The Pairwise Error Probability and Precoder Design for Amplify-And-Forward Half-Duplex Cooperative System

THE PAIRWISE ERROR PROBABILITY AND PRECODER DESIGN FOR AMPLIFY-AND-FORWARD HALF-DUPLEX COOPERATIVE SYSTEM

Yanwu Ding, Jian-Kang Zhang and K. Max Wong

Department of Electrical and Computer Engineering McMaster University, Hamilton, Ontario, Canada

ABSTRACT but also outperform the system for which Alamouti’s code is employed. In this paper, we derive the exact asymptotic pairwise error probability (PEP) of an AF half-duplex cooperative system in which the maximum likelihood detector is used. Based on our analysis and 2. SYSTEM MODEL observations, we propose a unitary precoder design to improve the system performance. We first design a precoder to achieve the full We now present the system model of the AF half-duplex protocol diversity gain function. Then for the case of a 4-QAM signal be- proposed in [9]. We assume that all the nodes in the system is ing transmitted, we further optimize the coding gain, and arrived equipped with single antenna and the channel is quasi-static, i.e., at a closed-form optimum precoder. Simulations indicate that our the channel remains unchanged during the period of observation. proposed precoder design greatly improves the performance of the Single Relay For a single relay system, the channel gain from system. the source to destination is denoted by hsd whereas those from the source to the relay and from the relay to the destination are de- 1. INTRODUCTION noted respectively by hsr and hrd. All channel gains are assumed to be independent and identically distributed (IID) zero mean cir- Diversity techniques have been employed in practical wireless cularly Gaussian with unit variance. We assume that the original communication systems to overcome the effects of channel fad- information signals are equally probable from a constellation set S ing. Multiple transmitter and receiver antennas is often desirable composed of either pulse amplitude modulation (PAM) or quadra- to obtain higher diversity gain. However, this is often impracti- ture amplitude modulation (QAM) signals which are processed by cal in some applications such as mobile communications. Another a unitary precoder before being transmitted from the source node. form of diversity called cooperative diversity, has recently been The transmission of the signal is carried out block by block, each proposed for mobile wireless communications [2–6]. New coop- block being of length 2P, P ≥ 1. We denote the original data s =[sT sT ]T s =[(1) ··· ( )]T s = erative protocols such as protocols with low-complexity and pro- block by I II ,where I s , ,s P , II T tocols achieving optimal diversity-multiplexing tradeoff [7] have [s(P +1), ··· ,s(2P )] , with s(p) being the original information been proposed [4,8,9]. An upper bound of the pairwise error prob- symbol at the pth time instant, p =1, ··· , 2P . The covariance H ability (PEP) has been presented for the Alamouti coded Amplify- of s is assumed to be an identity matrix, i.e. E[ss ]=I2P .The x =[xT xT ]T x =[ (1) ··· ( )]T and-Forward (AF) protocol [3]. In this paper, we focus our consid- precoded data block I II , I x , ,x P , T eration on the AF protocol proposed in [9]. We derive an exact ex- and xII =[x(P +1), ··· ,x(2P )] , can be expressed as x = Fs, pression for the asymptotic PEP when maximum likelihood (ML) where F is a 2P × 2P unitary matrix representing the precoder. detection is used at receiver. Unlike in the case of a traditional The input-output relation can be expressed as MIMO system, the “diversity gain” of the AF half-duplex coop- r = H x + n erative system is not simply an exponential function of signal to 1 Ep 1 1 (1) noise ratio (SNR) as it is in conventional MIMO systems. Rather, where the subscript 1 indicates that the quantities are asso- it involves the logarithm function of the SNR. This logSNR factor ciated with single relay (this transmittion scheme is referred as is also observed for orthogonal space-time modulated relay sys- Scheme 1 in the paper), and r1 is the received vector, Ep is the tem by Chang and Hua in [10]. We designate this characteristic average power for transmitting a symbol at each node, and b is the diversity gain function of the AF half-duplex cooperative sys- the amplification coefficient at the relay node and is chosen [3] as E E tem. On the other hand, similar to the case of conventional MIMO ≤ p = p n ∼N(0 2Σ ) b E[|h |2]E +σ2 E +σ2 ,and 1 ,σ 1 is the system, the coding gain is found to be proportional to the deter- sr p p minant of the autocorrelation of the coding error matrices. While equivalent noise vector at the destination and with the AF protocol in [9] provides the optimal diversity-multiplexing IP 0 tradeoff, it can not provide the full designated diversity gain func- Σ1 = 2 2 (2) 0 (1 + b |h | )I tion. We then propose a unitary precoder design to improve the rd P performance. First we design a precoder to achieve the maximum hsd IP 0 and H1 = (3) diversity gain function. Then we further optimize the precoder to bhsrhrdIP hsdIP maximize the minimum of the coding gain. Simulations indicate that our proposed precoders not only significantly improve the sys- being the channel matrix. For convenience in the analysis of PEP, tem performance over the system for which no precoder is used, we rewrite H1x as H1x = X1h1,whereX1 is the signal matrix

0-7803-9323-6/05/$20.00 ©2005 IEEE. 149 2 and h1 is the equivalent channel vector where ρ = Ep/σ represents the signal-to-noise ratio (SNR) when no relay is employed in the system, U1 is the error matrix after x 0 X = I h = hsd precoding, 1 x x and 1 . (4) II I bhsrhrd u 0 U = I = X − X Multiple Relay For a multiple relay system with N relay nodes, 1 u u 1 1 (8) we denote the channel from the source to the nth relay node by II I hsrn , and the channel from the nth relay to the destination by with u = x − x ,i = I, II, and X1 and X1 as deﬁned in Eq. =1··· i i i hrdn ,n , ,N. All channel gains are also assumed to be (4). The asymptotic PEP of te single relay is given in the following IID zero mean circularly Gaussian with unit variance. The relays theorem. take turns to assist the transmission from the source to the destination. At any instant, only one relay is active. The signal transmis- H Theorem 1 Suppose det(U1 U1) =0where U1 is given by active sion between the source and the relay at any instant assumes Eq. (8), then at high SNR, the average PEP for the single relay the mode of Scheme 1. We refer this multiple relay transmission system is given by scheme as Scheme M in the paper. For simplicity we assume that the block length between the source and each active relay is of 2. 6 | ln det(UH U )| T (s → s)= ( −2 ln )+O 1 1 We denote the original data vector by s =[s(1), ··· ,s(2N)] , Pe1 H ρ ρ 2 . H det(U1 U1) ρ with covariance being E[ss ]=I2N . The precoded signal vector x = F s is M M , where subscript M denotes entities associated with 1 (det(UH U )) ( −2 ln ) The terms 6 1 1 and ρ ρ are respectively desig- the multiple relays, and FM is a 2N × 2N unitary matrix, and x =[ (1) ··· (2 )]T nated the coding gain and the diversity gain function of the system. M x , ,x N . The received signals at the destination are: r = X h + n M Ep M M M (5) Comparing the result in Theorem 1 with the asymptotic PEP for a conventional MIMO system [12], the following observations are where r is the received vector, X is the 2N ×(N +1)transmit- M M noted: ted signal matrix, and h is the 2N +1 channel vector respectively M (1) The “diversity gain” for an AF single relay system in- given by   volves a function of the logarithm of SNR. We therefore have x(1) 0 ··· · 0   changed the designation here to diversity gain function to fully  x(2) x(1) 0 ··· 0    characterize the diversity behavior of the relay system. This log  x(3) 0 0 0 ···    SNR factor in the diversity gain function had also be noted for a X = x(4) 0 x(3) 0 ···  M   , system of two relays with orthogonal space-time modulation [10].  . .. ..   . . .  (2) The condition to fully reach the diversity gain function is x(2N − 1) 0 ··· 00 to have non-zero coding gain, i.e., x(2N)0··· 0 x(2N − 1) H 4 det(U1 U1)=uI + β =0 (9) T hM =[hsd,bhsr1 hrd1 ,bhsr2 hrd2 , ··· ,bhsrN hrdN ] , = u 2u 2 − uH u uH u where β I II II I I II. Similar to that in the case n ∼N(0 2Σ ) 2 × 1 Σ and M ,σ M is the N noise vector, M is given of a MIMO system, this condition can be regarded as rank crite- Σ = Σ 1, ··· , Σ N by M diag M M ,with rion. However auto-correlation matrix of the error in the single 10 relay system, unlike that in a conventional MIMO system, is lower Σ = Mn 2 2 . triangular in its structure because the channel matrix in (3) is lower 01+b |hrdn | triangular in structure with equal diagonal elements. For the AF multiple relay system, the asymptotic PEP is given 3. PAIR-WISE ERROR PERFORMANCE by the following theorem.

Consider the case that a ML detector is employed at the receiver det(UH U )= Theorem 2 Suppose that M M after the noise is pre-whitened. Then, in either case of single or N ξ |u(2n − 1)|2 =0,whereU = X − X ,u(i)= multiple relay, for a given channel realization hm,m=1,M,and n=1 M M M 2P 2N ( ) − ( ) =1··· 2 = N | (2 − 1)|2 an original symbol block s ∈S or s ∈S the PEP is deﬁned x i x i ,i , , N, and ξ n=1 u n . as the probability of deciding in favor of s = s, s, s ∈S2P or Then, at high SNR, the average PEP of the multiple relay system S2N , and is given by is (2 +1)!!2(N+1) dm(s, s ) (s → s)= n −(N+1) lnN Pem(s → s |hm)=Q ,m=1,M (6) PeM ( +1)!det(UH U ) ρ ρ 2 N M M H π 2 | ln det(U U )| ( )= 1 2exp − x (s s) +O M M lnN−1 where [11] Q x π 0 2sin2 θ dθ and dm , is the (N+1) ρ . Euclidean distance between s and s at the ML detector.Inthe ρ case of a single relay, the average pairwise error probability is given by In this case, the diversity gain function is given by ρ−(N+1) lnN ρ (s s) (N+1)! det(UH U ) d1 , while the coding gain is given by (2n+1)!! 2(N+1) M M . Pe1(s → s )=Eh1 Q 2 Parallel to Theorem 1, we can look upon the condition of π H H −1 H 1 2 h U Σ U h det(U U ) =0in Theorem 2 as the rank criterion under which = E exp − 1 1 1 1 1 M M h1 ρ 2 dθ, (7) full diversity gain function can be achieved. π 0 8sin θ

150 Scheme1 − no precoder 4. PRECODER DESIGN AND PERFORMANCE Scheme1 − precoded −1 Scheme2 − alamouti 10 We now design unitary precoders for Scheme 1 and Scheme M

−2 to improve system performance. Our design can be carried out 10 by ensuring that: (i) the rank criterion is satisﬁed to achieve full

−3 diversity, and then (ii) the coding gain is maximized to achieve a BER 10 minimum PEP.

−4 4.1. Precoder design for full diversity 10 4 −5 For the single relay system, the rank criterion is uI + β = 10 0,withβ is non-negative but can be equal to zero (when uI ∝ uII). 3 6 9 12 15 18 21 24 27 30 SNR in dB Thus, a necessary and sufficient condition for the rank criterion 4 written as uI =0. A sufficient condition to guarantee that is : |u(k)|2 =0if and only if s = s, ∀k ∈ [1, 2P ]. For the multiple Fig. 1. BER Performance — single relay relay system, this sufficient condition also guarantees the rank cri-

terion of Scheme M. A precoder which ensures this condition can 0 10 Multilpe relay without precoder be readily obtained by applying the design scheme in [13–16] as Multiple relay−with precoder stated in the following Lemma: −1 10 Lemma 1 We deﬁne the Cyclotomic ring [17] as Z[ζr]= r i −2 10 { i=1 ciζr : ci ∈ Z,i=1, ··· ,r − 1} where Z denotes {· · · 0 ± 1 ± 2 ···} =exp(j2π ) the integer ring , , , , and ζr r . BER K n −3 = k 10 Let L k=1 k ,wherek is prime, nk is a positive integer, K mk and k =1, ··· ,K. Form the integer L1 = L2 k=1 k where −4 10 mk ≥ 1 with L2 being prime to L and let Q = LL1. Deﬁne the L × L precoder matrix −5 10 H L−1 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Fod = WL diag(1,ζQ, ··· ,ζQ ) (10) SNR in dB where WL is a normalized discrete Fourier transform matrix of u = F e e ∈ ZL[ ] e = 0 size L.If od , ζL1 and , then all the entries Fig. 2. BER performance — 2-relays in vector u are nonzero.

Our precoder design for full diversity for Scheme 1 and ∈ [0 2 ] ∈ [0 ] Scheme M is chosen as F = F with L =2P for single relay where α1,α2,θ , π ,andφ ,π . Now, the value of od | (1)|4 = | (1) cos + (2) jφ sin |4 and L =2N for multiple relays. u e θ e e θ does not depend on the In Fig. 1, we compare the bit error rate (BER) of the single choice of α1 and α2. Therefore, θ and φ are the variables for the relay system without a precoder to that equipped with the pro- optimization problem. The optimal values of θ and φ are given by posed full diversity precoder. We employ a data block length of following theorem. 2, and BPSK is used for transmitting the signal. We also plot Theorem 3 For the single relay transmission system having a the BER performance of the the Alamouti [18] coded scheme [3] data block length of 2 and transmitting 4-QAM signals, the opti- (designated as Scheme 2 in this paper. we employ 4-QAM sig- mal precoder F as given in Eq. (11) that maximizes the minimum nalling so that the transmission is carried out at the same bit rate oc coding gain has values of θ and φ given respectively by as Scheme 1. It can be seen that the designed precoder provides   approximately a 6dB gain at moderate-to-high SNR. It also out- √ performs the Alamouti coded relay protocol at moderate-to-high −1  3 − 3  θ0 =sin ,φ0 = π/12. SNRs. Fig. 2 shows the BER performance of the 2-relay systems 6 with and without the precoder. The precoder is given by Eq. (10) with L =4and Q =16. The signals are transmitted using QPSK. Fig. 3 shows the performances of single relay system with the The precoded scheme again provides substantial gain over the un- precoder of optimal coding gain and with that proposed in Sec- precoded scheme. tion 4.1 to achieve full diversity. It can be seen that the precoder of optimal coding gain provides 1-2dB gain over the precoder of full 4.2. Precoder design for maximizing coding gain diversity at moderate-to-high SNR, and the unprecoded transmission suffers badly in performance, needing more than 8dB SNR For this criterion we consider the single relay system in which the compensation for a BER of 10−4. data block length is of 2, i.e., P =1and the transmitting sig- H nals are from 4-QAM constellation. In this case, det(U1 U1)= |u(1)|4. Thus, we seek an optimal 2 by 2 unitary precoder to max- 5. CONCLUSION imize the minimum of the coding gain. A general 2 by 2 unitary matrix group can be expressed as [19], In this paper, we have analyzed the performance of an AF cooperative half-duplex relay transmission system. Our derived PEP ejα1 0 cos θejφ sin θ analysis results suggest two criteria for the precoder design: One F = jα −jφ (11) c 0 e 2 −e sin θ cos θ being the rank criterion to achieve the maximum diversity gain,

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