), σ

2sr

). (26)

For Model 2 the e2e BER conditioned on γsr and γrd isequal to

BER(2)e2e(γsr , γrd, σ

2sd) = P{Esd|σ2

sd}, γsr < f2(γrd, σ2sd);

and

BER(2)e2e(γsr, γrd, σ

2sd)=P{Esr|γsr}

(P{Eprop|γrd, σ

2sd}

−P{Ecoop|γrd, σ2sd})

+ P{Ecoop|γrd, σ2sd}, otherwise.

The average e2e BER for Model 2 is given by

BER(2)e2e(σ

2sr , σ

2rd, σ

2sd)=Eγsr,γrd

[BER(2)

e2e(γsr, γrd, σ2sd)]

=Eγrd

[Eγsr

[BER(2)

e2e(γsr, γrd, σ2sd)]]

.

Then, Eγsr

[BER(2)

e2e(γsr , γrd, σ2sd)]

is calculated following

the same steps as in (24), and BER(2)e2e(σ

2sr , σ

2rd, σ

2sd) is found



as

BER(2)e2e(σ

2sr , σ

2rd, σ

2sd) = Eγrd

[P{Esd|σ2

sd}×(1 − exp(−f2(γrd, σ

2sd)/σ2

sr))+(P{Eprop|γrd, σ

2sd} − P{Ecoop|γrd, σ

2sd})

×h(f2(γrd, σ2sd), σ

2sr)

+P{Ecoop|γrd, σ2sd} exp(−f2(γrd, σ

2sd)/σ2

sr)],(27)

which is approximately equal to

BER(2)e2e(σ

2sr , σ

2rd, σ

2sd)

≈∫ ∞

0

[12

(1 −

√σ2

sd

1 + σ2sd

)(1 − exp(−f2(γrd, σ

2sd)/σ2

sr))

+((1 − e−γrd/σ2

sd) − eγrd/σ2sdh(γrd, σ

2sd))

h(f2(γrd, σ2sd), σ

2sr)

+(1 − e−γrd/σ2sd) exp(−f2(γrd, σ

2sd)/σ2

sr)]

1σ2

rd

eγrd/σ2rddγrd,

(28)

after substituting (3), (12) and (19) into (27).In a similar manner, the e2e BER for Model 3 and Model

4 are calculated as

BER(3)e2e(σ

2sr, σ

2rd, σ

2sd)

≈∫ ∞

0

[Q(√

2γsd)(1 − exp(−f3(σ2rd, γsd)/σ2

sr))

+(e−γsd/σ2

rd − eγsd/σ2rdh(γsd, σ

2rd))

h(f3(σ2rd, γsd), σ2

sr)

+eγsd/σ2rdh(γsd, σ2

rd) exp(−f3(σ2rd, γsd)/σ2

sr)]

×e−γsd/σ2sd

σ2sd

dγsd, (29)

and

BER(4)e2e(σ

2sr , σ

2rd, σ

2sd)

=∫ ∞

0

∫ ∞

0

[Q(√

2γsd)(1 − exp(−f4(γrd, γsd)/σ2sr))

+

(Q

(γsd − γrd√

(γsd + γrd)/2

)− Q(

√γrd + γsd)

)

×h(f4(γrd, γsd), σ2sr)

+Q(√

γrd + γsd) exp(−f4(γrd, γsd)/σ2sr)]

×e−γsd/σ2sde−γrd/σ2

rd

σ2sdσ

2rd

dγrd dγsd. (30)

APPENDIX CTHRESHOLD MINIMIZING E2E SYMBOL ERROR RATE FOR

MPSK MODULATION

Consider the case where the source and the relay modulatetheir signals using MPSK. Let the symbols be denoted byx0, . . . , xM−1, where xi = ej2πi/M . The symbol sent by thesource and the relay are denoted by xs and xr, respectively.The received signals are given by ysr = αsr

√Es,s xs + nsr,

ysd = αsd

√Es,s xs + nsd, and yrd = αrd

√Es,r xr + nrd,

where Es,s is the energy per symbol spent by the source and

Es,r is the energy per symbol spent by the relay. Let γij andσ2

ij denote the instantaneous and average SNR per symbol.Consider Model 1, where the relay makes decisions based

on S1 = {γsr, σ2rd, σsd}. The decision rule to minimize e2e

symbol error rate (SER) is P{Ee2e|S1, a0}a1

a0≷P{Ee2e|S1, a1},

where Ee2e represents the e2e symbol error event.P{Ee2e|S1, a0} = P{Esd|σ2

sd} and is given in [19,Eqn. (8.112)]. Without loss of generality, let us assumethat the source transmits x0. Then the term P{Ee2e|S1, a1}can be decomposed as

P{Ee2e|S1, a1}=P{xr = x0|xs = x0}P{Ecoop|σ2rd, σ

2sd}

+M−1∑i=1

P{xr = xi|xs = x0}

×P{Eprop|xs = x0, xr = xi, σ2rd, σ

2sd} (31)

The term P{Ecoop|σ2rd, σ

2sd} is given in [19, Eqn. (9.14)]. The

term P{xr = xi|xs = x0} is obtained in integral form in [20,Eqn. (4.198.b)], and [19, Eqn. (8.29)]. We observe that in M-ary modulation, unlike in BPSK, there are M − 1 ways ofmaking an incorrect decision and their impacts on detectionat the destination are not necessarily the same. After MRCthe decision variable is given by y = γsd + γrde

j2πi/M + n(derivation is given in Section IV-B). As in Section IV-B, weassume that an incorrectly detected symbol sent by the relayconstitutes the dominant cause of detection errors at the desti-nation. That is, the term P{Eprop|xs = x0, xr = xi, σ

2rd, σ

2sd}

can be approximated by the probability that γsd +γrdej2πi/M

falls in the decision region of symbol xi, denoted as Ri.Exploiting the geometry of the MPSK constellation, one caneasily derive that γsd + γrde

j2πi/M ∈ Ri if and only ifγsd − ci,M γrd < 0, where

ci,M �{

sin(π(2i−1)/M)sin(π/M) , i = 1, 2, . . . , �M/2;

− sin(π(2i+1)/M)sin(π/M) , i = �M/2 + 1, . . . , M − 1.

Then, as in (20), we can calculate an approximate expressionfor P{Eprop|x0, xi, σ

2rd, σ

2sd}:

P{Eprop|x0, xi, σ2rd, σ

2sd}≈P{γsd − ci,M γrd < 0|σ2

rd, σ2sd}

=∫ ∞

0

∫ ci,M γrd

0

1σ2

sdσ2rd

e−γsd/σ2sde−γrd/σ2

rddγsd dγrd

=σ2

rdci,M

σ2sd + σ2

rdci,M. (32)

By substituting (32) and the other terms into (31), the decisionrule can be determined. Since P{Ee2e|S1, a1} decreases withγsr, the decision rule is a threshold rule on γsr, where theoptimal threshold is a function of σ2

rd and σ2sd. Obtaining

a closed-form expression for the optimal threshold is quitedifficult. However, bounds such as union bound, can be usedto derive approximations for P{Ee2e|S1, a1}, thereby leadingto approximate closed-form expressions for the optimal thresh-old.

REFERENCES

[1] N. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wirelessnetworks: Efficient protocols and outage behavior,” IEEE Trans. Inform.Theory, vol. 50, pp. 3062–3080, Dec. 2004.



[2] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity–part I: system description,” IEEE Trans. Commun., vol. 51, pp. 1927–1938, Nov. 2003.

[3] S. Valentin, T. Volkhausen, F. A. Onat, H. Yanikomeroglu, and H. Karl,“Enabling partial forwarding by decoding-based one and two-stage se-lective cooperation,” in Proc. IEEE Cognitive and Cooperative WirelessNetworks (CoCoNET) Workshop, collocated with IEEE ICC, May 2008.

[4] Texas Instruments, CC2420 data sheet. 2.4 GHz IEEE802.15.4 / ZigBee-Ready RF Transceiver. Available at:http://focus.ti.com/lit/ds/symlink/cc2420.pdf.

[5] I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “A surveyon sensor networks,” IEEE Commun. Mag., vol. 40, pp. 102–114, 2002.

[6] M. Zorzi and R. R. Rao, “Coding tradeoffs for reduced energy consump-tion in sensor networks,” in Proc. IEEE Personal, Indoor and MobileRadio Communications (PIMRC), 2004.

[7] J. Boyer, D. D. Falconer, and H. Yanikomeroglu, “Multihop diversity inwireless relaying channels,” IEEE Trans. Commun., vol. 52, pp. 1820–1830, Oct. 2004.

[8] P. Herhold, E. Zimmermann, and G. Fettweis, “A simplecooperative extension to wireless relaying,” in Proc.International Zurich Seminar on Communications, pp. 36–39, 2004. Long version, available at http://wwwmns.ifn.et.tu-dresden.de/publications/2004/Herhold P IZS 04.pdf.

[9] A. Adinoyi and H. Yanikomeroglu, “Cooperative relaying in multi-antenna fixed relay networks,” IEEE Trans. Wireless Commun., vol. 6,no. 2, pp. 533-544, Feb. 2007.

[10] W. P. Siriwongpairat, T. Himsoon, W. Su, and K. J. R. Liu, “Opti-mum threshold-selection relaying for decode-and-forward cooperationprotocol,” in Proc. IEEE Wireless Communications and NetworkingConference (WCNC), Apr. 2006.

[11] J. N. Laneman and G. W. Wornell, “Distributed space-time-codedprotocols for exploiting cooperative diversity in wireless networks,”IEEE Trans. Inform. Theory, vol. 49, pp. 2415–2425, Oct. 2003.

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[14] T. Wang, A. Cano, and G. Giannakis, “Link-adaptive cooperativecommunications without channel state information,” in Proc. MilitaryCommunications Conference (MILCOM), pp. 1–7, Oct. 2006.

[15] F. A. Onat, Y. Fan, H. Yanikomeroglu, and J. S. Thompson, “AsymptoticBER analysis of threshold digital relaying in cooperative wirelesssystems.” IEEE Trans. Wireless Commun., to appear, 2008.

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[17] T. Wang, A. Cano, G. B. Giannakis, and J. N. Laneman, “High-performance cooperative demodulation with decode-and-forward re-lays,” IEEE Trans. Commun., vol. 55, no. 7, pp. 1427–1438, 2007.

[18] J. G. Proakis, Digital Communications. McGraw-Hill, 2000.[19] M. K. Simon and M. Alouini, Digital Communication over Fading

Channels: A Unified Approach to Performance Analysis. Wiley& Sons,Inc., 2000.

[20] M. K. Simon, S. M. Hinedi, and W. C. Lindsey, Digital CommunicationTechniques: Signal Design and Detection. Upper Saddle River, NJ:Prentice Hall, 1995.

Furuzan Atay Onat received the B.S. degree inElectrical and Electronics Engineering from MiddleEast Technical University, Ankara, Turkey and theM.S degree in Electrical and Computer Engineeringfrom Rutgers University, New Jersey, in 2000 and2004, respectively. During the summer of 2006, shewas an intern at Lucent Technologies, Bell Labs, NJ.Currently, she is a Ph.D. candidate in the departmentof Systems and Computer Engineering at CarletonUniversity, Ottawa, Canada. Her research interestsare in wireless communications and networking with

special emphasis on multi-hop networks and cooperative communications.

Abdulkareem Adinoyi received a B.Eng degreefrom the University of Ilorin, Nigeria, in 1992, M.Sdegree from the King Fahd University of Petroleumand Minerals (KFUPM), Dhahran, Saudi Arabia, in1998 and Ph.D degree from Carleton University,Ottawa, Canada, in 2006. All the degrees are inelectrical engineering. From April 1993 to August1995 Mr. Adinoyi was with Dubi Oil Limited, PortHarcourt, Nigeria as an instrument/electrical engi-neer. Between September 1995 and October 1998he was with KFUPM as a research assistant and

between January 1999 and August 2002 he held the position of a lecturer atthe department of electrical engineering, KFUPM. He is currently a seniorresearch associate at the Department of Systems and Computer Engineering(SCE) at Carleton University. Between January 2004 and December 2006,he participated in the European Union 6th Framework integrated project- the WINNER. From January 2007 to August 2007, he was briefly atQassim University, Saudi Arabia where he held the position of an assistantprofessor of electrical and electronic engineering. Since September 2007,under the auspices of SCE, he has been working on Samsung’s advanced radioresource management project for OFDMA-based relay-enhanced broadbandwireless networks. Dr. Adinoyi’s research interest covers infrastructure-basedmultihop and relay networks, cooperative diversity schemes and protocols,and efficient radio resource management techniques for the next-generationwireless networks.

Yijia (Richard) Fan received his BEng degreein electrical engineering from Shanghai Jiao TongUniversity (SJTU), Shanghai, P.R. China, in July2003, and PhD degree from the Institute for Dig-ital Communications, University of Edinburgh, UK,March, 2007. His PhD project was fully funded byEngineering and Physical Sciences Research Coun-cil (EPSRC), UK. He is currently a postdoctoralresearch associate in Department of Electrical Engi-neering, Princeton University. His research interestsinclude signal processing, information theory and

their applications in future wireless networks.

Halim Yanikomeroglu was born in Giresun,Turkey, in 1968. He received a B.Sc. degree in Elec-trical and Electronics Engineering from the MiddleEast Technical University, Ankara, Turkey, in 1990,and a M.A.Sc. degree in Electrical Engineering (nowECE) and a Ph.D. degree in Electrical and Com-puter Engineering from the University of Toronto,Canada, in 1992 and 1998, respectively. He was withthe R&D Group of Marconi Kominikasyon A.S.,Ankara, Turkey, from January 1993 to July 1994.

Since 1998 Dr. Yanikomeroglu has been with theDepartment of Systems and Computer Engineering at Carleton University,Ottawa, where he is now an Associate Professor; he is also the Associate Chairfor Graduate Studies in the department. Dr. Yanikomeroglu’s research interestscover many aspects of the physical, medium access, and networking layersof wireless communications with a special emphasis on multihop/relay/meshnetworks and cooperative communications.

Dr. Yanikomeroglu has been involved in the steering committees andtechnical program committees of numerous international conferences incommunications; he has also given 15 tutorials in such conferences. He hasbeen involved in the organization of the IEEE Wireless Communications andNetworking Conference (WCNC) over the years. He is a member of theWCNC Steering Committee, and he was the Technical Program Co-Chair ofWCNC 2004. He is the Technical Program Committee Chair of the IEEEWCNC 2008.

Dr. Yanikomeroglu was an editor for IEEE Transactions on WirelessCommunications [2002-2005] and IEEE Communications Surveys & Tutorials[2002-2003], and a guest editor for Wiley Journal on Wireless Communica-tions & Mobile Computing. He was an Officer of the IEEE’s Technical Com-mittee on Personal Communications (Chair: 2005-06, Vice-Chair: 2003-04,Secretary: 2001-02), and he was also a member of the IEEE CommunicationsSociety’s Technical Activities Council (2005-06). Dr. Yanikomeroglu is alsoa registered Professional Engineer in the province of Ontario, Canada.



John S. Thompson received his BEng and PhDdegrees from the University of Edinburgh in 1992and 1996, respectively. From July 1995 to August1999, he worked as a postdoctoral researcher atEdinburgh, funded by the UK Engineering and Phys-ical Sciences Research Council (EPSRC) and NortelNetworks. Since September 1999, he has been alecturer at the School of Engineering and Electronicsat the University of Edinburgh. In October 2005,he was promoted to the position of reader. Hisresearch interests currently include signal processing

algorithms for wireless systems, antenna array techniques and multihopwireless communications. He has published approximately 100 papers to dateincluding a number of invited papers, book chapters and tutorial talks, as wellas co-authoring an undergraduate textbook on digital signal processing. He iscurrently editor-in-chief of the IET Signal Processing journal and was recently

a technical programme co-chair for the IEEE International Conference onCommunications (ICC) 2007, held in Glasgow in June 2007.

Ian D. Marsland received a B.Sc.Eng. (Hon-ours) degree in Mathematics and Engineering fromQueen’s University in Kingston in 1987. From 1987to 1990 he was with Myrias Research Corp., Ed-monton, and CDP Communications Inc., Toronto,where he worked as a software engineer. He receiveda M.Sc. in 1994 and a Ph.D. in 1999, both inElectrical Engineering from the University of BritishColumbia. Since 1999 he has been an assistant pro-fessor in the Department of Systems and ComputerEngineering at Carleton University. His research

interests fall in the area of wireless digital communication.


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