Broadband BeamformingProject ReportMultidimensional Digital Signal ProcessingKalpana SeshadrinathanLaboratory for Image and Video EngineeringThe University of Texas at AustinAbstractBroadband wireless channels, where high data rates are transmitted, are extremely dispersive in nature.A fundamental challenge in the design of equalizers for the broadband case lies in reducing complexity.Broadband finite impulse response (FIR) beamformers employ a space-time antenna array which reducesthe multipath delay spread to narrowband levels. The beamformers should additionally preserve thewhiteness of the channel noise at the beamformer output to allow for the application of trellis basedequalizers. The power complementarity property has been used to address this issue in the literature.Techniques to design FIR filters that preserve the whiteness of the channel noise when the receivedsignal is oversampled are studied in this project.1 IntroductionOver the last decade, we have witnessed an explosive growth in cellular communications and the In-ternet. These trends indicate a strong potential in the future for mobile broadband wireless datacommunications. A key obstacle to reliable wireless communications is frequency-selective multipathchannels. For narrowband channels, trellis based decoding represents an effective method to combatintersymbol interference (ISI) in frequency-selective multipath channels [1]. However, in the broadbandcase, multipath dispersion is quite severe and results in the channel memory increasing linearly with thedata rate. Since the size of the trellis grows exponentially with the channel memory, the direct applica-tion of trellis based decoding algorithms becomes unfeasible due to their high complexity. Techniquesto overcome this effect include channel shortening equalizers and other equalization techniques that arenot trellis based. Co-channel interference (CCI) from adjacent users is a serious issue in cellular systemsand interference cancellation is another important factor in equalizer design.12 BackgroundSeveral approaches have been adopted to reduce equalizer complexity without sacrificing too much inperformance in terms of ISI mitigation. Multiple Input Multiple Output (MIMO) systems use space-time antenna arrays at both the transmit and receive ends to enhance diversity. A known drawback ofsymbol-spaced equalizers is that they are highly sensitive to the phase of the sampling at the receiver[1, 2]. Fractionally spaced equalizers, where the equalizer taps are placed closer together in time thanthe symbol interval are used to overcome this effect. Fractionally-spaced equalizers have been shownto be effective in equalizing MIMO channels [3] and can be designed using the theory of biorthogonalpartners [4]. Design of equalizers for MIMO channels is discussed in detail in [3, 5].Adaptive frequency-domain equalizers for broadband wireless communications have been proposed in[6]. Frequency-domain equalizers exhibit linear complexity growth with increase in channel memory andare well-suited for broadband channels. A feasible alternative is hence to use an adaptive equalizer thatoperates in the spatial-frequency domain and uses either least mean square (LMS) or recursive leastsquares (RLS) adaptive processing [6]. Reduced-complexity techniques for broadband wireless channelshave also been investigated [7]. Methods to allow the receiver to find burst and symbol timing and amodified decision-feedback equalizer structure are proposed.Another approach that has been considered is to employ a broadband beamformer followed by a finiteimpulse resp onse (FIR) filterbank as the front end of a communications receiver followed by a maximuma posteriori (MAP) sequence detector as part of the back end [8]. Trellis based decoders are based onthe maximum likelihood sequence estimation (MLSE) criterion and are optimum from a probability oferror viewpoint [1]. However, the application of MLSE algorithms becomes unfeasible in the broadbandcase due to their high complexity. The ISI can be reduced to narrowband levels by using a broadbandbeamformer where the antenna array observations are processed by an FIR filterbank [9]. OptimalMAP equalization is then performed at the receiver output. The FIR filter coefficients are chosen tominimize interference [10]. However, the noise at the output of such a receiver is colored and hence, theresultant signal cannot be applied to a trellis-based equalizer.Space-time receivers can be designed to preserve the whiteness of the channel noise while reducing ISI[8]. To ensure that the noise at the beamformer output remains white, the filterbank is required to havethe power complementarity property [11]. An N -channel FIR filterbank {W1(z), W2(z) . . . , WN(z)} is2said to be power complementary ifNXi=1Wi(z)˜Wi(z) = 1 (1)The tilde on transfer functions stands for complex conjugation followed by reciprocation of functionalargument, i.e.,˜W (z) = W∗(z−1). Design procedures for beamformers with the power complementarityconstraint have been proposed [8]. This design assumes that the noise at the input of the beamformerfilterbank is white. However, oversampling at the pulse shaping receive filter colors the noise and thiscoloring has to be incorporated into the power complementarity constraint. In this paper, filters aredesigned taking into account this coloring of the noise.This paper is organized as follows. Section 3 describes the signal model and the beamforming opti-mization problem. Section 4 presents simulation results and compares this design to previous designmethods. Finally, in section 5, conclusions and future work are presented.3 Problem FormulationWe consider a digital communication system where a symbol sequence is transmitted using a pulseshaping waveform f (t). The modulated signal has the complex baseband representation given bys(t) =Xmf(t − mT )xm(2)where T is the symbol period. This signal is passed through a frequency-selective wireless channel thatis modeled by an L-ray complex impulse response given byg(t) =LXl=1alδ(t − τl) (3)where aldenotes the complex reflection coefficient specifying the amplitude and phase of the lth rayand τlrepresents the associated time delay. We assume that the channel is an Additive White GaussianNoise (AWGN) channel so that the signal at the input of the antenna element is given byu(t) =XmLXl=1alf(t − mT − τl)xm+ νi(t)where the additive noises νi(t) are independent with 0 mean.3The