[ieee icc 2011 - 2011 ieee international conference on communications - kyoto, japan...

Maximum-Likelihood Detectors for Full-RateCooperative Communication Systems

Hala Mostafa, Mohamed Marey, Mohamed Ahmed, and Octavia DobreFaculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, Canada

Emails {[email protected], [email protected], [email protected] and [email protected]}

Abstract—A key issue in the full-rate two-relay cooper-ative communication systems is the interference which iscaused by the simultaneous transmission of the source andone of the relays at any time. In this paper, we exploit theinterference signal at the destination to develop a maximum-likelihood (ML) detector for decode and forward full-ratecooperative systems. It is shown that the Viterbi algorithmcan be employed to find the ML solution. To reduce the com-plexity of the proposed ML detector, a sub-optimal detectoris also introduced. Further, we propose a ML interferencecancellation scheme at the relays. The performance of theproposed schemes is evaluated through Monte Carlo simula-tions. Results indicate that the proposed schemes outperformdirect transmission and conventional relaying schemes interms of their bit error rate.

Index Terms—Cooperative systems, ML principle, Viterbialgorithm.

I. INTRODUCTION

Cooperative communication has received intense in-terest from the research community during the pastfew years. This is a simple way to improve error rateperformance and capacity by forming a virtual antennaarray among single antenna terminals, distributed inspace [1], [2]. The development of cooperative commu-nication can be seen in the current and emerging net-work architectures. For instance, the use of cooperationbetween nodes has been implemented in mesh typenetworks of the IEEE 802 standards (WiMAX, Wi-Fi,Bluetooth) [3], [4]. However, cooperative communica-tion is not without drawbacks. An important drawbackis the reduction in the spectral efficiency due to

• half-duplex constraint at the relays [2] and• orthogonal relay transmission in either frequency,

code, or time domain.

This has spurred researchers to investigate cooperativestrategies to fully recover the spectral efficiency loss.Such strategies can be classified into two main cate-gories, as follows.

One category supposes that each source transmits a’superimposed’ signal, which consists of its own dataand relaying information. This superposition can be

performed in code [5], [6] or in modulation domain[7], [8]. Obviously, if the relay does not have its owndata, a full-rate transmission can not be achieved.

Another category utilizes ’two-branch’ two relayswhich alternatively transmit and receive [9], [10], [11].The interference occurred at the relays and destinationrepresents a drawback in this case, though, as wewill see in the next section. The interference term canbe treated as additive noise; however, this leads toa severe performance degradation [12]. In order toavoid the interference, beam-forming/smart antennasor code-division multiple access (CDMA) techniqueshave been proposed to separate the interference signalsat the relays and destination [9], [13]. The formertechnique comes at the cost of complexity, where asthe latter comes at the cost of wasting resources. Thismotivates researchers to develop new algorithms forinterference cancellation at relays and destination. Thework in [10] and [14] proposes interference cancellationschemes at the destination for amplify and forwardcooperative systems. However, this does not considerthe direct link from the source to the destination, andthus, the spatial diversity is not achieved.

In this paper, we exploit the interference signal at thedestination to develop the maximum-likelihood (ML)detector for decode and forward full-rate two-relaycooperative systems. Starting from the ML principle,it is shown that the optimal detector can be imple-mented by using the Viterbi algorithm. The proposeddetector does not require interference cancellation atthe destination. Further, to reduce the complexity ofthe proposed detector, a sub-optimal detector is alsointroduced. In addition, ML interference cancellationtechnique is proposed at the relays.

The remainder of the paper is organized as follows:In Section II, the system model and problem formu-lation is presented. In Section III, the ML and sub-optimal detectors are proposed. The performance ofthe proposed detectors is evaluated through computersimulations in Section IV. Finally, we conclude thepaper in Section V.

978-1-61284-233-2/11/$26.00 ©2011 IEEE

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

Figure 1. System model with two relays.

II. SYSTEM MODEL AND PROBLEM FORMULATION

Figure 1 shows the cooperative communication sys-tem considered in this paper, where there are tworelays (R1, R2) between the source and destination. Thesource is always broadcasting data to destination. Ateach time slot, one relay (Rx) transmits what it receivedfrom the source at the previous time slot, while theother relay (Ry) listens to what the source and relayRx send. Here x, y ∈ {1, 2} and x �= y. The destinationreceives the signal transmitted by the source and oneof the relays. Unlike the single relay scheme, this co-operative diversity technique can support one symboltransmission per time slot. As a result, the bandwidthefficiency is not sacrificed, and full-rate transmission isretained.

In the following, we assume that the channels fromthe source and relays to the destination are flat fad-ing and the channel between relays is reciprocal, i.e.,hR1R2 = hR2R1 [10]. Further, we assume that perfectchannel state information is available at the destinationand relays. Based on figure 1, at the time slot n, thereceived signal at the destination yD(n), and relay R2,yR2(n), can be respectively expressed as

yD(n) = v(n)hSD(n) + vR1(n − 1)hR1D(n) + wD(n),(1)

yR2(n) =v(n)hSR2(n) +vR1(n − 1)hR1R2(n) + wR2(n),(2)

where v(n) is the nth symbol transmitted by the source,hAB(n) is the channel coefficient between nodes A andB at time slot n, vR1(n − 1) is the n − 1th detectedsymbol at relay R1, and wD(n) and wR2(n) are additive

white Gaussian (AWGN) contributions at destinationand relay R2, respectively. Further, at the time slotn + 1, the received signal at the destination and relayR1 can be respectively written as

yD(n+1)=v(n+1)hSD(n+1)+vR2(n)hR2D(n+1)+wD(n+1),(3)

yR1(n+1)=v(n+1)hSR1(n+1)+vR2(n)hR1R2(n+1)+wR1(n+1),(4)

where wR1(n + 1) is the AWGN contribution at relayR1, and vR2(n) is the nth detected symbol at relay R2.Our goal is to develop the ML detectors at relays anddestination.

III. MAXIMUM-LIKELIHOOD (ML) DETECTORS

A. ML Detector at Relays

From (2) and (4), one can observe that data receivedat a relay from the source is interfered by data sentfrom the other relay. This is because at any time slot,there is always a relay transmitting data simultane-ously with the source. In this sub-section, we proposea novel ML detector at the relay for a decode andforward cooperative scheme based on averaging outthe interference signal. Without loss of generality, wecan write the received signal at relay Rx (x ∈ {1, 2})as

yRx (n)=v(n)hSRx(n)+vRy(n −1)hRxRy(n) + wRx(n), (5)

where vRy(n− 1) is the n− 1th detected symbol at relayRy (y ∈ {1, 2} and x �= y). Based on the ML principle,the detected symbol, vRx (n), at relay Rx can be foundas

vRx (n) = arg maxv(n)

log p(

yRx (n), vRy(n − 1)∣∣∣v(n), hSRx (n), hRxRy(n))

, (6)

where log p (a, b|c) is defined as the logarithm of theprobability density function of jointly occurring eventsa and b given the event c, and v(n) is the trial valueof v(n). Since the symbol vRy(n − 1) is not known atrelay Rx, we remove its contribution by averaging outas

p(

yRx (n)∣∣∣v(n), hSRx (n), hRxRy(n)

)= ∑

vRy (n−1)∈Ω

p(

yRx (n)∣∣∣v(n), vRy(n − 1), hSRx (n), hRxRy(n)

)×p

(vRy(n − 1)

), (7)


Figure 2. The equivalent system of two-rely cooperative systems.

Figure 3. State diagram for the two-relay cooperative systems withQPSK modulation.

where Ω is the constellation of the transmitted symbolsand

p(

yRx (n)∣∣∣v(n), vRy(n−1), hSRx (n), hRxRy(n)

)= 1√

πσ2Rx

×

exp(−

∥∥∥yRx(n)−v(n)hSRx(n)−vRy(n−1)hRxRy(n)∥∥∥2

/σ2Rx

),

(8)where σ2

Rxis the AWGN noise variance at relay Rx.

We assume that the transmitted symbols are equallyprobable; thus, p

(vRy(n − 1)

)does not affect the opti-

mization.

B. ML Detector at Destination

From (1) and (3), one can observe that thesource and relays send their messages to the des-tination in a sequential form. For illustration, letus consider that the source transmits the sequence{· · · , v(n), v(n + 1), v(n + 2), · · · }. Accordingly, theinformation sent by the source and relays can be ex-pressed as {· · · , [v(n), vRx (n − 1)], [v(n + 1), vRy(n)],[v(n + 2), vRx (n + 1)], · · · }. To simplify the develop-ment of the proposed algorithm, we assume perfectdetection at the relays, i.e. v(n) = v(n). Later onthe performance of the proposed detector is assessed

under non-perfect detection at relays as well. Theequivalent block diagram of two-relay cooperative sys-tems can be represented as shown in figure 2. Based onthat, one can describe the transmitter of the equivalentsystem through a state diagram, with the state definedby the memory content. The number of states equalsthe modulation order, K, and are labeled by S1, ..., SK.The transmitter undergoes a state transition with everytransmitted symbol. For example, the state diagramof a transmitter using QPSK modulation is shownin figure 3, where {α1, α2, α3, α4} denote the QPSKsymbols. Each directed line between any two statesis labeled with the input/output pair. Accordingly, byusing this equivalent model, we can apply the Viterbialgorithm [15] with a small modification to derive theML detector.

An ML detector chooses V as the sequence thatmaximizes the log-likelihood function log p (Y |V, H )

V = arg maxV

log p(Y

∣∣V, H)

, (9)

where Y = [yD(0), yD(1), · · · , yD(L − 1)] isthe received sequence at the destination withyD(n) and yD(n + 1) defined in (1) and (3)respectively, and L is the length of the receivedsequence. V = [v(0), v(1), · · · , v(L − 1)] is thetrial value of the transmitted sequence andH =

[hSD(0), hRx D(0), hSD(1), hRyD(1), · · · ,

hSD(L − 1), hRx D(L − 1)] is the channel coefficients,with x, y ∈ {1, 2}. Since the probability of jointindependent events is simply the product ofprobabilities of the individual events, it followsthat

p (Y |V, H ) =L−1

∏l=0

p (yD(l) |v(l), h(l) ) , (10)

where v(l) = [v(l) v(l − 1)] and h(l) =[hSD(l) hRx D(l)] and the transition probabilityp (yD(l) |v(l), h(l) ) is defined as

p (yD(l) |v(l), h(l) ) = 1√πσ2

Dn

×

exp(− |yD(l) − v(l)hSD(l) − v(l − 1)hRx D(l)|2 /σ2

Dn

),

(11)where σ2

Dnis the noise variance at the destination. This

yields

log p (Y |V, H ) =L−1

∑l=0

log p (yD(l) |v(l), h(l) ) . (12)

The log-likelihood function log p (Y |V, H ), whichwe denote by G (Y |V, H ), represents the metricassociated with the sequence V. The probabilityp (yD(l) |v(l), h(l) ) is referred to as the branch metric,


and we denote it by G (yD(l) |v(l), h(l) ). We can nowexpress a partial path metric for the first t branches ofa path as

G ([Y |V, H ]t) =t−1

∑l=0

log p (yD(l) |v(l), h(l) ) . (13)

As one can easily observe, the proposed ML algorithmcan be implemented by using the Viterbi algorithm.To reduce the complexity of the optimal detector, wepropose the following sub-optimal detector.

C. Sub-Optimal Detector

The key principle of the sub-optimal detector is toemploy two consecutive received symbols, yD(n) andyD(n + 1) given in (1) and (3) respectively, to detectthe symbol, v(n). To simplify the derivation of theproposed sub-optimal algorithm, we assume that thepreviously transmitted symbol, vRx (n − 1), is perfectlydetected at the destination. However, in simulations,we evaluate the performance of the proposed detec-tor under non-perfect data detection. Accordingly, thecontribution of vRx (n − 1) can be removed from yD(n)forming zD(n) = yD(n) − vRx (n − 1)hRx D(n),

zD(n) = v(n)hSD(n) + wD(n). (14)

With (3) rewritten as

yD(n+1)=v(n+1)hSD(n+1)+vRy(n)hRyD(n+1)+wD(n+1),(15)

our aim is to detect the symbol v(n) from zD(n) andyD(n + 1). Based on the ML principle, the detectedsymbol v(n) can be obtained by averaging out thesymbol v(n + 1). Therefore, the ML detector of v(n)is

v(n)=arg maxv(n)

∑v(n+1)∈Ω

log p(y′(n)∣∣v′(n), h′(n))p(v(n+1)),

(16)where y′(n) = [yD(n) yD(n + 1)]T , v′(n) =[v(n) v(n + 1)]T , and h′(n) equals

h′(n) =[

hSD(n) 0hRx D(n + 1) hSD(n + 1)

], (17)

and

p(y(n) |v(n), h′(n) ) = 1πσ2

Dn×

exp(−‖y(n) − h′(n)v(n)‖2 /σ2

Dn

).

(18)

After detecting v(n), we can remove its contributionfrom yD(n + 1) forming zD(n + 1) = yD(n + 1) −v(n)hRx D(n + 1). Similarly, from zD(n + 1) and yD(n +2) and by averaging over v(n + 2), we can detectv(n + 1), and so on. As one can observe, this sub-

0 5 10 15 20 2510

−5

10−4

10−3

10−2

10−1

100

Eb / N

0 (dB)

BE

R

Interference treated as noiseProposed ML detectorPerfect interference cancellation

Figure 4. BER of the proposed ML detector at the relays.

0 5 10 15 20

10−5

10−4

10−3

10−2

10−1

100

Eb / N

0 (dB)

BE

R

One relay, 16−PSK, half−rateBest relay, 16−PSK, half−rateSub−optimal detector, QPSK, full−rateML detector, QPSK, full−rateDirect path, QPSK

Figure 5. BER of the proposed ML detectors at the destination.

optimal detector does not suffer from the complexityand delay problems associated with the exact MLdetector.

IV. SIMULATION RESULTS

In this section, we present simulation results toillustrate the performance of our proposed detectors.QPSK modulation is used. We model the channelcoefficients as independent complex Gaussian randomvariables with zero mean, which makes the marginaldistributions of the phase and amplitude uniform andRayleigh, respectively. To capture the effect of the pathloss on the performance, we consider E

[|hAB|2

]=

(dSD/dAB)ε [2], where dAB the distance between nodesA and B, ε is the path-loss exponent, and Ε[.] isthe statistical average operator. Further, we assumethat all nodes have equal additive white Gaussiannoise variance of σ2. The path-loss exponent equals 4.


The distance between the source and the two relaysequals 0.4, the distance between the two relays and thedestination equals 0.6208, and the distance between thetwo relays is 0.2. All these distances are normalized tothe source and destination distance dSD. The minimuminstantaneous signal to noise ratio per bit (Eb/N0)for which the relay decodes and forwards the signalreceived from the source is set to 4 dB.

First, we investigate the bit error rate (BER) perfor-mance at the relays. Figure 4 shows the BER perfor-mance of the proposed ML detector. For the sake ofcomparison, we also show the BER when the interfer-ence signal could perfectly be removed. In the sequelthis is referred to as the perfect interference cancel-lation. Further, we show the BER performance whenthe interference signal can be treated as additionalnoise. As one can observe, the BER that correspondsto treating the interference as additional noise leads tounacceptable BER degradation when compared withthe perfect interference cancellation. A significant im-provement in performance is achieved by applying theproposed ML detector, for which the BER degradationreduces almost to 0.5 dB.

Second, we show the performance at the destination.Figure 5 depicts the BER performance of the proposeddetectors as a function of Eb/N0. Since two relaysare used in the the full-rate two-relay cooperativesystems, the BER performance of the half-rate one relayand best relay from a set of two available relays arealso shown. For a fair comparison between the half-rate and full-rate systems, we set the same data rate.Then, we apply 16-PSK modulation for the half-ratesystems and QPSK for the full-rate systems. As one canobserve, the BER of the proposed full-rate cooperativesystems outperforms that of the half-rate cooperativesystems. For example, at BER equals 10−4, the full-rate system associated with the proposed ML detectoroutperforms the half-rate best relay by 5 dB. Moreover,the ML detector outperforms the sub-optimal detectorby around 2 dB.

V. CONCLUSIONS

In this paper, maximum-likelihood (ML) detectorswere developed for the two-branch decode and for-ward cooperative systems. We illustrated that the MLdetector at the destination can be implemented in arecursive manner. Further, to reduce the computationalcomplexity of the receiver, we proposed a sub-optimaldetector. The performance of the proposed detectorswere evaluated in terms of bit error rate (BER) throughMonte Carlo simulations. We showed that, at the same

data rate, the BER performance of the proposed detec-tors for full-rate cooperative systems outperforms thatof half-rate cooperative systems.

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ACKNOWLEDGMENT

The authors would like to thank Dr. Salama Ikkifor constructive discussions on an early draft of thispaper. Further, the authors gratefully acknowledge theEgyptian government for its financial support.


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