Adaptive Minimum Symbol Error Rate Beamforming Assisted Receiver for Quadrature Amplitude Modulation Systems S. Chen, H.-Q. Du and L. Hanzo School of Electronics and Computer Science University of Southampton, Southampton SO17 1BJ, U.K. E-mails: fsqc,lh,[email protected] ABSTRACT window density estimation [16]-[18]. In this sense, the proposed adaptive MSER technique is an extension of the method proposed The paper considers adaptive beamforming assisted receiver for in [14] to the interference-limited multiuser communication system multiple antenna aided multiuser systems that employ the band- employing the QAM scheme. width efficient quadrature amplitude modulation scheme. A min- imum symbol error rate (MSER) design is proposed for the beam- II. SYSTEM MODEL forming assisted receiver, and it is shown that this MSER approach provides significant performance enhancement, in terms of achiev- The system consists of S users, and each user transmits an M- able symbol error rate, over the standard minimum mean square er- QAM signal on the same carrier frequency ! = 2πf. The receiver ror design. A sample-by-sample adaptive algorithm, referred to as is equipped with a linear antenna array consisting of L uniformly the least symbol error rate, is derived for adaptive implementation spaced elements. Assume that the channel is narrow-band which of the MSER beamforming solution. does not induce intersymbol interference. Then the symbol-rate re- ceived signal samples can be expressed as I. INTRODUCTION S j!tl(θi) xl(k) = Aibi(k)e + nl(k) = x¯l(k) + nl(k); (1) The ever-increasing demand for mobile communication capacity i=1 X has motivated the development of adaptive antenna array assisted for 1 l L, where tl(θi) is the relative time delay at element l spatial processing techniques [1]–[10] in order to further improve ≤ ≤ for source i with θi being the direction of arrival for source i, nl(k) the achievable spectral efficiency. A particular technique that has 2 2 is a complex-valued Gaussian white noise with E[ nl(k) ] = 2σ , shown real promise in achieving substantial capacity enhancements j j n Ai is the channel coefficient for user i, and bi(k) is the kth symbol is the use of adaptive beamforming with antenna arrays. Through of user i which takes the value from the M-QAM symbol set appropriately combining the signals received by the different ele- ments of an antenna array to form a single output, adaptive beam- =4 bl;q = ul + juq; 1 l; q pM (2) forming is capable of separating signals transmitted on the same B f ≤ ≤ g carrier frequency, and thus provides a practical means of supporting with ul = 2l pM 1 and uq = 2q pM 1. Source 1 is − − − − multiusers in a space division multiple access scenario. Classically, the desired user and the rest of the sources are interfering users.The 2 2 2 the beamforming process is carried out by minimizing the mean desired-user signal to noise ratio is SNR= A1 σb =2σn and the j2 j 2 square error (MSE) between the desired output and the actual array desired signal to interferer i ratio is SIRi = A1=Ai , for 2 i S, 2 ≤ ≤ output. For a communication system, however, it is the bit error where σb denotes the M-QAM symbol energy. The received signal T rate (BER) or symbol error rate (SER) that really matters. Adaptive vector x(k) = [x1(k) x2(k) xL(k)] is given by beamforming based on directly minimizing the system's BER has · · · been proposed for binary phase shift keying and quadrature phase x(k) = Pb(k) + n(k) = x¯(k) + n(k); (3) shift keying modulation schemes [11],[12]. T where n(k) = [n1(k) n2(k) nL(k)] , the system matrix P = · · · For the sake of improving the achievable bandwidth efficiency, [A1s1 A2s2 AS sS ] with the steering vector for source i si = j!t (θ ) j!· t· ·(θ ) j!t (θ ) T high-throughput quadrature amplitude modulation (QAM) schemes [e 1 i e 2 i e L i ] , and the transmitted QAM sym- · · · T bol vector b(k) = [b1(k) b2(k) bS (k)] . [13] have become popular in numerous wireless network standards. · · · Adaptive minimum SER (MSER) equalization has been investi- A linear beamformer is employed, whose soft output is given by gated for the single-antenna single-user system with the pulse- amplitude modulation scheme [14] and with the QAM scheme [15]. y(k) = wH x(k) = wH (x¯(k) + n(k)) = y¯(k) + e(k) (4) In this paper, we derive the MSER beamforming design for the mul- T where w = [w1 w2 wL] is the beamformer weight vector tiple antenna assisted multiuser system with QAM signalling. We · · · 2 and e(k) is Gaussian distributed with zero mean and E[ e(k) ] = show that the MSER design can provide significant performance 2 H j j 2σnw w. Define the combined impulse response of the beam- gains, in terms of smaller SER, over the traditional minimum MSE H H former and system as w P = w [p1 p2 pS ] = [c1 c2 cS ]. (MMSE) design. An adaptive implementation of the MSER beam- · · · · · · forming solution is proposed based on a stochastic gradient algo- The beamformer's output can alternatively be expressed as rithm, which we refer to as the least symbol error rate (LSER). S Our proposed technique is very different to the method proposed y(k) = c1b1(k) + cibi(k) + e(k): (5) in [15]. The adaptive LSER algorithm has its root in the Parzen i=2 X Im[ y ] The noise-free part of the beamformer's output y¯(k) only takes val- 4 H c (uq +1) ues from the scalar set = y¯i = w x¯i; 1 i Nb , and R1 Y f ≤ ≤ g Y can be divided into the M subsets conditioned on the value of b1(k) l;q =4 y¯i : b1(k) = bl;q ; 1 l; q pM: (11) Y f 2 Y g ≤ ≤ cR uq Lemma 1: The subsets l;q, 1 l; q pM, satisfy the shifting 1 Y ≤ ≤ properties l+1;q = l;q + 2c1; 1 l pM 1; (12) Y Y ≤ ≤ − cR (uq −1) l;q+1 = l;q + j2c1; 1 q pM 1; (13) c u 1 Y Y ≤ ≤ − R1 l c (u −1) c (u +1) l+1;q+1 = l;q + (2 + j2)c1; 1 l; q pM 1: (14) R1 l R1 l Re[ y ] Y Y ≤ ≤ − The proof of Lemma 1 is straightforward. Lemma 2: The points of l;q are distributed symmetrically Fig. 1. Decision thresholds associated with point c1bl;q assuming cR1 > 0 and Y Y around the symbol point c1bl;q. cI1 = 0, and illustrations of symmetric distribution of l;q around c1bl;q . Lemma 2 is a direct consequence of symmetric distribution of the symbol constellation (2). This symmetric property is also illustrated Provided that c1 = cR1 + jcI1 satisfies cR1 > 0 and cI1 = 0, the ^ ^ ^ in Fig. 1. Note that the distribution of l;q is symmetric with respect symbol decision b1(k) = bR1 (k) + jbI1 (k) can be decoupled into Y to the two vertical decision thresholds cR (ul 1) and with respect 1 u1; if yR(k) cR (u1 + 1) to the two horizontal decision threshold cR (uq 1). ≤ 1 1 ul; if cR1 (ul 1) < yR(k) cR1 (ul + 1) ^bR (k) = − ≤ For the beamformer with weight vector w, denote 1 8 for 2 l pM 1 > ≤ ≤ − ^ < up ; if yR(k) > cR1 (up 1) PE (w) = Prob b1(k) = b1(k) ; (15) M M − f 6 g (6) ^ > PER (w) = Prob bR1 (k) = bR1 (k) ; (16) : u1; if yI (k) cR1 (u1 + 1) f 6 g ≤ ^ uq; if cR1 (uq 1) < yI (k) cR1 (uq + 1) PEI (w) = Prob bI1 (k) = bI1 (k) : (17) ^bI (k) = − ≤ f 6 g 1 8 for 2 q pM 1 > ≤ ≤ − It is then easy to see that the SER is given by < u ; if yI (k) > cR (u 1) pM 1 pM − (7) PE (w) = PE (w) + PE (w) PE (w)PE (w): (18) > R I R I where y(k):= yR(k)+jyI (k) and ^b1(k) is the estimate for b1(k) = − bR1 (k) + jbI1 (k). Fig. 1 depicts the decision thresholds associated The conditional probability density function (PDF) of y(k) given H with the decision ^b1(k) = bl;q. In general, c1 = w p1 is complex- b1(k) = bl;q is a Gaussian mixture defined by valued and the rotating operation (l;q) N jy−y¯ j2 sb i 1 − 2σ2 wH w old p(y bl;q) = e n ; new c old 2 H (19) w = d w (8) j Nsb2πσnw w old i=1 cd X (l;q) (l;q) (l;q) where Nsb = Nb=M is the size of l;q, y¯i = y¯R + jy¯I can be used to make c1 real and positi ve. This rotation is a linear Y i i 2 l;q, and y = yR + jyI . Noting that c1 is real-valued and pos- transformation and does not alter the system's SER. Thus the de- Y itive and taking into account the symmetric distribution of l;q sired user's channel A1 and steering vector s1 are required at the Y (lemma 2), for 2 l pM 1, the conditional error probability receiver in order to apply the decision rules (6) and (7). ≤ ≤ − of ^bR (k) = ul given bR (k) = ul can be shown to be 1 6 1 III. MINIMUM SYMBOL ERROR RATE BEAMFORMING Nsb 2 (l;q) PER;l(w) = Q(gR (w)); (20) Nsb i The classical MMSE solution for the beamformer (4) is given by i=1 X 1 2 − where H 2σn 2 wMMSE = PP + IL p1; (9) 1 1 z 2 − 2 σb Q(u) = e dz; (21) p2π u Z Since the SER is the true performance indicator, it is desired to con- (l;q) y¯ cR (ul 1) (l;q) Ri 1 sider the optimal MSER Beamforming solution.