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 em- ployed. In this paper, we derive the exact asymptotic pairwise error proba- bility (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 destina- tion. 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 destina- tion 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.