This letter shows that the wireless communication system capacity is greatly enhanced by employing nonlinear beamforming and the optimal Bayesian beamformer outperforms the standard linear beamformer significantly in terms of a reduced bit error rate, at a cost of increased complexity. Block-data adaptive implementation of the Bayesian beamformer is realized based on an orthogonal forward selection procedure with Fisher ratio for class separability measure

Chen, S.

Hanzo, L.

Wolfgang, A.

English

Southampton (e-Prints Soton)

478 IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 5, MAY 2004Kernel-Based Nonlinear Beamforming ConstructionUsing Orthogonal Forward Selection With the FisherRatio Class Separability MeasureS. Chen, Senior Member, IEEE, L. Hanzo, Fellow, IEEE, and A. WolfgangAbstract—This letter shows that the wireless communicationsystemcapacityisgreatlyenhancedby employingnonlinearbeam-forming and that the optimal Bayesian beamformer outperformsthestandard linearbeamformer significantly interms ofa reducedbiterrorrate,atacostofincreased complexity.Ablock-dataadap-tive implementation of the Bayesian beamformer is realized basedonanorthogonalforwardselectionprocedurewiththeFisherratiofor class separability measure.Index Terms—Bayesian classification, Fisher ratio for classseparability measure, nonlinear beamforming, orthogonal leastsquares.I. INTRODUCTIONSPATIAL processing with adaptive antenna arrays hasshown real promise for substantial capacity enhancementin mobile communication [1]–[5]. Adaptive beamforming canseparate signals transmitted on the same carrier frequency,provided that they are separated in the spatial domain. Thebeamforming processing is classically done by forming a linearcombination of the signals received from the different elementsof an antenna array. We refer to this classical beamformingas linear beamforming. Recent work [6] has investigated alinear beamforming technique based directly on minimizingthe system bit error rate (BER) and developed an adaptivealgorithm for realizing the linear minimum BER (LMBER)beamforming. The results in [6] have demonstrated that theLMBER beamforming provides considerable performancegains in terms of a reduced BER over the usual linear minimummean-square error (LMMSE) beamforming.Thespatialseparationinanglesofarrivalbetweenthedesiredsignal and the closest interfering signal determines the systemperformance and hence capacity. When this separation is belowa certain threshold, linear beamforming ultimately fails becausethe system becomes linearly inseparable, a situation that issimilar to the single-user channel equalization [7], [8]. For thesake of notational simplicity and for highlighting the basicconcepts, we assume that the modulation scheme is binaryphase shift keying (BPSK), the channel is nondispersive withadditive white Gaussian noise, and narrowband beamformingManuscript received May 1, 2003; revised September 9, 2003. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Prof. Jonathon A. Chambers.The authors are with the School of Electronics and Computer Sci-ence, University of Southampton, Southampton SO17 1BJ, U.K. (e-mail:sqc@ecs.soton.ac.uk).Digital Object Identifier 10.1109/LSP.2004.826509is considered. We derive the optimal solution for nonlinearbeamforming, which we refer to as the Bayesian beamformingsolution. A block-data kernel-based adaptive beamformer isproposed to realize the optimal Bayesian beamformer solutionusing an orthogonal forward selection (OFS) procedure withthe Fisher ratio for class separability measure [9]. The proposednonlinear beamformer construction algorithm is compared withthe state-of-art sparse kernel modeling based on the relevancevector machine (RVM) for classification [8], [10].II. SYSTEM MODELIt is assumed that the system consists of users (sources),and each user transmits a BPSK signal on the same carrier fre-quency . The baseband signal of user is given by(1)where the complex-valued is the channel coefficient foruser multiplying by the transmitted signal amplitude of user(therefore denotes user received signal power) andis the th bit of user . Without the loss of generality, source 1is assumed to be the desired user, and the rest of the sourcesare interfering users. The linear antenna array is considered,which consists of uniformly spaced elements, and signals atthe -element antenna array are given by(2)for , where is the relative time delay at elementfor source , is thedirectionof arrivalforsource , andis a complex-valued white Gaussian noise with zero mean and. The desired SNR is defined as SNR, and the desired signal-to-interferer ratio is givenby SIR for . In vector form, thearray input can be written as(3)where with denoting theidentity matrix, the system matrix is defined by(4)1070-9908/04$20.00 © 2004 IEEECHEN et al.: KERNEL-BASED NONLINEAR BEAMFORMING CONSTRUCTION 479with the steering vector for source being, andthe bit vector .Traditionally, a linear beamformer is used, whose output isgiven by(5)where is the complex-valued beamformer weight vector. Thedecision for the transmitted bit is made according to(6)where denotes the real part of . The clas-sical LMMSE beamforming solution is given by, with being thefirst column of . Re-cently, we have developed the LMBER beamforming solution[6]. For the linear beamformer to work adequately, the systemmust be linearlyseparable inthe noise-freecase. When themin-imum spatial separation in angles of arrival between the de-sired user and interfering users is below a certain threshold, thesystem inevitably becomes linearly inseparable. In such a sit-uation, the linear bermformer exhibits a high irreducible BERfloor, and a nonlinear processing has to be adopted.III. BAYESIAN BEAMFORMING SOLUTIONGiventheobservationvector ,theoptimalsolutiontothebeamforming problem is the maximum a posteriori probabilitysolution, which we derive as follows. Denote thepossible sequences of as . Further, denotethe first element of , corresponding to the desired user, as. Obviously, only takes values from the signal state setdefined as . The state set canbe divided into two subsets conditioned on(7)where .Theposteriorprobabilitiesordecisionvari-ables for given are(8)where are a priori probabilities of and. The optimal decision is given byotherwise(9)Let us redefine a single decision variable as(10)where sgn . Then, the optimal deci-sion (9) is equivalent to(11)IV. BLOCK-DATA KERNEL-BASED NONLINEARBEAMFORMER CONSTRUCTIONGiven a block of training data , considerthe nonlinear beamformer of the form(12)where are the real-valued weights, andare chosen kernel basis functions. In our application, canbe chosen as the Gaussian kernel function of the form(13)where the kernel variance is related to the noise variance. The RVM method [8], [10] can be applied to construct asparse beamformer of terms from (12). A drawback of theRVM method is its high computational complexity. The algo-rithm contains two loops, with the inner loop for updating thekernel weights and the outer loop for the associated hyperpa-rameters. Both loops involve expensive nonlinear optimization.Furthermore,theRVMmethodstartswiththefullmodelsetandremoves those kernel terms that have large values in their asso-ciated hyperparameters. Because the Hessian matrix associatedwith the full model set is typically ill-conditioned and may evenbe noninvertible, the RVM method is inherently ill-conditioned,and its iterative procedure generally converges with slow rateand may suffer from numerical instability.An alternative way of constructing a sparse kernel modelfrom the full model (12) is the OFS procedure based on Fisherratio class separability measure [9], which is computationallyattractive and numerically robust. Define the modeling residualas . Then, the kernelmodel (12) over the training dataset can be collected together as(14)where the target vector, the regression matrixwith(15)for , the kernel weight vector ,and the residual vector . Let an orthog-onal decomposition of the regression matrix be ,where.... . .. . .... ...(16)480 IEEE SIGNAL PROCESSING LETTERS, VOL. 11, NO. 5, MAY 2004. . .. . .. . .. . .(17)with orthogonal columns that satisfy if . Thekernel model (14) can alternatively be expressed as(18)where satisfiesthetriangularsystem .A sparse -term model can be selected by incrementallymaximizing a class separability measure in an OFS procedure[9]. Define the two class sets ,and let the numbers of points in be , respectively, with. The means and variances of training samplesbelonging to classes in the direction of basis are given by(19)(20)respectively,where for and for .The Fisher ratio, defined as the ratio of the interclass differenceand the intraclass spread, in the direction of is given by [11](21)BaseonthisFisherratio,significantkerneltermscanbeselectedin an OFS procedure. At the th stage, a term is chosen as the thterm in the selected model if it produces the largest amongthe candidate terms . The procedure is terminatedwith a sparse -term model when(22)where the threshold determines the sparsity of the selectedmodel. We have found out empirically that the appropriatevalues for is in the range of 0.005–0.01. The least squaresolution for the corresponding sparse model weight vectoris readily available given the least square solution of.The modified Gram–Schmidt orthogonalization procedure[12] is first summarized. Denote .F o r(23)The last stage is simply . The elements of arecomputed by transforming in a similar way(24)Next, defineand give a very small positive number . With the notation, the th stage of theselection procedure is given as follows.Step 1) For :Test: Conditioning number check. If, the th candidate is notconsidered.Compute the following:Let the index set be:and passes .Step 2) Find: .Then the th column of is interchangedwith the th column of , and the th columnof is interchanged with the th column of up tothe th row. This selects the th candidate asthe th kernel term in the subset model.Step 3) Perform the orthogonalization as indicated in (23) toderive the th row of and to transform into. Calculate and update into in theway shown in (24).V. SIMULATION EXAMPLEThe example consisted of four signal sources and a two-el-ement antenna array. Fig. 1 shows the locations of the desiredsource and three interfering sources graphically. The simulatedchannel conditions were , and all thefour users had equal signal power. The minimum spatial separa-tion was the difference in angles of arrival between the desireduser 1 and the interferer 2, which was . Fig. 2 comparesthe BERs of the LMMSE, LMBER, and Bayesian beamformersforthetwocasesof and ,respectively.Itisseenfrom Fig. 2(a) that for , the LMMSE beamformer couldnot achieve linear separability and exhibited a high BER floor,but the LMBER beamformer achieved linear separability andCHEN et al.: KERNEL-BASED NONLINEAR BEAMFORMING CONSTRUCTION 481Fig. 1. Locations of the desired and interfering sources with respect tothe two-element linear antenna array having ￿=2 spacing, where ￿ is thewavelength.(a) (b)Fig. 2. Comparison of the bit error rates of three theoretical beamformers.(a) ￿ =3 0. (b) ￿ =1 0.(a) (b)Fig. 3. Performance comparison of the Bayesian beamformer with the RBFbeamformers constructed by the RVM algorithm and the OFS with the Fisherratio, respectively. (a) ￿ =3 0. (b) ￿ =1 0.had a much better BER performance than the LMMSE beam-former. The Bayesian beamformer provided the optimal perfor-mance and had a 4-dB improvement in SNR at the BER level of, compared with the LMBER beamformer. When the spa-tial separation was reduced to , the system became in-herently linearly inseparable, and while the linear beamformerfailedinthissituation,theBayesianbeamformerstillperformedadequately. This is demonstrated clearly in Fig. 2(b).The OFS algorithm with the Fisher ratio and the RVMalgorithm were used to construct a RBF beamformer. Thenumber of training data for each given SNR was .The Gaussian kernel variance was determined empirically,and the appropriate values for were found to be in the rangeof to , depending on the SNR. The numbers of RBFcenters or kernel terms identified by the two algorithms overthegiven SNR values were similar, ranging from towith the typical value of . The BERs of the RVM andOFS beamformers are compared in Fig. 3. It can be seen thatboth kernel-based beamformers have similarly good perfor-mance with similar model sparsity. The OFS algorithm basedon the Fisher ratio, however, has considerable computationaland numerical advantages during the construction process.VI. CONCLUSIONThe optimal nonlinear beamforming assisted receiver hasbeen derived, and it has been shown that this optimal Bayesianbeamformer outperforms the linear beamformer significantly interms of a reduced bit error rate. This demonstrates the poten-tial of system capacity enhancement by employing nonlinearbeamforming. Block-data kernel-based adaptive implemen-tation of the optimal Bayesian beamformer is investigatedusing the OFS algorithm based on the Fisher ratio for classseparability measure. Empirical results have demonstrated thatthis construction algorithm has excellent performance similarto that of the RVM algorithm, but it is computationally muchsimpler and numerically much more robust.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.Commun., vol. 42, pp. 1740–1751, Feb./Mar./Apr. 1994.[2] J. Litva and T. K. Y. Lo, Digital Beamforming in Wireless Communica-tions. Norwood, MA: Artech House, 1996.[3] L. C. Godara, “Applications of antenna arrays to mobile communica-tions, part I: Performance improvement, feasibility, and system consid-erations,” Proc. IEEE, vol. 85, pp. 1031–1060, July 1997.[4] J. H. 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A clustering technique for digital communications channel equalization using radial basis function networks,”

Adaptive minimum bit error rate beamforming assisted receiver for wireless communications,” in

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

Orthogonal least squares methods and their application to nonlinear system identification,”

Pattern Classification and Scene Analysis.

RBF neural network center selection based on Fisher ratio class separability measure,”

Smart antennas for wireless systems,”

Sparse Bayesian learning and the relevance vector machine,”

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

The relevance vector machine technique for channel equalization application,”

Kernel-based nonlinear beamforming construction using orthogonal forward selection with Fisher ratio class separability measure

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Kernel-based nonlinear beamforming construction using orthogonal forward selection with Fisher ratio class separability measure

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