An adaptive beamforming assisted receiver is proposed for multiple antenna aided multiuser systems that employ bandwidth efficient quadrature amplitude modulation (QAM). A novel minimum symbol error rate (MSER) design is proposed for the beamforming assisted receiver, where the system’s symbol error rate is directly optimized. Hence the MSER approach provides a significant symbol error ratio performance enhancement over the classic minimum mean square error design. A sample-by-sample adaptive algorithm, referred to as the least symbol error rate (LBER) technique, is derived for allowing the adaptive implementation of the system to arrive from its initial beamforming weight solution to MSER beamforming solution

Chen, S.

Du, H.Q

Hanzo, L.

English

Southampton (e-Prints Soton)

Adaptive Minimum Symbol Error Rate Beamforming Assisted Receiver forQuadrature Amplitude Modulated SystemsS. Chen, H.-Q. Du and L. HanzoSchool of ECS, University of Southampton, SO17 1BJ, U.K.E-mails: {sqc,lh,hd404}@ecs.soton.ac.ukAbstract An adaptive beamforming assisted receiver isproposed for multiple antenna aided multiuser systemsthat employ bandwidth efﬁcient quadrature amplitudemodulation (QAM). A novel minimum symbol error rate(MSER) design is proposed for the beamforming assistedreceiver, where the system’s symbol error rate is directlyoptimized. Hence the MSER approach provides a sig-niﬁcant symbol error ratio performance enhancementover the classic minimum mean square error design. Asample-by-sample adaptive algorithm, referred to as theleast symbol error rate (LBER) technique, is derived forallowing the adaptive implementation of the system toarrive from its initial beamforming weight solution toMSER beamforming solution.I. INTRODUCTIONThe ever-increasing demand for mobile communicationcapacity has motivated the development of adaptive antennaarray assisted spatial processing techniques [1]–[10] in or-der to further improve the achievable spectral efﬁciency ofwireless systems. A particular technique that has shown realpromise in achieving substantial capacity enhancements isthe employment of adaptive beamforming with the aid of an-tenna arrays, which create angularly selective beam maximatowards the desired user and a null towards a limited num-ber of dominant interferers. By appropriately combining thesignals received by the different elements of an antenna ar-ray to form a single output, adaptive beamforming becomescapable of separating signals transmitted on the same carrierfrequency, provided that they arrive from sufﬁciently differ-ent angular directions. Thus bemaforming becomes capableof supporting multiple users in an ’angular division multi-ple access’ scenario. Classically, the beamforming processis carried out by adjusting the beamforming array weightsupon minimizing the mean square error (MSE) between thedesired output of the array, which is typically the most likelylegitimate transmitted symbol and the actual array output. Inother words, we set the partial derivative of the array outputto zero with respect to the array weights.However, in most communications systems it is the bit er-The ﬁnancial support of the EPSRC, Swindon UK and that of the EUunder the auspices of the PHOENIX as well as NEWCOM projects is grate-fully acknowledged.ror ratio (BER) or symbol error ratio (SER) that really mat-ters. Hence adaptive beamforming based on directly mini-mizing the system’s BER has been proposed for both binaryphase shift keying and quadrature phase shift keying modu-lation schemes in [11],[12].In recent years the family of high-throughput quadratureamplitude modulation (QAM) schemes [13] has become pre-dominant in numerous wireless local area network (WLAN)standards, such as the IEEE 802.11 standards. Adaptive min-imum SER (MSER) equalization has been investigated in thecontext of a single-antenna single-user system, when usingeither a pulse-amplitude modulation scheme [14] or a QAMscheme [15]. Against this backcloth, in this paper, we de-rive the MSER beamforming design for a multiple antennaassisted multiuser system employing QAM signalling. Weshow that the MSER design is capable of providing signif-icant SER performance gains over the traditional minimumMSE (MMSE) design. An attractive adaptive implementationof the MSER beamforming solution is also proposed, whichstep-by-step adjusts the array weights, commencing from anadequate initial solution using the classic stochastic gradi-ent algorithm (SGA), which we refer to as the least symbolerror rate (LSER) technique. Our proposed solution is sub-stantially different from the method proposed in [15], sincethe adaptive LSER algorithm invoked has its roots in the so-called Parzen window based density estimation [16]-[18]. Inthis sense, the proposed adaptive MSER technique is an ex-tension of the method proposed in [14] for an interference-limitedmultiusersystemtoamorebandwidth-efﬁcient QAMscheme.II. SYSTEM MODELThe system supports S users, and each user transmits anM-QAM signal on the same carrier frequency ω =2 πf.The receiver is equipped with a linear antenna array consist-ing of L uniformly spaced elements. We assume that thechannel does not induce intersymbol interference (ISI). Thenthe symbol-rate received signal samples can be expressed asxl(k)=S  i=1Aibi(k)ejωtl(θi)+nl(k)=¯ xl(k)+nl(k), (1)for 1 ≤ l ≤ L, where tl(θi) is the relative time delay atelement l for source i with θi being the direction of arrival0-7803-9392-9/06/$20.00 (c) 2006 IEEEfor source i, nl(k) is the complex-valued Additive WhiteGaussian Noise (AWGN) with E[|nl(k)|2]=2 σ2n, Ai is thecomplex-valued non-dispersive channel coefﬁcient of user i,while bi(k) is the kth M-QAM symbol of user iB△= {bl,q = ul + juq, 1 ≤ l,q ≤√M} (2)with ul =2 l −√M − 1 and uq =2 q −√M − 1.Source 1 is the desired user and the rest of the sources areinterfering users.The desired-user’s signal to noise ratio isSNR= |A1|2σ2b/2σ2n and the desired signal to interferer i ra-tio is SIRi = A21/A2i,f o r2 ≤ i ≤ S, where σ2b denotesthe M-QAM symbol’s energy. The received signal vectorx(k)=[ x1(k) x2(k)   xL(k)]T is given byx(k)=Pb(k)+n(k)=¯ x(k)+n(k), (3)where n(k)=[ n1(k) n2(k)   nL(k)]T, the system ma-trix P =[ A1s1 A2s2    ASsS] with the steering vector ofsource i given by si =[ ejωt1(θi) ejωt2(θi)    ejωtL(θi)]Tand that of the transmitted QAM symbol vector by b(k)=[ b1(k) b2(k)   bS(k)]T.A linear beamformer’s soft output is given byy(k)=wHx(k)=wH(¯ x(k)+n(k)) = ¯ y(k)+e(k) (4)where w =[ w1 w2    wL]T is the beamformer’s weightvector and e(k) is Gaussian distributed with zero mean andE[|e(k)|2]=2 σ2nwHw. We deﬁne the combined impulseresponse of the beamformer and the channel as wHP =wH[p1 p2    pS]=[ c1 c2    cS]. The beamformer’s out-put can alternatively be expressed asy(k)=c1b1(k)+S  k=2cibi(k)+e(k). (5)Provided that c1 = cR1 + jcI1 satisﬁes cR1 > 0 and cI1 =0, the symbol decision ˆ b1(k)=ˆ bR1(k)+jˆ bI1(k) can bedecoupled intoˆ bR1(k)=    u1, if yR(k) ≤ cR1(u1 +1 )ul, if cR1(ul − 1) <y R(k) ≤ cR1(ul +1 )for 2 ≤ l ≤√M − 1u√M, if yR(k) >c R1(u√M − 1)(6)ˆ bI1(k)=    u1, if yI(k) ≤ cR1(u1 +1 )uq, if cR1(uq − 1) <y I(k) ≤ cR1(uq +1 )for 2 ≤ q ≤√M − 1u√M, if yI(k) >c R1(u√M − 1)(7)where y(k)=yR(k)+jyI(k) and ˆ b1(k) is the estimate ofb1(k)=bR1(k)+jbI1(k). Fig. 1 depicts the decision thresh-olds associated with the decision ˆ b1(k)=bl,q. In general,c1 = wHp1 is complex-valued and the rotating operationwnew =coldd    coldd   wold (8)c  ul R c  uc  (u +1) R l Rc  (u −1)RRqqlRe[  ]Im[  ] yR qc  (u −1)yc  (u +1) 111111Fig. 1. Decision thresholds associated with point c1bl,q assuming cR1 > 0andcI1 =0 , illustratingthesymmetryofthedistributionofYl,q aroundc1bl,q.can be used to render c1 real-valued and positive. This rota-tion is a linear transformation and does not alter the system’sSER. Thus the desired user’s channel A1 and steering vectors1 are required at the receiver in order to apply the decisionrules of (6) and (7).III. MINIMUM SYMBOL ERROR RATE BEAMFORMINGThe classic MMSE solution for the beamformer of (4) isgiven bywMMSE = PPH +2σ2nσ2bIL −1p1, (9)Since the SER is the ultimate performance indicator, it is de-sirable to ﬁnd the optimal MSER beamforming weight so-lution. We denote the Nb = MS number of possible se-quences of b(k) as bi, 1 ≤ i ≤ Nb. Then ¯ x(k) canonly assume values from the ﬁnite signal set deﬁned byX△= {¯ xi = Pbi, 1 ≤ i ≤ Nb}.T h e s e t X can be par-titioned into M subsets, depending on the value of b1(k)Xl,q△= {¯ xi ∈X: b1(k)=bl,q}, 1 ≤ l,q ≤√M. (10)The noise-free component of the beamformer’s output ¯ y(k)only assumes values from the scalar set Y△= {¯ yi =wH¯ xi, 1 ≤ i ≤ Nb}, and Y can be divided into M sub-sets conditioned on the value of b1(k)Yl,q△= {¯ yi ∈Y: b1(k)=bl,q}, 1 ≤ l,q ≤√M. (11)Lemma 1: The subsets Yl,q, 1 ≤ l,q ≤√M, satisfy theshifting propertiesYl+1,q = Yl,q +2 c1, 1 ≤ l ≤√M − 1, (12)Yl,q+1 = Yl,q + j2c1, 1 ≤ q ≤√M − 1, (13)Yl+1,q+1 = Yl,q +( 2+j2)c1, 1 ≤ l,q ≤√M − 1. (14)The proof of Lemma 1 is straightforward.Lemma 2: The points of Yl,q are distributed symmetri-cally around the symbol point c1bl,q.Lemma 2 is a direct consequence of the symmetry of thesymbol constellation (2). This symmetric property is alsoillustrated in Fig. 1. Note that the distribution of Yl,q is sym-metric with respect to the two vertical decision thresholdscR1(ul ± 1) and with respect to the two horizontal decisionthreshold cR1(uq ± 1).For the beamformer having a weight vector w we intro-duce the notationPE(w)=Prob{ˆ b1(k)  = b1(k)}, (15)PER(w)=Prob{ˆ bR1(k)  = bR1(k)}, (16)PEI(w)=Prob{ˆ bI1(k)  = bI1(k)}. (17)It is then readily seen that the SER is given byPE(w)=PER(w)+PEI(w) − PER(w)PEI(w). (18)The conditional probability density function (PDF) of y(k)given b1(k)=bl,q is a Gaussian mixture deﬁned byp(y|bl,q)=1Nsb2πσ2nwHwNsb  i=1e−|y−¯ y(l,q)i|22σ2nwHw , (19)where Nsb = Nb/M is the size of Yl,q, ¯ y(l,q)i =¯ y(l,q)Ri +j¯ y(l,q)Ii ∈Y l,q, and y = yR + jyI. Noting that c1 is real-valued and positive, as well as taking into account the sym-mety of the distribution of Yl,q (lemma 2), for 2 ≤ l ≤ √M − 1, the conditional error probability of ˆ bR1(k)  = ulgiven bR1(k)=ul can be shown to bePER,l(w)=2NsbNsb  i=1Q(g(l,q)Ri (w)), (20)whereQ(u)=1√2π  ∞ue− z22 dz, (21)g(l,q)Ri (w)=¯ y(l,q)Ri − cR1 (ul − 1)σn√wHw. (22)Furthermore, taking into account the shifting property(lemma 1), it is straightforward to show that we havePER(w)=γ1NsbNsb  i=1Q(g(l,q)Ri (w)), (23)where γ = 2√M−2 √M . It is seen that PER can be evaluatedusing (the real part of) any single subset Yl,q. Similarly, PEIcan be evaluated using (the imaginary part of) any single sub-set Yl,q asPEI(w)=γ1NsbNsb  i=1Q(g(l,q)Ii (w)) (24)withg(l,q)Ii (w)=¯ y(l,q)Ii − cR1 (uq − 1)σn√wHw. (25)Note that the SER is invariant to a positive scaling of w.The MSER solution wMSER is deﬁned as the one that min-imizes the upper bound of the SER given byPEB(w)=PER(w)+PEI(w), (26)that is,wMSER = argminw PEB(w). (27)The upper bound PEB(w) is very tight, i.e. very close to thetrue SER PE(w). The gradients of PER(w) and PEI(w)with respect to w can be shown to be∇PER(w)=γ2Nsb√2πσn√wHwNsb  i=1e− ¯ y(l,q)Ri−cR1(ul−1) 22σ2nwHw× ¯ y(l,q)Ri − cR1(ul − 1)wHww − ¯ x(l,q)i +( ul − 1)p1 , (28)∇PEI(w)=γ2Nsb√2πσn√wHwNsb  i=1e− ¯ y(l,q)Ii−cR1(uq−1) 22σ2nwHw× ¯ y(l,q)Ii − cR1(uq − 1)wHww + j¯ x(l,q)i +( uq − 1)p1 ,(29)where ¯ x(l,q)i ∈X l,q. With the gradient ∇PEB(w)=∇PER(w)+∇PEI(w), the optimization problem (27) canbe solved iteratively using a gradient optimization algorithm,such as the simpliﬁed conjugate gradient algorithm [11]. Therotating operation (8) should be applied after each iteration,to ensure that we have a real and positive c1 value.The PDF p(y) of y(k) can be estimated using the Parzenwindowestimatebasedonablockoftrainingdata. Thisleadsto an estimated SER for the beamformer. Minimizing this es-timated SER based on a gradient optimization yields an ap-proximated MSER solution. To derive a sample-by-sampleadaptive algorithm, consider a single-sample “estimate” ofp(y)˜ p(y,k)=12πρ2ne−|y−y(k)|22ρ2n (30)andthecorrespondingone-sampleSER“estimate” ˜ PEB(w,k).Usingtheinstantaneousstochasticgradientof∇ ˜ PEB(w,k)=∇ ˜ PER(w,k)+∇ ˜ PEI(w,k) with∇ ˜ PER(w,k)=γ2√2πρne−(yR(k)−ˆ cR1(k)(bR1(k)−1))22ρ2n×(−x(k)+( bR1(k) − 1)ˆ p1) (31)and∇ ˜ PEI(w,k)=γ2√2πρne−(yI(k)−ˆ cR1(k)(bI1(k)−1))22ρ2n×(jx(k)+( bI1(k) − 1)ˆ p1) (32)gives rise to the following stochastic gradient adaptive algo-rithm, which we refer to as the LSER algorithmw(k +1 )=w(k)+µ −∇ ˜ PEB(w(k),k) , (33)ˆ c1(k +1 )=wH(k +1 )ˆ p1, (34)w(k +1 )=ˆ c1(k +1 )|ˆ c1(k +1 ) |w(k +1 ) , (35)where ˆ p1 is an estimate of p1. The step size µ and the kernelwidth ρn are the two algorithmic parameters that should beset appropriately in order to attain an adequate performancein terms of both the convergence rate and steady-state SERmisadjustment.IV. SIMULATION STUDYStationary system. Our prototype system supported fourusers with the aid of a three-element antenna array. Fig. 2shows the locations of both the desired source and the inter-fering sources. The channel coefﬁcients were Ai =1+j0,1 ≤ i ≤ 4. Thus we had SIRi =0dB for 2 ≤ i ≤ 4.The modulation scheme was 16-QAM. Fig. 3 compares theSER performance of the MSER solution to that of the MMSEsolution under three different conditions: (a) the minimumspatial separation between the desired user 1 and the inter-fering user 4 was θ =3 2 ◦ (b) θ =3 0 ◦, and (c) θ =2 8 ◦.F o rλ/2 λ/21interfererinterferer4source 2−θ65(desired)interferer3−70ooFig. 2. Locations of the desired source and the interfering sources withrespect to the three-element linear array with λ/2 element spacing, λbeing the wavelength.-6-5-4-3-2-1 0 10  15  20  25  30  35  40log10(Symbol Error Rate)SNR (dB)MMSEMSER(a) θ =3 2 ◦-6-5-4-3-2-1 0 10  15  20  25  30  35  40log10(Symbol Error Rate)SNR (dB)MMSEMSER(b) θ =3 0 ◦-6-5-4-3-2-1 0 10  15  20  25  30  35  40log10(Symbol Error Rate)SNR (dB)MMSEMSER(c) θ =2 8 ◦Fig. 3. SER performance over non-fading channels. 1e-5 1e-4 1e-3 1e-2 0  500  1000  1500  2000Bit Error RatesampleMMSEDD: k=0trainingMSER(a) w(0) = wMMSE 1e-5 1e-4 1e-3 1e-2 1e-1 1 0  500  1000  1500  2000Bit Error RatesampleMMSEDD: k=250trainingMSER(b) w(0) = [0.1+j0.10 .1 − j0.01 0.1 − j0.1]TFig. 4. Learning curves of the stochastic gradient adaptive LSER algo-rithm for the stationary system averaged over 20 runs, given θ =3 0 ◦and SNR=2 6dB, where DD denotes decision-directed adaptation withˆ b1(k) substituting for b1(k). The step size µ =0 .001 and kernel widthρn = σn.thisexample, theMSERbeamformerachievedasigniﬁcantlybetter performance than the MMSE beamformer.The erformance of the adaptive LSER algorithm was in-vestigated using the system associated with θ =3 0 ◦ andSNR=2 6dB. Given w(0) = wMMSE, the step size ofµ =0 .001 and the kernel width ρn = σn, Fig. 4 (a) de-picts the learning curves of the LSER algorithm, where DDdenotes the decision-directed adaptation with ˆ b1(k) substi-tuting for b1(k). Fig. 4 (b) portrays the associated learningcurves of the LSER algorithm under the same conditions, ex-ceptforw(0) = [0.1+j0.10.1−j0.010.1−j0.1]T. Itcanbeseen from Fig. 4 that the LSER beamformer had a reasonableconvergence speed. It can also be seen that the initial condi-tion w(0) had some inﬂuence on the achievable convergencerate.V. CONCLUSIONSAn adaptive MSER beamforming technique has been pro-posed for multiple antenna aided multiuser wireless com-munication systems using 16QAM signalling. It has beendemonstrated that the MSER beamforming design is capa-ble of providing signiﬁcant SER performance improvementsover the standard MMSE beamforming design. An adap-tive implementation of the MSER beamforming solution hasbeen proposed, namely the stocastic gradient adaptive algo-rithm, which was referred to here as the LSER technique.Our future research will consider similar schemes employ-ing error correction codecs and iterative receivers.REFERENCES[1] J.H. Winters, J. Salz and R.D. Gitlin, “The impact of antenna diver-sity on the capacity of wireless communication systems,” IEEE Trans.Communications, Vol.42, No.2, pp.1740–1751, February/March/April1994.[2] J. Litva and T. K.Y. Lo, Digital Beamforming in Wireless Communica-tions. London: Artech House, 1996.[3] L. C. Godara, “Applications of antenna arrays to mobile communica-tions, Part I: Performance improvement, feasibility, and system con-siderations,” Proc. IEEE, Vol.85, No.7, pp.1031–1060, 1997.[4] A.J. Paulraj and C.B. Papadias, “Space-time processing for wirelesscommunications,” IEEE Signal Processing Magazine, Vol.14, No.6,pp.49–83, 1997.[5] R.Kohno, “Spatialandtemporalcommunicationtheoryusingadaptiveantenna array,” IEEE Personal Communications, Vol.5, No.1, pp.28–35, 1998.[6] P. Petrus, R.B. Ertel and J.H. Reed, “Capacity enhancement usingadaptive arrays in an AMPS system,” IEEE Trans. Vehicular Technol-ogy, Vol.47, No.3, pp.717–727, 1998.[7] J.H. Winters, “Smart antennas for wireless systems,” IEEE PersonalCommunications, Vol.5, No.1, pp.23–27, 1998.[8] P. Vandenameele, L. van Der Perre and M. Engels, Space DivisionMultiple Access for Wireless Local Area Networks. Boston: KluwerAcademic Publishers, 2001.[9] J.S. Blogh and L. Hanzo, Third Generation Systems and Intelli-gent Wireless Networking – Smart Antenna and Adaptive Modulation.Chichester: John Wiley, 2002.[10] A. Paulraj, R. Nabar and D. Gore, Introduction to Space-Time WirelessCommunications. Cambridge: Cambridge University Press, 2003.[11] S. Chen, N.N. Ahmad and L. Hanzo, “Adaptive minimum bit error ratebeamforming,” IEEE Trans. Wireless Communications, Vol.4, No.2,pp.341–348, 2005.[12] S. Chen, L. Hanzo, N.N. Ahmad and A. Wolfgang, “Adaptive min-imum bit error rate beamforming assisted QPSK receiver,” in Proc.ICC 2004, 2004, Vol.6, pp.3389–3393.[13] L. Hanzo, S.X. Ng, T. Keller and W. Webb, Quadrature AmplitudeModulation: From Basics to Adaptive Trellis-Coded, Turbo-Equalisedand Space-Time Coded OFDM, CDMA and MC-CDMA Systems. JohnWiley and IEEE Press: Chichester, UK, 2004.[14] S. Chen, L. Hanzo and B. Mulgrew, “Adaptive minimum symbol-error-rate decision feedback equalization for multi-level pulse-amplitude modulation,” IEEE Trans. Signal Processing, Vol.52, No.7,pp.2092–2101, 2004.[15] C.-C. Yeh and J.R. Barry, “Adaptive minimum symbol-error rateequalization for quadrature-amplitude modulation,” IEEE Trans. Sig-nal Processing, Vol.51, No.12, pp.3263–3269, 2003.[16] E. Parzen, “On estimation of a probability density function and mode,”The Annals of Mathematical Statistics, Vol.33, pp.1066–1076, 1962.[17] B.W. Silverman, Density Estimation. London: Chapman Hall, 1996.[18] A.W. Bowman and A. Azzalini, Applied Smoothing Techniques forData Analysis. Oxford: Oxford University Press, 1997.

Adaptive minimum bit error rate beamforming assisted QPSK receiver,” in

Adaptive minimum bit error rate beamforming,”

Adaptive minimum symbol-error rate equalization for quadrature-amplitude modulation,”

Adaptive minimum symbolerror-rate decision feedback equalization for multi-level pulseamplitude modulation,”

Applications of antenna arrays to mobile communications, Part I: Performance improvement, feasibility, and system considerations,”

Applied Smoothing Techniques for Data Analysis.

Capacity enhancement using adaptive arrays in an

Density Estimation.

Introduction to Space-Time Wireless Communications. Cambridge:

On estimation of a probability density function and mode,”

Quadrature Amplitude Modulation: From Basics to Adaptive Trellis-Coded, Turbo-Equalised and Space-Time Coded OFDM, CDMA and MC-CDMA Systems.

Smart antennas for wireless systems,”

Space Division Multiple Access for Wireless Local Area Networks.

Space-time processing for wireless communications,”

Spatialandtemporalcommunicationtheoryusingadaptive antenna array,”

The impact of antenna diversity on the capacity of wireless communication systems,”

Third Generation Systems and Intelligent Wireless Networking – Smart Antenna and Adaptive Modulation.

Adaptive minimum symbol error rate beamforming assisted receiver for quadrature amplitude modulation systems

Adaptive minimum symbol error rate beamforming assisted receiver for quadrature amplitude modulation systems

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