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: {sqc,lh,hd404}@ecs.soton.ac.uk
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 ω = 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 jωtl(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 | | 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 √M (2) forming is capable of separating signals transmitted on the same B { ≤ ≤ } carrier frequency, and thus provides a practical means of supporting with ul = 2l √M 1 and uq = 2q √M 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 |2 | 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 = jωt (θ ) jω· t· ·(θ ) jωt (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 | | 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 { ≤ ≤ } 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 √M. (11) Y { ∈ Y } ≤ ≤ cR uq Lemma 1: The subsets l,q, 1 l, q √M, satisfy the shifting 1 Y ≤ ≤ properties
l+1,q = l,q + 2c1, 1 l √M 1, (12) Y Y ≤ ≤ −
cR (uq −1) l,q+1 = l,q + j2c1, 1 q √M 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 √M 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 ˆ ˆ ˆ 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) ˆbR (k) = − ≤ For the beamformer with weight vector w, denote 1 for 2 l √M 1 ≤ ≤ − ˆ u√ , if yR(k) > cR1 (u√ 1) PE (w) = Prob b1(k) = b1(k) , (15) M M − { 6 } (6) ˆ PER (w) = Prob bR1 (k) = bR1 (k) , (16) u1, if yI (k) cR1 (u1 + 1) { 6 } ≤ ˆ uq, if cR1 (uq 1) < yI (k) cR1 (uq + 1) PEI (w) = Prob bI1 (k) = bI1 (k) . (17) ˆbI (k) = − ≤ { 6 } 1 for 2 q √M 1 ≤ ≤ − It is then easy to see that the SER is given by u , if yI (k) > cR (u 1) √M 1 √M − (7) PE (w) = PE (w) + PE (w) PE (w)PE (w). (18) R I R I where y(k)= yR(k)+jyI (k) and ˆb1(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 ˆb1(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 |y−y¯ |2 sb i 1 − 2σ2 wH w old p(y bl,q) = e n , new c old 2 H (19) w = d w (8) | 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 ∈ 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 √M 1, the conditional error probability receiver in order to apply the decision rules (6) and (7). ≤ ≤ − of ˆbR (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 ∞ z 2 − 2 σb Q(u) = e dz, (21) √2π 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. Denote the Nb = gR (w) = − − . (22) S i √wH w M number of possible sequences of b(k) as bi, 1 i Nb. σn ≤ ≤ Then x¯(k) can only take values from the finite signal set defined by Further taking into account the shifting property (lemma 1), it is straightforward to show that =4 x¯i = Pbi, 1 i Nb . The set can be partitioned into X { ≤ ≤ } X M subsets, depending on the value of b1(k) Nsb 1 (l,q) PE (w) = γ Q(g (w)), R Ri (23) 4 √ Nsb l,q = x¯i : b1(k) = bl,q , 1 l, q M. (10) i=1 X { ∈ X } ≤ ≤ X 2√M 2 where γ = − . It is seen that PE can be evaluated using ( x(k) + (bR (k) 1)pˆ1) (31) √M R × − 1 − (real part of) any single subset l,q. Similarly, PE can be evaluated Y I and using (imaginary part of) any single subset l,q as 2 Y y (k)−cˆ (k)(b (k)−1) ( I R1 I1 ) Nsb γ − 2ρ2 P˜E (w, k) = e n 1 (l,q) ∇ I √ PEI (w) = γ Q(gI (w)) (24) 2 2πρn Nsb i i=1 X (jx(k) + (bI1 (k) 1)pˆ1) (32) × − with (l,q) gives rise to a stochastic gradient adaptive algorithm, which we refer y¯I cR1 (uq 1) g(l,q)(w) = i − − . (25) to as the LSER algorithm Ii H σn√w w w(k + 1) = w(k) + µ P˜E (w(k), k) , (33) Note that the SER is invariant to a positive scaling of w. −∇ B H The MSER solution wMSER is defined as the one that minimizes cˆ1(k + 1) = w (k + 1)pˆ1, (34) the upper bound of the SER given by cˆ (k + 1) w(k + 1) = 1 w(k + 1), (35) w w w cˆ1(k + 1) PEB ( ) = PER ( ) + PEI ( ), (26) | | where pˆ1 is an estimated p1. The step size µ and the kernel width that is, ρn are the two algorithmic parameters that should be set appropri- wMSER = arg min PE (w). (27) w B ately in order to ensure an adequate performance in terms of con- vergence rate and steady-state SER misadjustment. The upper bound PEB (w) is very tight, i.e. very close to the true
SER PE (w). The gradients of PER (w) and PEI (w) with respect to w can be shown to be respectively IV. SIMULATION STUDY
(l,q) 2 N y¯ −cR (ul−1) Stationary system. The example consisted of four sources and sb Ri 1 γ − 2σ2 wH w a three-element antenna array. Fig. 2 shows the locations of the PE (w) = e n R H ∇ 2Nsb√2πσn√w w desired source and the interfering sources graphically. The simu- i=1 X lated channel conditions were Ai = 1 + j0, 1 i 4. Thus (l,q) ≤ ≤ y¯ cR (ul 1) SIRi = 0 dB for 2 i 4. The modulation scheme was 16- Ri 1 (l,q) ≤ ≤ − − w x¯ + (ul 1)p1 , (28) QAM. Fig. 3 compares the SER performance of the MSER solution × wH w − i − ! with that of the MMSE solution under three different conditions:
(l,q) 2 (a) the minimum spatial separation between the desired user 1 and y¯ −c (u −1) Nsb R q Ii 1 the interfering user 4 θ = 32◦ (b) θ = 30◦, and (c) θ = 28◦. For γ − 2σ2 wH w PE (w) = e n I H this example, the MSER beamformer achieved significantly better ∇ 2Nsb√2πσn√w w i=1 X performance than the MMSE beamformer. (l,q) y¯ cR (uq 1) Ii 1 (l,q) Performance of the adaptive LSER algorithm was investigated − − w + jx¯ + (uq 1)p1 , (29) H i using the system with θ = 30◦ and SNR= 26 dB. Given w(0) = × w w − ! wMMSE and with the step size µ = 0.001 and the kernel width (l,q) where x¯ l,q. With the gradient PE (w) = PE (w) + ρn = σn, Fig. 4 (a) depicts the learning curves of the LSER al- i ∈ X ∇ B ∇ R PE (w), the optimization problem (27) can be solved iteratively gorithm, where DD denotes the decision-directed adaptation with ∇ I ˆ using a gradient optimization algorithm, such as the simplified con- b1(k) substituting for b1(k). Fig. 4 (b) shows the learning curves of jugate gradient algorithm [11]. The rotating operation (8) should be the LSER algorithm under the same conditions except that w(0) = T applied after each iteration, to ensure a real and positive c1. [0.1 + j0.1 0.1 j0.01 0.1 j0.1] . It can be seen from Fig. 4 − − that the LSER beamformer had a reasonable convergence speed. It The PDF p(y) of y(k) can be estimated using the Parzen win- can also be seen that the initial condition w(0) had some influence dow estimate based on a block of training data. This leads to an on convergence rate. estimated SER for the beamformer. Minimizing this estimated SER based on a gradient optimization yields an approximated MSER so- lution. To derive a sample-by-sample adaptive algorithm, consider source 1 (desired) a single-sample “estimate” of p(y) interferer4
|y−y(k)|2 1 2 interferer − 2ρn −θ 3 p˜(y, k) = 2 e (30) interferer2 o 2πρn o 65 −70 ˜ and the corresponding one-sample SER “estimate” PEB (w, k). Using the instantaneous stochastic gradient of P˜E (w, k) = ∇ B P˜E (w, k) + P˜E (w, k) with ∇ R ∇ I λ /2 λ/2 2 y (k)−cˆ (k)(b (k)−1) ( R R1 R1 ) γ 2 ˜ − 2ρn Fig. 2. Locations of the desired source and the interfering sources with respect to the PER (w, k) = e ∇ 2√2πρn three-element linear array with λ/2 element spacing, λ being the wavelength. 1e-2 MMSE 0 DD: k=0 MMSE MSER training MSER -1 1e-3
-2
-3 Bit Error Rate 1e-4
-4
log10(Symbol Error Rate) 1e-5 -5 0 500 1000 1500 2000 sample -6 10 15 20 25 30 35 40 (a) w(0) = wMMSE SNR (dB) 1 (a) θ = 32◦ MMSE DD: k=250 0 MMSE 1e-1 training MSER MSER -1 1e-2
-2 1e-3 Bit Error Rate -3 1e-4 -4
log10(Symbol Error Rate) 1e-5 -5 0 500 1000 1500 2000 sample -6 (b) w(0) = [0.1 + j0.1 0.1 j0.01 0.1 j0.1]T 10 15 20 25 30 35 40 − − SNR (dB) Fig. 4. Learning curves of the stochastic gradient adaptive LSER algorithm for the stationary system averaged over 20 runs, given θ = 30◦ and SNR= 26 dB, where (b) θ = 30◦ DD denotes decision-directed adaptation with ˆb1(k) substituting for b1(k). The step size µ = 0.001 and kernel width ρ = σ . 0 n n MMSE MSER -1 V. CONCLUSIONS
An adaptive MSER beamforming technique has been proposed -2 for multiple antenna aided multiuser wireless communication sys- tems with QAM signalling. It has been demonstrated that the MSER -3 beamforming design can provide significant performance enhance- ment, in terms of the system SER, over the standard MMSE beam- -4 forming design. An adaptive implementation of the MSER beam- forming solution has been realized using the stocastic gradient adap- log10(Symbol Error Rate) -5 tive algorithm known as the LSER technique.
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