Diversity-Multiplexing-Delay Tradeoff inDecode-And-Forward MIMO Relay Network with

CC-ARQ

Praveen PawarDept. of ICT

ABV-IIITM

Gwalior, India-474 015

powersati@gmail.com

Aditya TrivediDept. of ICT

ABV-IIITM

Gwalior, India-474 015

atrivedi@iiitm.ac.in

Karm Veer AryaDept. of ICT

ABV-IIITM

Gwalior, India-474 015

kvarya@iiitm.ac.in

Abstract—In this paper, three dimensional tradeoff be-tween diversity, multiplexing, and delay is explored, wherethe delay considered is introduced only by retransmissions.Multiple antennas at the transceiver stations provide betterreliability of reception and it can also increase the datarate of communication to improve the system performance.Channel controlled automatic retransmission request (CC-ARQ) design is used on the channel link to achieve thebetter diversity gain at the cost of increased retransmissionrequest. Further, theoretical results are verified using thenumerical simulation which shows the tradeoff betweendiversity (i.e., reliability), multiplexing (i.e., data rate), anddelay (i.e., no. of ARQ rounds).

Index Terms—MIMO, decode-and-forward, linearequalizers, ARQ, diversity-multiplexing tradeoff.

I. INTRODUCTION

Wireless MIMO relay networks have become animportant area due to its ability to reduce the fadingpropagation effect and to improve the reliability ofthe communication system [1]. In the relay networks[2], intermediate node, i.e., relay between sourceand destination forwards the information receivedfrom source to destination. Relay node providesmultiple copies of information at the destinationin addition to the source. In [3], MIMO relayingnetworks on the channel capacity is studied whichincorporates with different signalling schemes androuting methods.

MIMO relaying network has been proposed toachieve diversity at the cost of increased complex-ity [3]. Most of the MIMO relay system adoptsmaximum-likelihood equalizers (MLEs) which have

high decoding complexity to enjoy the better diver-sity which is infeasible for low complexity practi-cal communication system. Zhang et al. [4] haveproposed technique which adopts linear equalizers(LEs) and decision-feedback equalizers (DFEs) toreduce decoding complexity. However, they fail tocollect the receive diversity and unable to give bettersystem performance. Therefore, It is required toredesign MIMO relay system which achieve jointdiversity with low complexity equalizers.

Zorzi et al. [5] have presented an automaticrepeat request (ARQ) method to reduce the fadingpropagation effect from data link layer. In thiswork, they have introduce throughput performancefor ARQ protocols such as selective-repeat andGo-Back-N. This ARQ error control protocols arelimited to buffer capacity at the receiver whichrequires large memory at the cost of long delay.Hunter et al. [6] have presented coded cooperationtechnique, cooperation with channel coding is pre-sented while full diversity gain is achieved. Receiversends negative acknowledgment (NACK) when thepacket is dropped, lost or corrupted which degradesthe system performance. Cyclic redundancy check(CRC) codes are used as error detection codes at thereceiver, but with CRC it is difficult to completelyremove the errors from the information receivedat the receiver. In [7], link-adaptive relay (LAR)scheme is proposed in which CRC is not required.In this scheme, power scaling design is appliedat the relay before forwarding the message to thedestination. This power efficient strategy achieves

2013 5th International Conference on Computational Intelligence and Communication Networks

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DOI 10.1109/CICN.2013.46

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full diversity and practical implementation is alsosimple, compared to other existing relaying strate-gies.

MIMO relay design gives high data rate andlow complexity have been proposed to achievejoint diversity, i.e., cooperative diversity and spa-tial (receive) diversity [8]. Cooperative diversity isdue to the multiple access at the receiver and thespatial diversity is due to the multiple antennas atthe receive station. Multiple antennas equipped atthe transceiver stations, enhance the data transmis-sion rate and the reliability of the communicationchannel. Zero Forcing linear equalizers (ZF-LEs)have low complexity amongst all other existingtechniques, thus ZF-LE is applied at the receiverto reduce decoding complexity. In the relay net-works, decode and forward (DF) method is adoptedat the relay station. The received information iscompletely decoded at the relay and re-encodedmessage is forwarded to next subsequent nodesthrough half duplex relaying strategy is used toavoid the overlapping of the message. In [8], powerscaling design is applied to achieve cooperativediversity, while channel controlled ARQ techniqueis introduced to achieve receive diversity.

The diversity-multiplexing-delay tradeoff(DMDT) is introduced [9], as an extension ofDMT to include the delay as another parameter.The asymptotic DMT was discussed by Zheng andTse [10] to focus on the asymptotically high SNRregime. The DMDT for ARQ cooperative broadcastchannels has been investigated in [11]. Diversity(i.e., reliability) as well as multiplexing (i.e., datarate) both are important parameter to boost thesystem performance. The packet may drop whenthe ARQ rounds limit is reached and the messageis not decoded by the receiver. Therefore, the delayshould be in the range of threshold level.

Message reaches at the destination through dif-ferent paths provides diversity. On the other hand,if message transmits through single channel giveshigh multiplexing. So there is a tradeoff betweendiversity and multiplexing. In this paper, anotherparameter is also considered, i.e., delay which oc-curs due to the retransmission requests. Orthogonaldeficiency leads to more orthogonal channels whichcan mitigate errors in the received information.Furthermore, Delay due to retransmissions should

Fig. 1. Single MIMO relay network model

be in acceptable range. Consequently, optimal ARQprotocol is required so that error-probability ofinformation can be minimized and better systemperformance can be achieved.

The outline of this paper is organized as follows.The basic MIMO relay network model is intro-duced in Section II. Section III represents the Zeroforcing MIMO relay network design with CC-ARQtechnique. Diversity-multiplexing-delay tradeoff isestablished in Section IV. Simulation results aredrawn in Section V. Finally, conclusion and futurework are given in Section VI.

II. SYSTEM MODEL

Fig. 1 shows a single relay network model hasthree terminals: source, relay, and destination. Eachterminal is deployed with N number of antennas.All channels are independent and identically dis-tributed and half duplex relaying strategy is adopted,one complete transmission is divided into two timeslots. In the first time slot, source transmits theinformation which reaches at the relay as well asidentical copy reaches at the destination. In the nexttime slot, relay forward the received informationto the destination. Decode and forward techniqueis applied at the relay which completely decodesthe received information and forward the re-encodedsymbols vector to the destination. Formulation for

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basic relay network is as follows [8]:

ys,r = Hs,rxs + ns,r,

ys,d = Hs,dxs + ns,d,

yr,d = Hr,d

√αx̂r + nr,d, (1)

where, (s,r), (s,d), and (r,d) indicates the source-relay link, source-destination link, and relay-destination link, respectively. The N × 1 vectorsx and y represents the transmitted symbol vectorand received symbol vector respectively. MatrixHi,j is the N × N random channel matrix, whoseall the components are independent and identicallycomplex gaussian distributed with zero mean andunit variance. The N × 1 noise vector n is inde-pendent and identically distributed with zero meanand variance σ2 while α is the power scaling factor(PSF) at the relay. The average power transmitted bysource and relay is normalized to 1. In this model,it is assumed that complete information for channelmatrices available with receiver.

The zero forcing equalizer is adopted wheneverdecoding is required for reducing decoding com-plexity. The estimation of transmitted symbol vectory by adopting ZF-LE is as [8],

xest. = H†s,rys,r = x+H†

s,rns,r, (2)

where, H† =(HHH

)−1HH denotes the Moore-

Penrose pseudoinverse of the channel matrix H andHH represents the hermitian of the channel matrixH . Channel controlled ARQ technique has beenintroduced to reduce the error probability on thecommunication channel link. The ZF MIMO relaywith CC-ARQ technique is discussed in the nextsection.

III. ZF MIMO RELAY WITH CC-ARQ

ZF-LE is a low complexity equalizer directlyassociated to the channel realization. Orthogonaldeficiency(od) [8] for any M × N channel matrixH = [h1, h2, ., hN ], where hi represent the ith

column of H, calculated as,

od (H) = 1− det(HHH

)∏N

n=1 ‖hn‖2, (3)

where, the norm of the nth column of H is repre-sented as ‖hn‖ for all 1 ≤ n ≤ N .

Orthogonal deficiency provides more orthogonalchannels which reduces the probability of gettingerroneous information. Clearly, (3) shows that oddepends only on the channel matrix H . According tochannel controlled ARQ technique which is directlylinked to the realization of the channel matrix. Thisod is bounded with the constraint εth which has thevalue strictly less than 1. Assuming, all channelmatrices are complex variables independent andidentically Gaussian distributed.

In the first step of transmission, at the relaynode decoding and forward technique is applied ifod (Hs,r) ≤ εth, otherwise relay sends NACK or inother words retransmission request to source node.

After successful completion of first step, trans-mitted signal is estimated at the destination adopt-ing ZF-LE, if od (Hequ.) ≤ εth, where Hequ. =HH

sdHsd + αHHrdHrd. Otherwise, destination sends

retransmission request to the source. As long as thecondition not fulfilled, many times retransmissionrequest reaches at the source node. Large number ofretransmission request causes delay in transmission.Hence channel realization increases, consequentlysystem performance degrades. In the next section,tradeoff between diversity (i.e., reliability), multi-plexing (i.e., data rate), and delay is established.

IV. DIVERSITY-MULTIPLEXING-DELAY

TRADEOFFS

The diversity-multiplexing tradeoffs has been ex-plained by Zheng and Tse [10] in which the diver-sity gain d and the multiplexing gain r are definedas,

d = − limρ→∞

log Pe(ρ)

log ρ, and r = lim

ρ→∞R(ρ)

log ρ,

(4)where, Pe (ρ) is the error probability as the functionof SNR (ρ). Considering, space-time codes Cp hasrate R (ρ) bits per-channel unit.

Consider, M and N are transmit and receiveantennas, respectively, maximum diversity gain canbe achieved of MN . For the same number oftransceiver antennas maximum multiplexing gain ofmin {M,N} can be achieved. Both full diversitygain and full multiplexing gain can not achievetogether. According to Zheng and Tse, the optimaldiversity gain d for MIMO channel, as the function

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of multiplexing gain r is given as,

dDM (r, 1) = f (r) , (5)

where f(.) is the piecewise linear function connect-ing the points,

(k, (M − k) (N − k)) for k = 0, .....,min {M,N} .(6)

In particular, dDM (r, 1) is achieved by the ran-dom Gaussian i.i.d. code ensemble for all blocklengths. Assumed block length is T ≥M +N − 1.

In dDM(r, 1), 1 states for the maximum trans-mission delay L=1. The ARQ diversity gaindDMD (r, L) [9] is defined below, which dependson the multiplexing gain r and the maximum trans-mission delay L.

dDMD (r, L) =

{f(rL

), 0 ≤ r < min {M,N}

0, r ≥ min {M,N} .(7)

The optimal DMD tradeoff: dDMD (r) = f (r/L); assumed block length is LT ≥M +N − 1. Gen-erally, during one ARQ round channels are staticwith short term power constraint, power level issame in all the retransmissions. Multiplexing gain isconstant in all retransmissions while diversity gainis increased along with retransmissions. So then, forgetting better diversity gain, there is a compromisewith delay in transmission. Now obtained theoreticalresults are verified through simulation results in thenext section.

V. SIMULATION RESULTS

Simulation results are obtained using MATLAB.Quadrature phase shift keying (QPSK) is applied asmodulation technique. [2× 2] MIMO relay networkis considered and channels are complex Gaussiandistributed. Generally, the value of εth choosesslightly less than 1, because of better channel re-alizations. The ZF MIMO relay network with CC-ARQ design has been carried out in Fig. 2, whichplots BER versus SNR in dB for arbitrary values ofεth = 0.984, 0.928, and 0.856, respectively. It hasbeen found that at high SNR regime smaller valuesof εth gives better performance in terms of BER,hence gives better diversity gain. That is to say, theod of the channel H is bounded with constraint εth,gives better reliability to the system.

Fig. 2. ZF MIMO relay network with CC-ARQ design

On the other hand, as diversity gain increases thepercentage of using of zero forcing equalization isalso increases which causes increment in retrans-mission requests. The ZF MIMO relay network andCC-ARQ design with εth = 0.984, 0.928, and 0.856is considered and found that ZF equalizer decodes80%, 50%, and 20% of the channel realizations,

Fig. 3. Diversity-Multiplexing-Delay Tradeoffs with CC-ARQ

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respectively. System gives best performance for εth= 0.856 but only 20% of the channel realizationis used. As decreasing the value of εth, channelrealization is also decreases which is unreliable thatretransmission request increases with increasing de-lay. The diversity-multiplexing-delay tradeoff withCC-ARQ technique is shown in Fig. 3, it depicts thatas curve reaches to flat, gives better performance.

The diversity-multiplexing-delay tradeoff hasbeen analyzed and observed. The simulation resultsshow that at higher multiplexing gain, as high aspossible diversity gain is achieved within the limitof maximum number of ARQ rounds. The threedimensional tradeoff between diversity, multiplex-ing, and delay is established in Fig. 4 with variousmaximum number of transmission delay L.

However, ZF equalizer has low complexity. CC-ARQ design, od gives perfect orthogonal channelenhance the performance while channel realizationincreases computational complexity. Furthermore,another tradeoff emerges between system perfor-mance and computational complexity.

VI. CONCLUSION

In this paper, performance of ZF MIMO relaynetwork with CC-ARQ design has been carried out.It has been observed that orthogonal deficiency is

Fig. 4. Three dimensional diversity-multiplexing-delay tradeoffs

bounded by the constraint εth. By decreasing thevalue of εth, probability of error decreases whichresults in better diversity gain. Further, ZF MIMOrelay network with three dimensional tradeoff hasbeen established. For given maximum multiplexinggain better diversity gain is achieved but delay inthe transmission is also increased. As retransmis-sion request increases, loss of information occurswhich degrades the system performance. In futurework, optimal ARQ protocol can be improved whichincludes channel state information and messagearriving rate at the receiver.

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