IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 7, APRIL 1, 2013 1657

Source Transmit Antenna Selection for MIMODecode-and-Forward Relay Networks

Xianglan Jin, Jong-Seon No, and Dong-Joon Shin

Abstract—Transmit antenna selection (TAS) is usually applied to mul-tiple-input multiple-output (MIMO) systems because it does not requireadditional radio frequency (RF) chains which are quite expensive. InMIMO decode-and-forward (DF) relay networks, both source-destinationand source-relay-destination paths should be simultaneously considered tofind an effective source TAS (STAS). In this paper, a new STAS is proposedbased on both channel state information and transmission scheme for theMIMO DF relay networks. It is also shown that the proposed STAS whichselects antennas among transmit antennas at the source canachieve full diversity regardless of the value of . Simulation resultsshow that the proposed STAS has better average bit error probability(BEP) performance than other STASs. Also, the proposed STAS with

has lower cost, complexity, overhead, and BEP than the STASwith using full-rate full-diversity space-time block codes withthe same total transmit power.

Index Terms—Decode-and-forward (DF), diversity, multiple-input mul-tiple-output (MIMO), relay network, transmit antenna selection (TAS).

I. INTRODUCTION

When multiple antennas are used at the source, transmit diversitycan be achieved by using space-time block codes (STBCs). However,STBCs require multiple antennas associated with radio frequency (RF)chains which are costly in terms of size, power, and hardware [1].To solve this problem, low-cost and low-complexity antenna selec-tion schemes have been studied [1]–[5], and transmit antenna selec-tions (TASs) with STBCs have also been considered [6]–[8].Unlike point-to-pointMIMO systems, cooperative relay systems uti-

lize two independent source-destination (SD) and source-relay-desti-nation (SRD) paths. To select good transmit antennas at the source,we have to consider both the SD and SRD paths simultaneously. Foramplify-and-forward (AF) relay networks, the optimal and suboptimalTASs at the source were investigated based on maximizing signal-to-noise ratio (SNR) at the destination [9], [10]. Unfortunately, contrary tothe AF relaying case, the exact SNR for the decode-and-forward (DF)relaying is very difficult to derive. For the DF relay networks, a jointrelay-and-antenna selection schemewhich selects the best relay and thebest antenna at both source and the selected relay was studied withoutconsidering the SD link in [11]. In [12], a suboptimal TAS of selectingtwo antennas at the source was proposed such that one maximizes the

Manuscript received October 31, 2011; revised June 14, 2012 and October13, 2012; accepted December 31, 2012. Date of publication January 18, 2013;date of current version March 08, 2013. The associate editor coordinating thereview of this manuscript and approving it for publication was Dr. Josep Vidal.This work was partly supported by Basic Science Research Program throughthe National Research Foundation of Korea(NRF) funded by the Ministry ofEducation, Science and Technology(2012-0004375) and the Korea Communi-cations Commission (KCC), Korea, under the R&D program supervised by theKorea Communications Agency(KCA) (KCA-2012-08-911-04-003).X. Jin is with the Department of Information and Communication En-

gineering, Dongguk University-Seoul, Seoul 100-715, Korea (e-mail:jinxl77@gmail.com).J.-S. No is with the Department of Electrical Engineering and Computer

Science, INMC, Seoul National University, Seoul 151-744, Korea (e-mail:jsno@snu.ac.kr).D.-J. Shin is with the Department of Electronic Engineering, Hanyang Uni-

versity, Seoul 133-791, Korea (e-mail: djshin@hanyang.ac.kr).Digital Object Identifier 10.1109/TSP.2013.2241053

SNR of source-relay (SR) link, and the other maximizes the SNR of theSD link. Also, the maximum diversity was achieved by using Alamouticode [13]. However, this scheme can be used only for two-antenna se-lection and has not been extended to general multiple-relay networks.In this paper, we consider DF relay networks of one source, one des-

tination, and relays with , , and antennas, respectively.We assume that the relay-destination (RD) channels are orthogonal,which decreases the data transmission rate. The reason for this assump-tion is that if the relays transmit signals via the same channel, the po-tential maximum diversity may be difficult to achieve. To achieve suchmaximum diversity, joint coding for multiple relays should be investi-gated and it is far from the scope of this paper.In this paper, we propose a criterion of source TAS (STAS) of se-

lecting antennas among transmit antennas at the source basedon the upper bound on the pairwise error probability (PEP) derivedin [14]. The proposed STAS can be performed at the destination, andthen the information on the selected transmit antennas is fed back tothe source. In the first phase, the source transmits an uncoded singlesymbol ( ) or a codeword of a full-diversity STBC with

transmit antennas. During the second phase,the relays decode, re-encode, and re-transmit signals from an-tennas, and so the relays may transmit erroneous signals. Finally, thedestination decodes the received signals from the source and the relaysby using the near-maximum-likelihood (near-ML) decoding scheme[15].We prove that the proposed STAS which selects antennas

among transmit antennas achieves the maximum diversityin the DF relay networks. We also

compare the average bit error probability (BEP) of the proposed STASwith those of other STASs through Monte Carlo simulation. Thesimulation results show that the proposed STAS has better averageBEP than other STASs [12]. Moreover, with the same total transmitpower, the proposed STAS with has lower cost, complexity,overhead, and BEP than the STAS with using full-ratefull-diversity STBCs.The following notations are used in this paper: the capital letter de-

notes a matrix; denotes the identity matrix; denotes aset of complex matrices; represents the Frobenius norm ofa matrix; denotes the expectation; the superscript denotes thecomplex conjugate transpose. For ,denotes that the elements of are independent and identically dis-tributed (i.i.d.) circularly symmetric Gaussian random variables withzero mean and variance .

II. SYSTEM MODEL AND SOURCE TRANSMIT ANTENNA SELECTION

A. System Model

A cooperative DF relay network with one source, one destination,and relays with , , and antennas, respectively, isconsidered as shown in Fig. 1. The half-duplex transmission and fre-quency-flat quasi-static fading channels are assumed. It is also assumedthat the relay knows the channel state information (CSI) of the cor-responding SR link and the destination knows the CSIs of all SR,SD, and RD links. Let and be the numbers of transmitted sym-bols at the source and the relay during the first and the second phases,respectively, and be a set of message symbols from the -ary signalconstellation. Let and

be the channel coefficient matricesof the SR link and the SD link, respectively, where and are

and channel vectors from the transmit antenna

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Fig. 1. A DF relay network with multiple relays. The solid line denotes the firstphase transmission and the dashed line denotes the second phase transmission.

at the source to antennas in the relay and antennas in thedestination, respectively.In the first phase, the source broadcasts a codeword

encoded from a full-diversity code with -tuple mes-sage vector to the relays and thedestination by using the selected an-tennas. Thus, possible source antenna subsets canbe selected. We define them as ,

. We assume that the antennas in the sourceantenna subset are selected. Then, the column vectors

and composethe channels matrix and , respectively. Hence, the receivedsignal at the relay and the destination can be written as

(1)(2)

respectively, where is the average transmit power at the source,is the noise matrix at the relay with dis-

tribution , andrepresents the noise matrix at the destination with distribution

.In the second phase, relays transmit the codewords reencoded

from their decoded symbols through orthogonal RD channels.Thus, the received signal at the destination through the orthogonalchannel is given as

(3)

where is the codeword constructed from -tuplemessage vector decoded by therelay in the first phase. A relay may decode correctly or incorrectly,and therefore may be different from . is the average transmitpower of each relay, is the channel coefficient matrixof the RD channel distributed as , and

is the noise matrix at the destination for theorthogonal channel with .

B. Source Transmit Antenna Selection

To select good source transmit antennas, we should consider boththe SD and SRD paths simultaneously. Unlike AF relay networks, it isdifficult to find the optimal solution for the STAS in the DF relay net-works due to the difficulty in deriving their error probabilities. Instead,the union bound on BEP can be used as a criterion of selecting goodsource antennas by deriving PEPs. However, it is still difficult to derivethe exact PEP. Therefore, for the MIMO relay networks with thesource antenna subset, we can use the following upper bound on PEPfor , ,

(4)

by adopting the result of Theorem 2 in [14], where means. As proved in [14], the expectation of the

upper bound in (4) taken over the random variables denoting CSIs isproportional to , where is the diversity of MIMO DF relaynetworks. Therefore, the union bound on BEP derived from the upperbounds for all pairs of and in (4) can be used as a performance cri-terion.Let

(5)

Then, selecting the antennas in the subset which satisfies

(6)

the union bound on BEP derived from the upper bounds on PEPs in (4)can be minimized. The performance of this proposed STAS is analyzedin the following sections.

III. DIVERSITY ANALYSIS OF THE PROPOSED STAS

In this section, we show that the proposed STAS achieves full diver-sity. Using (4), the upper bound on the average PEP of the proposedSTAS based on (6) can be written as

(7)Let be the minimum rank among the ranks of

for all , and and be theunitary matrices whose columns are the eigenvectors of

andfor any , respectively. We define an matrix

with , where

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 7, APRIL 1, 2013 1659

means the column of a matrix. Since the multiplication of the uni-tarymatrix does not change the statistical distribution of thematrix withcircularly symmetric complex Gaussian entries, the entries of ,

, and have the same distribution as the entries of ,and , respectively. Let and be the minimum valuesamong nonzero eigenvalues ofand for all , respectively.Since is a full-diversity code, we have

(8)

where is due to Fact 1 in Appendix A, and is fromand .

Let

and

The PDF of is very difficult to derive for general multiple-antennacases. However, by doing integration by parts, we can rewrite the lastpart of (8) as

(9)

Then, an upper bound on the average PEP can be derived by calcu-lating an upper bound on cumulative density function (CDF) of ,

which is derived in the following lemma.Lemma 1: The CDF of can be upper bounded as

where , , and.

Proof: See the Appendix B.Using Lemma 1, the following theorem for the achievable diversity

can be established.Theorem 1: The proposed STAS which selects antennas

among transmit antennas at the source can achieve the maximumdiversity in the MIMO DF relaynetworks of one source, one destination, and relays with , ,and antennas, respectively.

Proof: Let , , and. Then, by using the upper bound

in Lemma 1, the expectation in (9) can be upper bounded as

Byfrom the Appendix D in [14], the expectation can also be upperbounded as

Finally, an upper bound on the average PEP for the DF relay networkswith the proposed STAS is derived as

(10)

Therefore, the DF relay networks with the proposed STAS can achievethe diversity , and when

, the maximum diversity is achieved.

IV. SIMULATION RESULTS AND DISCUSSION

In this section, we compare the average BEPs of MIMO DF relaynetworks with the proposed STAS and other STASs, and also comparethe performance of the proposed STAS for various . For otherSTAS of selecting antennas among transmit antennas at thesource, the following schemes are considered. The first one selects an-tennas with largest SNRs of the SR link, called MAX-SR, and anotherone selects antennas with largest SNRs of the SD link, called MAX-SDas shown in [12]. Also, the random selection which selects antennasrandomly is considered. For the case of , the STAS whichselects one antenna with the maximum SNR of the SR link and theother antenna with the maximum SNR of the SD link [12] is also con-sidered, which will be calledMAX-SR-SD. For the simulation, quadra-ture phase shift keying (QPSK) is used under the channel condition of

, and total transmit powers at the sourceand at each relay are 1 and , respectively. Furthermore, the MLdecoder is used at each relay and the near-ML decoder [15] is used atthe destination.To begin with, we consider the case of ,

, , and . Fig. 2 compares the av-erage BEPs of various STASs in the DF relay network. It is easy tofind that the proposed STAS has better average BEP performance thanMAX-SD by about 1.5 dB at BEP= and much better performancethan random selection and MAX-SR for both cases of and

.In Fig. 3, we compare the average BEP of the proposed STAS with

those of other STASs for , , , andin the Alamouti-coded DF relay network withand , where

The proposed STAS also shows better average BEP thanMAX-SD andMAX-SR-SD, andmuch better average BEP than random selection andMAX-SR.

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Fig. 2. Comparison of average BEPs of various STASs with ,, and in DF relay networks with one relay.

Fig. 3. Comparison of average BEPs of various STASs with ,, and in Alamouti-coded DF relay networks with one relay.

Next, we discuss and compare the proposed STAS for various .The proposed STAS with requires one RF chain,feedback bits, and calculations of the metric in (5). Therefore,without considering the error correction performance, the proposedSTAS with is the most beneficial case. To fairly compare theerror probabilities on similar decoding complexity level, we considerthe proposed STAS with various by using single-symbol-decod-able full-rate full-diversity STBCs at the source.First, we consider the case of and with

. While the uncoded single symbol isused for the case of , Alamouti codeis used for the case of . Fig. 4 shows that the BEP curves ofthe proposed STAS with and with have the sameslope for the same and even though the case of hasbetter average BEP performance than the case of with thesame total transmit power.

Fig. 4. Comparison of average BEPs of the proposed STAS with variousin MIMO DF relay networks with and .

Fig. 5. Comparison of average BEPs of the proposed STAS with variousin MIMO DF relay networks with and .

Additionally, we consider the case of and within Fig. 5. For the same symbol

rate and diversity, an uncoded single symbol is used for, the Alamouti code is used for the

case of , and the coordinate interleaved STBC (CISTBC)[16],

is used for the case of , whereand with the optimal rotation angle . TheBEP curves of the proposed STAS for various also show the sameslope, and the case of shows the better average BEP perfor-mance than the cases of with the same total transmit power.Therefore, the proposed STAS with can be a good STASscheme at the source.

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V. CONCLUSION

In this paper, a new STAS for MIMO DF relay networks was pro-posed by considering the SD and SRD paths. It was also proved thatthe proposed STAS can achieve the maximum diversity

regardless of the number of the selected an-tennas. The simulation results showed that the proposed STAS hasbetter average BEP performance than other existing STASs (and ). Surprisingly, the proposed STAS with haslower cost, lower complexity, lower overhead, and better BEP perfor-mance than the cases of with full-rate full-diversity STBCs.Therefore, the proposed STAS with can be a good STASscheme.

APPENDIX AFACT 1

Fact 1: [15]: For an matrix , there exists a unitary ma-trix and a real diagonal matrix such that , where

, are the eigenvalues of , andthe columns of are the corresponding eigenvectors. Suppose that

are nonzero and the remaining eigenvalues are all zero, andis the minimum nonzero eigenvalue. Then, for any matrix

, the inequality always holds where is anmatrix constructed by using the column of as its

column, .

APPENDIX BPROOF OF LEMMA 1

The CDF of can be written as

Since the random variables ,are not statistically independent, the CDF of is very difficult toderive. However, we have

and thus, the CDF can be upper bounded as

(11)

where the row vectors andare statically independent and

the row vectorsare i.i.d. Therefore, we can rewrite the upper bound on the CDF of

in (11) as

(12)

Since

where means the element of row and column of a matrix,we have

(13)

where is due to

and the fact that and are i.i.d., andis due to the exponential distribution of random variable

with the rate parameter . Forthe last term in (12), we have

(14)

Since

by the similar derivation in (13), we have

(15)

where . Also, from the fact

we have

(16)

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where . By plugging (15) and (16) into (14),and plugging (13) and (14) into (12), the CDF of can be upperbounded as the equation at the top of the page.

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