3262 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL ...web.xidian.edu.cn/fkgong/files/20160902_160115.pdf2016/09/02 · 3262 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 8, AUGUST

3262 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 64, NO. 8, AUGUST 2016

Distributed Concatenated RecursiveAlamouti-Circulant STBC for

Two-Way Multi-Relay NetworksFeng-Kui Gong, Member, IEEE, Guo Li, Student Member, IEEE, Jianhua Ge, and Jinhong Yuan, Fellow, IEEE

Abstract— A general distributed multi-antenna two-wayrelaying network with multiple relays is considered in this paper.We first obtain a tight lower bound of pairwise error proba-bility (PEP) of a maximum likelihood detector for the generaldistributed linear dispersion code for a half-duplex amplify-and-forward two-way relaying network (TWRN) consisting of twosources with each having single antenna and N relays with eachhaving two antennas. Furthermore, by jointly considering signalprecoding at the sources and signal processing at the relays,a general distributed concatenated recursive Alamouti-circulantspace-time block code is proposed for the considered TWRNs.Our design ensures that the equivalent channel matrices at bothsource nodes are the so-called the recursive Alamouti-circulantmatrices, with each block being a product of the two Alamoutichannel matrices. Based on a lower–upper bound strategy and aninduction method, asymptotic PEP formula is attained to showthat given the optimal angle rotation matrix and the precodingmatrix, the code can meet the lower bound of the diversity gain,as well as the maximum coding gain. In addition, the proposedrate one code turns to be effectively decodable.

Index Terms— Two-way relaying networks (TWRNs),distributed space-time block code (DSTBC), pairwise errorprobability (PEP), recursive Alamouti-Circulant (RAC) matrices,diversity gain.

I. INTRODUCTION

W IRELESS relaying communication is widely recog-nized as a promising candidate for more reliable broadarea transmission in future communication systems [1], [2].Compared with the traditional one-way relaying with half-duplex relays, the two-way relaying with higher spectralefficiency is proposed in [3], which is accomplished bysimultaneously transmitting from sources to the relays inthe multi-access phases and broadcasting the processedinformation from relays to sources in the broadcast phases.Two-way relaying networks (TWRNs) has been widely studied

Manuscript received January 3, 2016; revised May 13, 2016; acceptedJuly 1, 2016. Date of publication July 9, 2016; date of current versionAugust 12, 2016. This work was supported in part by the National HightechR&D Program of China (2014AA01A704), the National Natural ScienceFoundation of China (61372067) and the 111 Project (B08038). The associateeditor coordinating the review of this paper and approving it for publicationwas M. Tao. (Corresponding author: Feng-Kui Gong.)

F.-K. Gong, G. Li, and J. Ge are with the State Key Laboratory ofISN, Xidian University, Xi’an 710071, China (e-mail: [email protected];[email protected]; [email protected]).

J. Yuan is with the School of Electrical Engineering andTelecommunications, University of New South Wales, Sydney,NSW 2052, Australia (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCOMM.2016.2589269

in remote communication, mobile communication and satellitecommunication [1], [2], [4], [5], etc. One of the principletasks in TWRNs is to enhance the transmission reliabilitybetween the sources and the relays. Using multiple antennas ordistributed multiple antennas come to effective solutions to thistask. With a competitive diversity performance, the space-timeblock code (STBC) was proposed for the multi-input multi-output (MIMO) systems, such as Alamouti STBC, orthog-onal and quasi-orthogonal STBC, block-orthogonal STBC,group-decodable STBC [6]–[9], etc. Furthermore STBC hasalso been wisely considered in the TWRNs [10], [11],i.e, distributed STBCs (DSTBCs). DSTBCs have attractedgreat attentions since they can reduce the complicated real-ization at the transmitting node and the receiving node whileachieving full diversity. Specially, in view of the promisingspectrum efficiency gain of TWRNs, many DSTBC-basedTWRN protocols have also been reported [12]–[28]. Thesesignificant studies have strongly characterize DSTBC-basedTWRNs as follows:

• Being great different with the MIMO system, the channelin the relay system is generally non-linear, and the statis-tics of the equivalent received noises at sources stronglydepend on the channels as well as on the signal processingat the relay nodes.

• Distributed space-time coding for the relay system isjointly performed through the source nodes and relaynodes with corrupted noisy signals. Specially, the signalsfrom different relay nodes cannot be exchanged.

• From the view of theoretical analysis, the diversity gainfor the relay system involves the logarithm of SNR [11],[24], [25], [29] if the amplify-and-forward (AF) protocolsare considered.

• Given the total network power, power loading amongthe source nodes and the relay nodes significantly affectsthe overall performance of the relay system [11], [25],[30]–[32].

The results regarding to the DSTBC relaying networksin the state of the art mostly come from the theoreticalperformance analysis and the feasible DSTBC design forsome specific scenarios. Generally speaking, because of theintractability of the theoretical performance derivation in thedistribution relaying networks, most accurate performanceevaluation in mathematically cannot be attained in technicallystraightforward manner by just following the MIMO tech-niques. To the best knowledge of the authors, only the upper

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GONG et al.: DISTRIBUTED CONCATENATED RECURSIVE ALAMOUTI-CIRCULANT STBC 3263

bound of the diversity gain for a general relaying network wasderived [11], [29], whereas asymptotic performance analysis isavailable only for some specific relaying protocols [12]–[14],[24], [25], [28], [33]–[39]. However, the upper bound cannottell us what is either the best diversity gain or the optimalcoding gain for a DSTBC be possibly provided. From the viewof the implementation, i.e. the concrete DSTBC design, it isstill not clear that which DSTBC design is optimal in manycases. Therefore, there exist several aspects to be exploredeven for conventional STBC. For example, although manySTBCs with rate-one have been proposed for as many as eighttransmit antennas [40]–[43], however, computer searching willbe required for those codes to obtain the optimal parametersfor the specific modulation constellations. In addition, to fullyexplore the advantages of multi-antenna deployment and toenhance the link reliability, two or more nodes may be requiredto work as relay nodes. However, as far as we know, onlythe specific system models with one or two relay nodes areconsidered in most of the current reported studies and theycannot be extended to more general cases straightforwardly.

In this paper, we focus on the half-duplex TWRNs as thoseconsidered in [24], [25], and [30], in which each source hassingle antenna and every relay is equipped with two antennas.By utilizing the received signals from the two antennas locatedon the same relay node, we investigate how to design anovel DSTBC to improve the diversity performance, whilstkeeping the low-complexity property of orthogonal STBCand quasi-orthogonal STBC. We mainly consider the moregeneral TWRNs with multiple dual-antenna relays and itsoptimal DSTBC design. The main contributions of our workare summarized as follows:

1) For the first time, a tight lower bound on PEP with theML detector for any distributed linear dispersion codewith a fixed-gain amplifier is established in Section IIfor the general TWRN with N dual-antenna relays,showing that the diversity gain cannot decay faster thanlnN SNR/SNR2N , whereas for the similar network with2N single-antenna relays, error performance decays withln2N SNR/SNR2N [11], [29].

2) A general design with its decoding delay being halfor less than that of the design in [30] is proposed inSection III. The transmission block having 2N complexsignals is firstly divided into two groups with eachincluding N symbols according to the parity of theindices. Then, both groups are rotated and precodedindividually before they are combined into the transmitsignal. We also give the optimal angle rotation matrixand precoding matrix to ensure the full rank of thedifference matrix and maximize its determinant whenM-QAM constellations are used.

3) The notable results in [24] and [25] are extended toa more general case, i.e., the TWRNs with 2b relays.Instead of obtaining the closed-form PEP by using thedirect analysis strategies [24], [25], we investigate theasymptotic PEP performance in Section IV by derivingthe upper bound and the lower bound, as well asutilizing the induction method. Asymptotic PEP analysisshows that our design meets the diversity-gain lower

Fig. 1. The TWRN model with N dual-antenna relays.

bound as well as achieves the maximum coding gainfor quadrature amplitude modulation (QAM).

4) The optimal power allocation that maximizes thereceived SNR of the worse link is attained in Section IIIand examined by measuring average bit error rate (BER)performance of the whole system through compre-hensive computer simulations for asymmetric relayingchannels.

Notation: Notation ‖r‖ denotes a 2-norm of the vector r,R (r) = [�(rT ),�(rT )]T , where � and � represent the realpart and imaginary part of a complex vector, respectively;AT , A∗, AH , det(A), vec(A) denote transpose, conjugate,conjugate transpose, determinant, and vectorization of thematrix A respectively; E[·] denotes the expected value ofthe expression in brackets; IN denotes the N × N identitymatrix. Hadamard(N) is a Hadamard matrix of order N ;A � B denotes that A and B are positive semi-definiteand A − B is also positive semi-definite; We use ⊗ and �denote the Kronecker product and the Schur product; Notationf (x) = O(g(x)) with g(x) ≥ 0 denotes that there exists a pairof constants, c1 and c2, independent of the variable x suchthat c1g(x) ≤ f (x) ≤ c2g(x); (n)!! is the double factorialnotation; We also use diag(a1, a2, · · · , an) to represent adiagonal matrix whose diagonal entries are a1, a2, · · · , an .

II. TWO-WAY COMMUNICATION WITH MULTIPLEDUAL-ANTENNA RELAYS

In this paper, we consider a half-duplex AF TWRN con-sisting of two sources with each having single antenna andN relays with each having two antennas. As shown in Fig. 1,the two-way relaying transmission can be described as follows.

In the 2N time slots of the first phase, Tk , k = 1, 2 transmittheir messages sk =

[sk,1, sk,2, · · · , sk,2N

]Tto all relay nodes,

with the transmission power being Pk . The received signalvector r j,l = [r (1)j,l , r (2)j,l ]T from the two antennas of R j at thel-th time slot can be written as

r j,l =√

P1h1, j s1,l +√

P2h2, j s2,l + n j,l, (1)where E[sksHk ] = I2N , l = 1, 2, · · · , 2N . Furthermore,hk, j = [hk,2 j−1, hk,2 j ]T denotes the the channel vectorbetween Tk and R j , j = 1, · · · , N . We assume hk,lare independent non-identically distributed (i.n.i.d) complexGaussian random variables with zeros-mean and variances �k .


We further assume hk,l remain constant in one transmissionblock. In addition, we also assume that perfect channel stateinformation (CSI) hk,l are known for two sources, whereasjust �1 and �2 are needed for the relays. The noise vectors

n j,l = [n(1)j,l , n(2)j,l ]T are assumed to be independent identicallydistributed (i.i.d) zero-mean complex Gaussian random vari-ables with E[n(i)j,l] = σ 2, i = 1, 2. Let the 4N by one vectorr j = [rTj,1, rTj,2, · · · , rTj,2N ]T , we have

r j =√

P1H1, j s1 +√

P2H2, j s2 + n j , (2)where Hk, j = I2N ⊗ hk, j and n j = [nTj,1, nTj,2, · · · , nTj,2N ]T .

In the second phase, R j combines the 4N received signalsfrom two antennas into new symbols and transmit themwith power Pr in consecutive 2N time slots simultaneously.Amplify-and-forward (AF) relaying protocol is used in all therelays and a power factor is used to constraint the relay’stransmission power. By using yk =

[yk,1, yk,2, · · · , yk,2N

]T

as the received signals of Tk during the second phase, wehave

yk =N∑

j=1

(hk,2 j−1t(1)j + hk,2 j t(2)j

)+ ηk, (3)

where t(i)j = [t(i)j,1, t(i)j,2, · · · , t(i)j,2N ]T , and t(i)j,l denotes the signaltransmitted in the l-th time slot from the i -th antenna of theR j node. Furthermore, ηk = [ηk,1, ηk,2, · · · , ηk,2N ]T is a 2Nby one complex Gaussian noise vector received by Tk withzero mean and covariance matrix σ 2I2N .

Define the transmitted signal at the l-th time slot of the j -th

relay as t j,l , i.e., t j,l = [t(1)j,l , t(2)j,l ]T , which is generated bylinearly combining r j and its conjugate r∗j as

t j,l =√

β(A j,lr j + B j,lr∗j ), (4)where (A j,l)2×4N and (B j,l)2×4N are the coefficient matricesadopted for the l-th time slot of the j -th relay. Eq. (4)is normalized to make the average transmitted power pertransmission used at every antenna of every relay be Pr .β denotes the AF factor which ensures that the transmittedrelay signals obey this energy constraint. Without loss ofgenerality, we further assume that the total power per symboltransmission used in the whole network is fixed as 1, i.e.,2N Pr + P1 + P2 = 1.

A. General System Model for Detection andPairwise-Error-Probability Analysis

In this section, we first present two equivalent expressionsfor the received signals at user nodes and then present the PEPperformance analysis. The received signals by Tk at the l-thtime slot of the second phase can be written as

yk,l =√

β

N∑

j=1(hTk, j A j,lr j + hTk, j B j,lr∗j ) + ηk,l . (5)

Without loss of generality, we only present the equivalentsignal model at T1. Substituting r j from (2) into (5) and then

eliminating the self-interference caused by s1, we obtain thel-th received signal at T1 as

z1,l =√P2βN∑

j=1

(hT1, j A j,lH2, j s2 + hT1, j B j,lH∗2, j s∗2

)+ ξ1,l ,

(6)

where ξ1,l = √β∑Nj=1(hT1, j A j,ln j +hT1, j B j,ln∗j )+η1,l . Then,letting z1 = [z1,1, z1,2, · · · , z1,2N ]T , we get

z1 =√

P2βAH1H2s2 +√

P2βBH1H∗2s∗2 + ξ1, (7)where (H1)8N2×4N2 = diag(I4N ⊗ h1,1, · · · , I4N ⊗ h1,N ),(H2)4N2×2N = [HT2,1, HT2,2, · · · , HT2,N ]T , A2N×8N2 = [a1,a2, · · · , a2N ]T and B2N×8N2 = [b1, b2, · · · , b2N ]T , inwhich (al)8N2×1 =[vecT (A1,l), vecT (A2,l), · · ·, vecT (AN,l )]T,(bl)8N2×1 = [vecT (B1,l), vecT (B2,l), · · · , vecT (BN,l )]T .In addition, the combined noise ξ1 can be written asξ1 =

√βAH1n + √βBH1n∗ + η1, in which n =

[nT1 , nT2 , · · · , nTN ]T . We can also rewrite Eq. (7) asz1 =

√P2βAS2Ȟ1h2 +

√P2βBS∗2Ȟ1h∗2 + ξ1, (8)

where (S2)8N2×4N = IN ⊗ (s2 ⊗ I4), (Ȟ1)4N×2N =diag(I2 ⊗ h1,1, I2 ⊗ h1,2, · · · , I2 ⊗ h1,N ) and h2 =[hT2,1, hT2,2, · · · , hT2,N ]T . For the convenience of PEP analysis,(8) can also be expressed as a real-valued form by vectorizingthe complex vectors in terms of their real and imaginary partsas follows

z̄1 =√

P2βKH2N MK2N h̄2 + ξ̄1, (9)where the noise vector

ξ̄1 =√

βKH2N NK4N2 n̄ + η̄1, (10)z̄1 = R (z1), h̄2 = R (h2), ξ̄1 = R (ξ1), n̄ = R (n),η̄1 = R (η1), Kn = 1√2

[In jInIn − jIn

], N =

[AH1 BH1B∗H∗1 A∗H∗1

], and

M =[

AS2Ȟ1 BS∗2Ȟ1B∗S2Ȟ∗1 A∗S∗2Ȟ∗1

].

B. PEP Analysis for the General Network

Eq. (9) can be written as a more common form for simplic-ity, i.e.,

z̄1 =√

P2βX(h1, s2)h̄2 + ξ̄1, (11)where X(h1, s2) = KH2N MK2N and it is a 4N × 4N signalmatrix. According to (10), for any fixed h1, ξ̄1 is an indepen-dently circularly-symmetric complex Gaussian noise with thezero mean and covariance matrix � = σ 2β2 KH2N NNH K2N +σ 2

2 I4N . Thus, given a channel realization, the probabilityP(s2 → s′2|h1, h2

)of transmitting s2 and deciding in favor

of s′2 �= s2 with the ML detector is given by [44]

P(s2 → s′2|h1, h2

) = Q(

d(s2, s′2)√2

), (12)

where d(s2, s′2) is defined by d2(s2, s′2) =P2βh̄H2 X

H (h1, e)�−1X(h1, e)h̄2 with e = s2 − s′2,and the Q function Q(x) = 1π

∫ π/20 exp

(− x2

2 sin2 θ

)dθ .


Now, taking the average of (12) over the random 4N by onevector h̄2, we have

P(s2 → s′2|h1

)

= 1π

∫ π/2

0

dθ

det12(I + β2(θ) X H (h1, e)�−1X(h1, e)

) , (13)

where β2(θ) =(4 sin2 θ

)−1P2�2β. By taking the expectation

of (13) over h1, we obtain the following theorem, which givesthe lower bound of the PEP.

Theorem 1: There exist two constants Ck,N , k = 1, 2,independent of SNR ρ, such that the PEPs at Tk satisfy

P(

sk̄ → s′̄k)

= Ehk[P(sk̄ → sk̄ ′|hk

)]

≥ Ck,N ρ−2N lnN ρ, (14)when SNR ρ is large, where

(k, k̄) ∈ {(1, 2) , (2, 1)}. �

Proof : The detailed proof is similar to that in [25].Remarks: Theorem 1 shows that the lower bound PEP of

our considered TWRNs. To achieve this lower bound, thecoefficient matrices at relays should be carefully designed.We call one design as the optimal design if its PEP can achievethis lower bound. We have presented two optimal designs forthe special cases with N = 1 in [24] and N = 2 in [25]. Formultiple relays, N > 2, we will use an upper-lower boundstrategy and the induction method to investigate the asymptoticPEP in view of the mathematic difficulty of the conventionaldirect methods [24], [25].

III. OPTIMAL DISTRIBUTED CONCATENATED RECURSIVEALAMOUTI-CIRCULANT STBCs

In this section, we proposed a feasible DSTBC design,which is verified optimal to achieve the PEP lower bound.Specially, we consider the orthogonal or quasi-orthogonalDSTBC for such a networks with N = 2b relays, where b is anonnegative integer. In addition, we use the Alamouti code asthe elementary code block as to expect achieving full-diversityand low-complexity detection. With these assumptions, we canfind the fact that the equivalent channel at each source node isthe so-called recursive Alamouti-circulant (RAC) matrix, andits each block is the product of the two Alamouti channelmatrices. Before we present our design, we first give thedefinition of the RAC matrix.

Definition 1: Let a two-by-two matrix Alamouti code

matrix be

[a b

−b∗ a∗]

, a, b ∈ C We call the Alamouti codematrix RACM1. Then the 2n × 2n RAC matrix set, denotedas RACMn , is given by

RACMn ={[

M1 M2M2 M1

], M� ∈ RACMn−1 for � = 1, 2

},

for any integer n larger than one. �From Definition 1, we have the following proposition, which

is crucial when we analyze the diversity performance of ourdesign.

Proposition 1: A RAC matrix Gb ∈ RACMb+1 can bedecomposed into

Gb = THb DbTb, (15)

Fig. 2. Diagram illustration of the signal design at Tk for the proposeddistributed concatenated recursive Alamouti-circulant STBC.

where Db is a block diagonal matrix, i.e., Db =√2bdiag(Tb,1gb, Tb,2gb, · · · , Tb,N gb), Tb, j denotes a

2 × 2b+1 matrix consisting of the (2 j − 1)-th and (2 j)-throws of Tb, and gb is a 2b+1 × 2 matrix consisting of the firsttwo columns of Gb. Here, Tb is a unitary block Hadamardmatrix, which is defined as Tb = 1√

2bHadamard(2b) ⊗ I2. �

Proposition 1 can be proved by using the induction on b.See Appendix B for more details.

A. Description of the Proposed Design

Let xk,l , l = 1, · · · , 2N , be the signals generated from astandard M-QAM constellation Q with unit symbol energy.At Tk , 2N complex constellation symbols xk are dividedinto two groups, i.e., xk,o = [xk,1, xk,3, · · · , xk,2N−1]T andxk,e = [xk,2, xk,4, · · · , xk,2N ]T . As shown in Fig. 2, to achievefull diversity, both groups are precoded individually by theangle rotation transformation D, the inverse discrete Fouriertransform (IDFT) transformation WHN , and the Hadamardtransformation. Then sk,o and s∗k,e are combined into the trans-mitted signal sk . The whole process can be expressed as sk,o =Pxk,o and s∗k,e = P∗x∗k,e, where P = 1√N Hadamard(N)WHN D.Here, WN is a DFT matrix, i.e., WN (p, q) = 1√N e− j2πpq/N ,p, q = 1, · · · , N . To ensure the full rank of the differ-ence matrix and maximize its determinant, N − 1 rotationangles are required, i.e., D = diag(1, e j π2N , · · · , e j (N−1) π2N ).By combining sk,o and s∗k,e, we can obtain

s̃k = Ediag(P, P∗)EH x̃k, (16)where E is a 2N × 2N elementary permutationmatrix which permutes [xTk,o, xTk,e]T into xk .In addition, x̃k = [xk,1, x∗k,2, · · · , xk,2N−1, x∗k,2N ]T ands̃k = [sk,1, s∗k,2, · · · , sk,2N−1, s∗k,2N ]T . In the first consecutive2N time slots, s̃1 and s̃2 are transmitted to the relayssimultaneously. Each relay then generates the broadcastingsignals by linearly combining the received 4N signals fromits two antennas and broadcasts them in the followingconsecutive 2N time slots.

Let ηk = [ηk,1, ηk,2, · · · , ηk,2N ]T still be a 2N by onecomplex Gaussian noise vector received by Tk with zeromean and covariance matrix σ 2I2N , the received signals atTk, k = 1, 2 are then denoted as

yk =√

βRT hk + ηk, (17)where R is the generated transmission signal matrix by theN = 2b relays. Recall (3), R is generated in a way as

R = (R j,i )N×N =[t(1)1 t

(2)1 · · · t(1)N t(2)N

]T

=

⎡

⎢⎢⎣

R1,1 R1,2 · · · R1,NR2,1 R2,2 · · · R2,N· · · · · · · · · · · ·

RN,1 RN,2 · · · RN,N

⎤

⎥⎥⎦, (18)


in which

R j,i =[t j,2i−1 t j,2i

]

=[

r (1)j,L j,i + r(2)∗j,L j,i+1 r

(1)j,L j,i+1 − r

(2)∗j,L j,i

r (2)j,L j,i − r(1)∗j,L j,i+1 r

(1)∗j,L j,i

+ r (2)j,L j,i+1

]

.

Here, L j,i denotes the received signal index during the(2i − 1)-th time slot of the j -th relay, j, i ∈ {1, · · · , N}. If letLb = (L j,i )N×N denote the index matrix, then for our design,Lb has the following structure.

Definition 2: If b = 0, L0 = 1, otherwise if b > 0, the2b × 2b matrix Lb is defined as

Lb =[

Lb−1 2b1b−1 + Lb−12b1b−1 + Lb−1 Lb−1

],

where 1� is a 2�×2� matrix of all ones, and b is a nonnegativeinteger. �

It should be pointed out that since the average transmissionenergy per symbol used at each antenna of the relay nodesis normalized to be Pr , the AF amplifier β must be accord-ingly chosen in such a way that β = Pr

2(�1 P1+�2 P2+σ 2) ≈Pr

2(�1 P1+�2 P2) from (17). In addition, according to the generaldefinition given by (4), i.e., t j,l = √β(A j,lr j + B j,lr∗j ), weget

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

A j,2i−1(1, 2L j,i − 1) = 1A j,2i−1(2, 2L j,i ) = 1A j,2i(1, 2L j,i + 1) = 1A j,2i(2, 2L j,i + 2) = 1,

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

B j,2i−1(1, 2L j,i + 2) = 1B j,2i−1(2, 2L j,i + 1) = −1B j,2i (1, 2L j,i ) = −1B j,2i (2, 2L j,i − 1) = 1,

where all the remaining elements of A and B are zero. It canbe seen that

1) A j,lATj,l′ = O and B j,lBTj,l′ = O for l �= l ′;2) A j,lATj,l = I2 and B j,lBTj,l = I2;3) A j,2i−1BTj,2i = A j,2iBTj,2i−1 =

[0 1

−1 0]

for i =1, · · · , N , otherwise A j,lBTj,l′ = O.

Compared with [25], we can see that our proposed design ismore general for N = 2b. However, the work in [25] is just aspecial case for N = 2 which can not be directly extended toour scenario without the precoding precessing and the conceptof recursive Alamouti-circulant (RAC) matrix.

B. Equivalent Received Signal Model for Detection

We assume the self-interference caused by the transmittedsignal sk at Tk can be cancelled ideally. By using some basicmathematic transformations, we can rewrite (17) into

z̃1 =√

P2βGbs2 + �1, (19a)z̃2 =

√P1βGTb s1 + �2, (19b)

where z̃k = [zk,1, z∗k,2, zk,3, z∗k,4, · · · , zk,2N−1, z∗k,2N ]Tdenotes the received equivalent signal vector at Tk

after self-interference cancelling from yk and taking theconjugation for the terms with even indices. Specifically,Gb is a 2N × 2N RACM, in which the first two row is[H1,1H T2,1 H1,2H T2,2 · · · H1,N H T2,N ]. Furthermore,

�k =√

β

⎡

⎢⎢⎢⎣

∑Nj=1 Hk, j w j,1∑Nj=1 Hk, j w j,2

· · ·∑Nj=1 Hk, j w j,N

⎤

⎥⎥⎥⎦

+

⎡

⎢⎢⎣

η̃k,1η̃k,2· · ·

η̃k,N

⎤

⎥⎥⎦,

where η̃k, j = [ηk,2 j−1, η∗k,2 j ]T , w j,i = [w(1)j,i , w(2)j,i ]T =[n(1)j,L j,i + n

(2)∗j,L j,i+1, n

(1)∗j,L j,i+1 − n

(2)j,L j,i

]T and Hk, j =[

hk,2 j−1 −hk,2 jh∗k,2 j h∗k,2 j−1

]. Due to sk = Ediag(P, P)EH xk , (19) can

be also written as

z̃1 =√

P2βGbEdiag(P, P)EH x2 + �1, (20a)z̃2 =

√P1βGTb Ediag(P, P)E

H x1 + �2. (20b)

C. Equivalent Received Signal Model for PEP Analysis

Since (19) is not easy to be used to derive the theoretic PEPperformance, in this part, we present an equivalent formulationfor the diversity analysis in the next section. For simplicity andthe symmetric property of two users, we still focus on T1 only.

By using the Proposition 1, (19a) can be written as

Tb̃z1 = √P2βDbχ2 + ζ 1, (21)where Db = TbGbTHb , χk = Tbsk = [χk,1, χk,2, · · · , χk,2N ]Tand ζ k = Tb�k = [ζk,1, ζk,2, · · · , ζk,2N ]T . Conjugating theterms with even indices and using some basic mathematicaltransformations, we have

Tbz1 = √2P2βS2F1K2N h̄2 + ζ̃ 1, (22)where Sk = diag(Sk,1, Sk,2, · · · , Sk,N ) with Sk, j =[χk,2 j−1 −χk,2 jχ∗k,2 j χ∗k,2 j−1

], h̄k = R (hk) and the matrix Fk is given by

Fk =

⎡

⎢⎢⎣

Fk,1 Fk,2Fk,3 Fk,4· · · · · ·

Fk,2N−1 Fk,2N

⎤

⎥⎥⎦,

in which Fk,2l =[

d2l � hTk0

]and Fk,2l−1 =

[0

d2l−1 � ĥTk

],

l = 1, · · · , N . In addition, dl denotes the l-th rowof the Hadamard matrix of order 2N and ĥk =[hk,2, hk,1, · · · , hk,2N , hk,2N−1]T . Notice that the noisevector ζ̃ k can be expressed as

ζ̃ k =√

βTbW hk + Tbηk, (23)

where W = (W j i )N×N and W j i =[

w(1)j,i −w(2)j,i

w(2)∗j,i w

(1)∗j,i

]

. Due to

the fact that W j1i1 is independent with W j2i2 for any possible{ j1, i1} ∈ { j, i} �= { j2, i2} ∈ { j, i}, we have

�k = E[̃ζ k ζ̃ Hk ] = βE[TbW hkhHk W H Tb] + σ 2I2N= (2β‖hk‖2 + 1)σ 2I2N . (24)


Thus, (22) can be transformed into the following expression:

z̄T,1 = √P2βKH2N diag(S2, S∗2 )F̄1K2N h̄2 + ζ̄ 1, (25)where z̄T,k = R (Tbzk), ζ̄ k = R (̃ζ k), and F̄k = [Fk; F∗kB] inwhich B =

[O I2N

I2N O

].

D. Comments and Examples1) From (20), we can see that the equivalent channel

matrix Gb and its transpose are the specific RACmatrices. It should be mentioned that the structure ofthe channel matrix Gb is mimic the structure of therate one STBC proposed in [45]. However, the majordifference here is that each 2×2 block entry of Gb is theproduct of two Alamouti matrices. It is for this reasonthat we call our code as a distributed recursive block-circulant STBC.

2) Power loading among source nodes and relay nodessignificantly affects the overall performance of the wholerelay system [11], [31]. One solution of the optimalpower allocation can be obtained by maximizing thereceived SNR of the weak user link, which is given inthe following proposition.

Proposition 2: The optimal power loading to maximizethe average received SNR of the weak user link isdetermined as follows:

Pr = 14N

,

P1 =√

�2

2(√

�1 + √�2) , P2 =√

�1

2(√

�1 + √�2) .�

The proof of Proposition 2 follows that we used in [25],so it is omitted. It reveals that the optimal total powerassigned to all relays is half of the total network power,regardless of the channel variances.

3) It is known that when CSI is perfectly available at thesource nodes, the optimal detector for estimation of thetransmitted signals is the ML detector, which is to solvethe following two optimization problems:

x′2 = arg minx2 ‖̃z1 −√

P2βGbEdiag(P, P)EH x2‖2,(26a)

x′1 = arg minx1 ‖̃z2 −√

P1βGTb Ediag(P, P)EH x1‖2.

(26b)

Due to the Proposition 1, the minimization of (26a) isequivalent to minimizing the following two expressionsindividually:

x′2,o = arg mins2,o ‖̃z1 −√

P2βGb,oPx2,o‖2, (27a)x′2,e = arg mins2,e ‖̃z1 −

√P2βGb,ePx2,e‖2, (27b)

where Gb,o and Gb,e denote the 2N × N matricesincluding only odd columns and even columns of Gbrespectively.

4) It is very interesting to see that (25) is equivalent to (22)in a sense that is given by the following lemma.

Lemma 1: The following two determinants are equal

det(

I4N + αk F̄Hk̄ diag(S Hk , STk )diag(Sk , S∗k )F̄k̄) 1

2

= det(

I2N + αkFk̄FHk̄ S Hk Sk), (28)

where αk is a constant independent with sk and hk ,k = 1, 2. �The proof of Lemma 1 can be completed by usingSylvester’s determinant theorem [46] and the fact thatFk,2mFHk,2n−1 = O, m, n = 1, · · · , N .

For better understanding of our design, we consider the full-rate distributed space-time transmission schemes and furtherpresent three typical examples in the following.

Example 1: Consider the TWRN with single dual-antennarelay, i.e., b = 0, we have A1 = (I2, O2), A2 = (O2, I2), B1 =(O2, J2), and B2 = (−J2, O2), where J2 = [0, 1; −1, 0].These lead to G0 = H1,1H T2,1, E = I2, and P turns into 1.Thus, we get sk = xk , which means the transmitted signals skbelong to the constellation set Q .

Example 2: For the TWRN with 2 dual-antenna relays,i.e., b = 1, we have the coefficients as A1,1 = A2,3 =(I2, O2, O2, O2), A1,2 = A2,4 = (O2, I2, O2, O2), A1,3 =A2,1 = (O2, O2, I2, O2), A1,4 = A2,2 = (O2, O2, O2, I2),B1,1 = B2,3 = (O2, J2, O2, O2), B1,2 = B2,4 =(−J2, O2, O2, O2), B1,3 = B2,1 = (O2, O2, O2, J2), andB1,4 = B2,2 = (O2, O2,−J2, O2). Hence, we have

G1 =[

H1,1H T2,1 H1,2HT

2,2

H1,2H T2,2 H1,1HT

2,1

]

.

In addition, P = D = diag(1, e j π4 ), and E = EH is obtainedby exchanging the second row and the third row of I4.Furthermore, sk = diag(1, e j π4 , 1, e j π4 )xk .

Example 3: In this case, we present a more complex TWRNwith 4 dual-antenna relays, i.e., b = 2. Similarly, we have theRAC matrix

G2 =

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

H1,1H T2,1 H1,2HT

2,2 H1,3HT

2,3 H1,4HT

2,4

H1,2H T2,2 H1,1HT

2,1 H1,4HT

2,4 H1,3HT

2,3

H1,3H T2,3 H1,4HT

2,4 H1,1HT

2,1 H1,2HT

2,2

H1,4H T2,4 H1,3HT

2,3 H1,2HT

2,2 H1,1HT

2,1

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

and E = [I8(1, :); I8(5, :); I8(2, :); I8(6, :); I8(3, :); I8(7, :);I8(4, :); I8(8, :)], in which I8(i, :) denotes the i -th row vectorof I8. As a consequence, s2,o = 12 Hadamard(4)WH4 Dx2,oand s2,e = 12 Hadamard(4)WH4 Dx2,e, where D =diag(1, e j

π8 , e j

2π8 , e j

3π8 ).

IV. PEP ANALYSIS FOR THE PROPOSED DESIGN

In this section, we will show that the PEP of our designeddistributed concatenated recursive Alamouti-circulant STBCscheme proposed in Section III can achieve the lower boundgiven by Theorem 1.


A. Derivation of the PEP Formulations

By using (25) and taking the average of P(s2 → s′2|h1, h2

)

over the random vector h2, we have

P(s2 → s′2|h1

)

= 1π

∫ π/2

0

dθ

det(

I4N + 2P2�1β4sin2θ KH2N FH1 �̃

H�−11 �̃F1K2N

) 12

,

(29)

where �̃= diag(�S2,�S∗2 ), the difference matrix�S2 = S2 − S′2. According to (24), the noise covariancematrix �1 = (2β‖h1‖2 + 1)σ 2I4, which yields

P(s2 → s′2|h1

)

= 1π

∫ π/2

0

dθ

det(

I4N + 2β2(θ)ρ2βh12+1 FH1 �̃

H�̃F1

) 12

(30)

where β2(θ) =(4 sin2 θ

)−1P2�2β and SNR is defined as

ρ = 1σ 2

. As shown in Lemma 1, (30) is equivalent to

P(s2 → s′2|h1

)

= 1π

∫ π/2

0

dθ

det(

I2N + 2β2(θ)ρ2Pr ‖h1‖2+1 F1FH1 �S

H2 �S2

). (31)

By taking the average of (31) over h1, we attain the followinglemma, which will help to give our final Theorem 2.

Lemma 2: For a 2N × 2N semi-definite matrix Q with fullrank, if we let J (h1, Q) = det(I4N + α(θ, ρ)�Q�H )−1 andF(ρ, Q) = Eh1 [J (h1, Q)], then F(ρ, Q) has the followingasymptotic formula when SNR is large,

F(ρ, Q) = lnN ρ

det(I2N + �1α(θ, ρ)Q) + O(

lnN−1 ρρ2N

).

where � = diag(h1,1, h1,1, h1,2, h1,2, · · · , h1,N , h1,N ) is a4N × 2N channel matrix, N = 2b and b is a nonnegativeinteger. We assume that all h1,l , l = 1, · · · , 2N , are i.i.d.complex Gaussian random variables, each of which has zeromean and variance �1. In addition, α(θ, ρ) = O(ρ) withρ being the SNR. �

Proof : See Appendix C.Theorem 2: By using the proposed distributed concatenated

recursive Alamouti-circulant STBC scheme, the PEP at Tk̄ forthe TWRN with N = 2b relays with each having two antennasis given by

P(sk → s′k

)

= 22N−1(4N − 1)!!

(4N)!!N2N P2Nk �2N1 �2N2 β2N det(�S Hk �Sk)lnN ρ

ρ2N

+ O(

lnN−1 ρρ2N

). (32)

�Proof : See Appendix D.

We would like to make the following observations onTheorem 2. Firstly, the code design presented in this paperenables the ML detector of the receiver to achieve thediversity-gain lower bound given in Theorem 1. Secondly, in

addition to the maximum diversity gain, the asymptotic PEPperformance is dominated by min det(�S Hk �Sk), which, fol-lowing the concept from the MIMO system, is called codinggain. Furthermore, to check the correctness of the PEP per-formance presented in Theorem 2, we perform the numericalsimulations and compare them with theoretical results aspresented in Section V.

B. Design of the Transmission Signal forAchieving the Minimum PEP

The PEP leads to two design criteria for space-timecodes [47]: Full Rank Criterion (maximize the mini-mum rank of the difference matrix over all pairs ofdistinct codewords) and Product Criterion (maximize theminimum value of the determinant over all pairs ofdistinct codewords). Since det(�S H2 �S2) = det2(�S2)and �S2 = diag(�S2,1,�S2,2, · · · ,�S2,N ), �S2, j =[�χk,2 j−1 −�χk,2 j�χ∗k,2 j �χ∗k,2 j−1

], we define the cost function as

λ(s2, s′2) = det(�S2) =N∏

j=1det(�S2, j )

=N∏

j=1

(|�χ2,2 j−1|2 + |�χ2,2 j |2

). (33)

From (33), we can see that the determinant can be zeroif N > 1 and sk is chosen directly from the M-QAMconstellation sets. To solve this problem, we properly choosethe rotation angle φi of the 2N transmitted signals and theprecoding matrix W to ensure that the worst case of the deter-minant is maximized. Thus, the solution to this optimizationproblem can be obtained as following.

d2 = maxφi ,i=1,··· ,N−1

( minx2 �=x′2

λ(φi , x2, x′2))

= maxφi ,i=1,··· ,N−1

( minx2 �=x′2

det(�S2)). (34)

Due to χ2 = Tbs2 = TbEdiag(P, P)EH x2, which is dividedinto χ2,o = 1√N Hadamard(N)s2,o = WHN Dx2,o and χ2,e =WHN Dx2,e, we can see that

1) When N = 1, λ(s2, s′2) = λ(x2, x′2) = |�x2,1|2 +|�x2,2|2. It can be obtained that d2 = d2min , where dmindenotes the minimum Euclidean distance of M-QAMconstellation Q with unit power.

2) When N = 2, the determinant is simplified tomaxφ1

( minx2 �=x′2

λ(φ1, x2, x′2))

= 14

maxφ1

minx2 �=x′2

((|�x2,1 + e jφ1�x2,3|2

+ |�x2,2 + e jφ1�x2,4|2) · (|�x2,1 − e jφ1�x2,3|2+|�x2,2 − e jφ1�x2,4|2)

). (35)

It turns to be the same optimization problem as theconventional STBC signal design in [48]. It can beproved that, for M-QAM, φ1 = π/4 is the optimal angleto maximize the worst case of the determinant.


3) When N = 2b, b ≥ 2, as analyzed in [49],the transmitted signal s2 should be precoded bys2,o = 1√N Hadamard(N)WHN Dx2,o and s2,e =

1√N

Hadamard(N)WHN Dx2,e, where xl ∈ Qis the constellation symbol vector. D =diag(1, e jφ1, · · · , e j (N−1)φ1) and WN is a discreteFourier matrix, i.e., WN (p, q) = 1√N e− j2πpq/N ,p, q = 1, · · · , N . To ensure the full rankof the difference matrix and maximize itsdeterminant, N − 1 rotation angles are required,i.e., φi = i × φ1, i = 2, · · · , N − 1. Thus, the solutionto this optimization problem becomes

maxφi

( minx2 �=x′2

λ(φ1, x2, x′2))

= maxφi ,i=1,··· ,N−1

minx2 �=x′2

N∏

p=1

(|wpD�x2,e|2+|wpD�x2,o|2),

(36)

where wp denotes the p-th row of the matrix WHN . ForM-QAM, the optimal rotation angle is determined asφ1 = π2N . The detailed proof can refer to the Corollary 1and [49, Th. 5].

V. SIMULATIONS

Throughout the simulations, we assume that both of the twosource nodes know perfect CSI, whereas the relay nodes onlyknow the first and second order statistics values of the source-relay channels. Firstly, we carry our computer simulations andexamine error performance by comparing the following twohalf-duplex TWRNs, which have almost the same complexityat relays:

(a) The TWRN composed of two source nodes and2N relay nodes with each employing oneantenna [15]–[17], [29], where the transmitted signal atthe i -th relay is also designed to be a linear function ofits received signal and its conjugate. For the networkswith 2 relays and 4 relays, the corresponding DSTBCshave been presented in [29].

(b) The TWRN assisted by N dual-antenna relays using thecode design proposed in Section III, where the optimizedangle rotation and precoding are adopted to ensure theexpected performance.

Fig. 3 gives the bit error rate (BER) comparison ofnetwork (a) and network (b) by using optimal power alloca-tion (OPA) over the symmetric channels with �1 = �2 = 1,where the BER curves are obtained by averaging the BER val-ues at two source nodes. It can be observed from Fig. 3 thatnetwork (b) outperforms network (a) in the whole SNR regionconsidered when the two networks have the same antennanumber. The superiority of our network (b) over network (a)becomes more obvious when SNR is increasing, owing tothe fact that the slopes for network (b) always outperformthose for network (a). This observation is consistent withour asymptotic PEP analysis, i.e., the PEP function for thenetwork (a) is proportional to ρ−2N ln2N ρ, whereas that forthe network (b) is proportional to ρ−2N lnN ρ when 2N relay

Fig. 3. Average BER performance comparison of network (a) andnetwork (b) over symmetric channels �1 = �2 = 1, where N = 1, 2, 4, 8and 4-QAM are considered. OPA: P1 = P2 = 14 , Pr = 14N ;EPA: P1 = P2 = 13 , Pr = 16N .

Fig. 4. BER of T1 → T2, where N = 1, 2, 4, 8, �1 = 1,�2 = 3 and4-QAM. OPA: P1 = 3−

√3

4 , P2 =√

3−14 and Pr = 14N , EPA: P1 = P2 = 13

and Pr = 16N .

antennas are used. In addition, we can also observe that theBER performance of OPA outperforms that of conventionalequal power allocation (EPA).

Secondly, we further illustrate the impact of power alloca-tion on the BER performance of the proposed network (b) overasymmetric channels. Fig. 4 and Fig. 5 demonstrate the BERat T2 and T1 respectively under a typical asymmetric channel,i.e., �1 = 1 and �2 = 3. We can observe from these twofigures that the OPA given by Proposition 2 indeed enhancesthe error performance of the whole network, compared withconventional equal power allocation (EPA). Specifically, at theBER of 10−4, the SNR gains of OPA over EPA are about0.3 dB−0.8 dB for T1 → T2 link, whilst 1.5 dB−2.5 dB forthe reverse T2 → T1 link. Power loading is more important forthe terminal having poor channel condition with relay nodes.In addition, it can be concluded that power loading doesn’taffect the achieved diversity gain due to the fact that all theBER curves with the same relay number are parallel in highSNR region.

Thirdly, from the view of the pure STBC design, we can seethat our proposed concatenated recursive Alamouti-circulant


Fig. 5. BER of T2 → T1, the same conditions as Fig. 4 are adopted.

15 20 25 30

SNR (dB)

10-6

10-5

10-4

10-3

10-2

10-1

BE

R

Diagonal STBCKarmakar&Rajan, CUW-DSD, [43]Khan&Rajan&Lee, DSDD-STBC, [40]Yuen&Guan&Tihung, MDC-QOSTBC, [42]Proposed RAC-STBC

Fig. 6. BER performance comparison of different STBCs for MISO systemwith eight transmit antennas, 4-QAM.

Fig. 7. Comparisons of the simulated PEP and the theoretical PEP givenin Theorem 2 over channel �1 = �2 = 1, where OPA and 4-QAM areconsidered.

STBC (RAC-STBC) can also used for MISO systems. In thispart, we carry the computer simulations and examine error per-formance compared with the existed four rate-one STBCs [40],[42], [43], where MISO system with eight transmit antennas isconsidered. From Fig. 6, we can observe that in this case, ourproposed RAC-STBC scheme outperforms the other STBCsin the considered SNR region, e.g., about 1dB performanceimprovement at least at BER 10−4. Furthermore, our designis a general rate-one design for any transmit antenna number

with a power of two, i.e., it can achieve the full diversity andthe maximum coding gain without computer searching.

Finally, we perform the numerical PEP simulations underthe assumption of s′k being the nearest one of the transmittedvector sk with only one different adjacent constellation ele-ment. From the simulated PEP given in Fig.7, we can see thatthe theoretical derivation in Theorem 2 can accurately reflectthe PEP performance when SNR is large.

VI. CONCLUSION

In this paper, we have first derived a tight diversity-gainlower bound with the ML receiver for the general distributedlinear dispersion coded TWRN consisting of the two sourceswith each equipped with a single antenna and N relays witheach employing two antennas. This lower bound reveals thatthe PEP cannot decay faster than lnN SNR/SNR−2N whenSNR is high. Then, by carefully making use of the Alamouticoding scheme to combine the received signals at the relaynodes, we have presented an optimal DSTBC design for suchnetworks with N = 2b relays, resulting in the fact that theequivalent channel at each source node is the RAC matrix,of which each block being the product of the two Alamoutichannel matrices, and the equivalent noise at each sourcenode is white Gaussian. Asymptotic PEP formulae have beenobtained to show that the maximum diversity gain function,i.e., the diversity-gain lower bound, lnN SNR/SNR2N , as wellas the maximum coding gain are achieved by using our design.In addition, to further improve error performance, we haveattained the optimal power allocation maximizing the worst-link received SNR. Comprehensive computer simulations havedemonstrated that the code presented in this paper, alongwith the optimal power loading, provides substantial errorperformance gains over the currently-available DSTBCs inliterature. Overall, because of the product of the two Alamouticoding matrices in the equivalent channel matrices and becauseof all those aforementioned optimality, we have called sucha code optimal distributed concatenated recursive Alamouti-circulant STBC.

APPENDIX

A. Some Necessary Properties

In order to prove our results, we first recall some basicproperties on matrix theory [46] which will be required in ourderivation.

Proposition 3: Sylvester’s determinant theorem states thatfor A, an m-by-n matrix, and B, an n-by-m matrix [46]:

det (Im + AB) = det (In + BA) .Proposition 4: For any two N×N positive definite matrices

A and B, if A � B, then,det(A) ≥ det(B).

Proposition 5: Let K =(

K11 K12K21 K22

)be a positive semi-

definite matrix. Then, the following two statements are true.1) If K11 is invertible, then, its Schur complementary

matrix K22−K21K−111 K12 is also a positive semi-definitematrix and det

(K) = det(K11) det

(K22 −K21K−111 K12

).


2) If K22 is invertible, then, its Schur complementarymatrix K11−K12K−122 K21 is also a positive semi-definitematrix and det

(K) = det(K22) det

(K11 −K12K−122 K21

).

B. Proof of Proposition 1

1) When b = 0, i.e., N = 1, we directly have T0 =T0,1 = I2 and g0 = G0, thus D0 = TH0 D0T0 =diag(T0,1g0) = G0.

2) Now, we assume that Gn = THn DnTn holds for anyRACM Gn ∈ RACMn+1. Then, for b = n + 1,we prove that a RACM Gn+1 ∈ RACMn+2 can beexpressed as Gn+1 = THn+1Dn+1Tn+1. To prove thisequation, we firstly recall that Gn+1 =

[G1n G

2n

G2n G1n

],

where G�n ∈ RACMn+1 for � = 1, 2. Accordingto our assumption, G�n = THn D�nTn , where D�n =√

2ndiag(Tn,1g�n, Tn,2g�n, · · · , Tn,2n g�n), in which g�n is

the first two columns of G�n . Since Tn+1 can be writtenas Tn+1 = 1√2

[Tn TnTn −Tn

], we readily have

Dn+1 =√

2n+1diag(Tn+1,1gn+1, Tn+1,2gn+1, · · · ,Tn+1,2n+1 gn+1)

=[

D1n + D2nD1n − D2n

]

due to gn+1 =[g1n; g2n

]and Tn+1, j ={[

Tn, j Tn, j], if j ≤ 2n ,

[Tn, j−2n − Tn, j−2n

], else if j > 2n.

Now, Gn+1can be rewritten as

Gn+1 =[

THn D1nTn T

Hn D

2nTn

THn D2nTn T

Hn D

1nTn

]

= 12

[THn T

Hn

THn −THn] [

D1n + D2nD1n − D2n

]

·[

Tn TnTn −Tn

]

= THn+1Dn+1Tn+1.This completes the proof of Proposition 1. �

C. Proof of Lemma 2

In the following, we let t j = 1�1 (|h1,2 j−1|2 + |h1,2 j |2), i.e.,the probabilistic density function of t j is t j e−t j . We prove thislemma by the induction on b.

1) When b = 0, i.e., N = 1, Q = q11I2, D = ‖h1,1‖2I2 andhence, F(ρ, Q) becomes

F0(ρ, Q) =∫ ∞

0

t1e−t1(1 + �1α(θ, ρ)q11t1)2 dt1

= ln ρ + O(1)�21α

2(θ, ρ)q211

= ln ρdet(I2 + �1α(θ, ρ)Q) + O

(1

ρ2

),

where we use 1ρ = 11+ρ + O( 1ρ2 ).

2) Assume that when b = n, i.e., N = 2n relays, we have

Fn(ρ, Q) = lnN ρ

det(I2N + �1α(θ, ρ)Q + O(

lnN−1 ρρ2N

). (37)

3) When b = n + 1, i.e., 2N relays, we proveFn+1(ρ, Q)

= ln2N ρ

det(I4N + �1α(θ, ρ)Q) + O(

ln2N−1 ρρ4N

). (38)

By Proposition 5

Jn+1(h1, Q) = det−1(I8N + α(θ, ρ)�Q�H )= det−1

(I8N + α (θ, ρ)

[�1Q11�H1 �1Q12�

H2

�2Q21�H1 �2Q22�H2

])

(39)

= det−1(I4N + α(θ, ρ)�2Q22�H2 )·det−1(I4N + α(θ, ρ)�1Q̄22�H1 ), (40)

where Q11 = Q22, Q12 = Q21, and Q̄22 = Q11 −α(θ, ρ)Q12�H2 (I4N + α(θ, ρ)�2Q22�H2 )−1�2Q21. We canuse the assumption Fn(ρ, Q) to obtain that

Fn+1(ρ, Q) = Eh1[Jn+1(h1, Q)]= E�2 [det−1(I4N + α(θ, ρ)�2Q22�H2 )

·E�1[det−1(I4N + α(θ, ρ)�1Q̄22�H1 )]

]

= E�2[

1

det(I4N + α(θ, ρ)�2Q22�H2 )

·(

lnN ρ

det(I2N + �1α(θ, ρ)Q22)+ O

(lnN−1ρ

ρ2N

))]

= E�2[

1

det(I4N + α(θ, ρ)�2Q22�H2 )

· lnN ρ

det(I4N + �1α(θ, ρ)CQ22CH )

]

+ O(

ln2N−1ρρ4N

),

where C = 1√2

I2N ⊗ e2 and e2 = [1, 1]T . According toProposition 5, Fn+1(ρ, Q) arrives at

Fn+1(ρ, Q) = E�2[

lnN ρ

det(I8N + �1α(θ, ρ)�̄Q�̄H )]

+O(

ln2N−1 ρρ4N

), (41)

where �̄ = diag(√�1C,�2). Again, we have

Fn+1(ρ, Q) = E�2[

lnN ρ

det(I4N + �1α(θ, ρ)CQ11CH )

· 1det(I4N + α(θ, ρ)�2Q11�H2 )

]

+O(

ln2N−1ρρ4N

)

where Q̄11 = Q22 − �1α(θ, ρ)Q21CH (I4N +�1α(θ, ρ)CQ11CH )−1CQ12. By using the assumption (37)


again, we have

Fn+1(ρ, Q)

= ln2N ρ

det(I8N + �1α(θ, ρ)diag(C, C)Qdiag(CH , CH ))+ O

(ln2N−1 ρ

ρ4N

),

which is equivalent with

Fn+1(ρ, Q) = ln2N ρ

det(I4N + �1α(θ, ρ)Q) + O(

ln2N−1 ρρ4N

).

(42)

This completes the proof of Lemma 1. �

D. Proof of Theorem 2

We prove Theorem 2 by deriving the lower bound and theupper bound of PEP respectively.

1) Lower Bound: In this case, the conditional PEP givenby (31) satisfies

P(s2 → s′2|h1

)

≥ 1π

∫ π2

0

dθ

det(I2N + 2β2(θ)ρF1FH1 �S H2 �S2

)

= 1π

∫ π2

0

dθ

det(I2N + 2β2(θ)ρ TbF1FH1 THb︸︷︷︸D

Tb�S H2 �S2THb︸︷︷︸

Q

)

= PLB(s2 → s′2|h1

), (43)

where �S2 = diag(�S2,1, �S2,2, · · · , �S2,N ). Let � =diag(h1,1, h1,1, h1,2, h1,2, · · · , h1,N , h1,N ), D = TbF1FH1THb = N�H � = Ndiag(‖h1,1‖2I2, ‖h1,2‖2I2, · · · ,‖h1,N ‖2I2) and Q = Tb�S H2 �S2THb are also semi-positivematrices. According to Proposition 3, (43) becomes

PLB(s2 → s′2|h1

) = 1π

∫ π/2

0

dθ

det(I4N + 2Nβ2(θ)ρ�Q�H

) .

(44)

By using Lemma 2, we can write the lower bound as

PLB(s2 → s′2

)

= 1π

∫ π/2

0Eh1

[1


)]

dθ

= 1π

∫ π/2

0

lnN ρ

det (I2N + 2N�1β2(θ)ρQ)dθ + O(

lnN−1 ρρ2N

).

(45)

In addition, since we assume Q is a full rank matrix, we noticethat det(I2N + 2N�1β2(θ)ρQ) =∏2N�=1(1 + 2N�1β2(θ)ρλ�)and 11+2N�1β2(θ)ρλ� = 12N�1β2(θ)ρλ� + O(ρ−2) with λ� is the�-th non-zero eigenvalue of Q, then we have

PLB(s2 → s′2

) = 1π

∫ π/2

0

lnN ρdθ

22N N2N �2N1 β2N2 (θ)

∏2N�=1 λ�ρ2N

+ O(

lnN−1 ρρ2N

). (46)

Replacing β2(θ) withP2�2β4 sin2 θ

, we obtain

(46) = 22N−1(4N − 1)!!

(4N)!!N2N P2N2 �2N1 �2N2 β2N∏2N

�=1 λ�lnN ρ

ρ2N

+ O(

lnN−1 ρρ2N

), (47)

where we use∫ π/2

0 sin4N θdθ = (4N−1)!!π(4N)!!2 .

2) Upper Bound: To derive the upper bound of PEP, theconditional PEP given by (31) can be written as

P(s2 → s′2|h1

)

= 1π

∫ π2

0

(2β‖h1‖2 + 1)2N dθdet((2β‖h1‖2 + 1)I2N + 2β2(θ)ρFFH �S H2 �S2

)

≤ 1π

∫ π2

0

(2β‖h1‖2 + 1)2N dθdet(I2N + 2β2(θ)ρFFH�S H2 �S2

) . (48)

Similarly, we define the upper bound of PEP as

PUB(s2 → s′2

)

= 1π

∫ π/2

0Eh1

[(2β‖h1‖2 + 1)2N


)]

dθ. (49)

Notice that (2β‖h1‖2 + 1)2N can be expandedinto (2β‖h1‖2 + 1)2N = 1 + f (‖h1‖2) = 1 +∑2N

�=1 C�2N 2�β�‖h1‖2�, where C�m = m!�!(m−�)! . Substitutingthis expansion into (49) yields

(49) = 1π

∫ π/2

0Eh1

[1


)]

dθ

+ 1π

∫ π/2

0Eh1

[f (‖h1‖2)


)]

dθ

︸︷︷︸P1,h (ρ)

.

(50)

The first term of (50) is the same as (45), while the secondterm of (50) can be simplified as

P1,h(ρ)

≤ 1π

∫ π/2

0Eh1

[f (‖h1‖2)

det(I2N + 2Nβ2(θ)ρλmin�H�

)]

dθ

= 1π

∫ π/2

0Eh1

[ ∑2N�=1 C�2N 2�β�‖h1‖2�

∏Nj=1(1 + 2Nβ2(θ)ρλmin‖h1, j‖2

)2

]dθ.

(51)

Due to ‖h1‖2� = (∑Nj=1 ‖h1, j‖2)� =∑

�1,··· ,�N d�1,··· ,�N∏Nj=1 ‖h1, j‖2� j , where d�1,··· ,�N are constants which are inde-

pendent of h1, j for j = 1, · · · , N , � j are integers rangingfrom 0 to �, we obtain

Eh1

[ ‖h1‖2�∏N

j=1(1 + 2Nβ2(θ)ρλmin‖h1, j ‖2

)2

]

=N∏

j=1Eh1, j

[ ‖h1, j‖2� j(1 + 2Nβ2(θ)ρλmin‖h1, j‖2

)2

], (52)


which is obtained for that ‖h1, j‖2 are independent with eachother. Due to

Eh1, j

[ ‖h1, j ‖2� j(1 + 2Nβ2(θ)ρλmin‖h1, j‖2

)2

]

={

O(ρ−2), if � j > 0,O(ρ−2 ln ρ), else if � j = 0, (53)

and at least one of � j is not zero, combining (52), (53)with (51) results in

P1,h(ρ) = O(ρ−2N lnN−1 ρ). (54)Finally, substituting (54) and (47) into (50), we have

PUB(s2 → s′2

) = 22N−1(4N − 1)!!

(4N)!!N2N P2N2 �2N1 �2N2 β2N∏2N

�=1 λ�

× lnN ρ

ρ2N+ O

(lnN−1 ρ

ρ2N

). (55)

Combining (47) and (55) result in (32) with k = 2, which isthe PEP at T1. This completes the proof of Theorem 2. �

ACKNOWLEDGMENT

Part of the work was performed while F.-K. Gong was a visi-tor in the Department of Electrical and Computer Engineering,McMaster University, Canada. The authors would like to thankProf. Jian-Kang Zhang with McMaster University in Canadafor his valuable suggestions which significantly improved thequality of our work.

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Feng-Kui Gong (M’12) was born in Shandong,China, in 1979. He received the M.S. and Ph.D.degrees from Xidian University, Xian, China, in2004 and 2007, respectively. From 2011 to 2012,he was a Visiting Scholar with the Departmentof Electrical and Computer Engineering, McMasterUniversity, Hamilton, ON, Canada. He is currentlya Professor with the Department of CommunicationEngineering, State Key Laboratory of IntegratedServices Networks, Xidian University. His researchinterests include cooperative communication, distrib-

uted space-time coding, digital video broadcasting system, satellite commu-nication, and 4G/5G techniques.

Guo Li (S’14) was born in Shaanxi, China, in 1989.He received the B.Sc. and M.Sc. degrees in com-munications engineering from Xidian University,Xian, China, in 2011 and 2014, respectively. He iscurrently pursuing the Ph.D. degree in communi-cation and information system, Xidian University.His research interests include MIMO wirelesscommunications, cooperative communications, andlarge-scale antenna system.

Jianhua Ge was born in Jiangsu, China, in 1961.He received the B.Sc. and Ph.D. degrees in com-munications engineering from Xidian University,Xi’an, China, in 1982 and 1990, respectively.He is currently a Professor and the Deputy Directorof the State Key Laboratory of Integrated ServicesNetworks, Xidian University, Xi’an. He has beeninvolved in digital television (DTV) standardizationas a DTV Technical Expert. His research inter-ests include digital video broadcasting system andmultiple-input-multiple-output and mobile commu-

nication techniques.

Jinhong Yuan (M’02–SM’11–F’16) received theB.E. and Ph.D. degrees in electronics engineeringfrom the Beijing Institute of Technology, Beijing,China, in 1991 and 1997, respectively. From 1997to 1999, he was a Research Fellow with the Schoolof Electrical Engineering, University of Sydney,Sydney, Australia. In 2000, he joined the Schoolof Electrical Engineering and Telecommunications,University of New South Wales, Sydney, Australia,where he is currently a Telecommunications Profes-sor. He has authored two books, three book chapters,

over 200 papers in telecommunications journals and conference proceedings,and 40 industrial reports. He is a co-inventor of one patent on MIMOsystems and two patents on low-density-parity-check codes. He has co-authored three best paper awards and one Best Poster Award, including theBest Paper Award from the IEEE Wireless Communications and NetworkingConference, Cancun, Mexico, in 2011, and the best paper award fromthe IEEE International Symposium on Wireless Communications Systems,Trondheim, Norway, in 2007. His current research interests include errorcontrol coding and information theory, communication theory, and wirelesscommunications. He is currently serving as an Associate Editor of the IEEETRANSACTIONS ON COMMUNICATIONS. He served as the IEEE NSW Chairof Joint Communications/Signal Processions/Ocean Engineering Chapter from2011 to 2014.

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