SUBBAND ADAPTIVE GENERALIZED SIDELOBE CANCELLER FOR
`BROADBAND BEAMFORMING
`
`Wei Liu, Stephan Weiss, and Lajos Hanxo
`
`Communications Research Group
`Department of Electronics & Computer Science
`University of Southampton, SO17 lBJ, U.K.
`{w.liu, s.weiss, 1.hanzo)Qecs.soton.ac.uk
`
`ABSTRACT
`In this paper, we propose a novel subband adaptive
`broadband beamforming architecture based on the gen-
`eralised sidelobe canceller (GSC), in which we decom-
`pose each of the tapped delay-line signals feeding the
`adaptive part of the GSC and the reference signal into
`subbands and perform adaptive minimisation of the
`mean squared error in each subband independently.
`Besides its lower computational complexity, this new
`subband adaptive GSC outperforms its fullband coun-
`terpart in terms of convergence speed because of its pre-
`whitening effect. Simulations based on different kinds
`of blocking matrices with different orders of derivative
`constraints are presented to support these findings.
`
`1. INTRODUCTION
`
`Adaptive beamforming has found many applications in
`various areas ranging from sonar and radar to wireless
`communications. It is based on a technique where, by
`adjusting the weights of a sensor array with attached
`filters, a prescribed spatial and spectral selectivity is
`achieved. Fig. 1 shows a beamformer with M sensors
`receiving a signal of interest from the direction of ar-
`rival (DOA) angle 19.
`
`. . . . . . . . . . . . . . . . . . , x,bI
`. . . . fib
`\,,Ay
`, -
`,
`*
`
`(I
`
`,
`
`,
`
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`I
`
`
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`
`
`Fig. 1: A signal impinging from an angle 29 onto a beam-
`former with M sensors.
`
`To perform beamforming with high interference re-
`jection and resolution, arrays with a large number of
`sensors and filter coefficients have to be employed. To
`facilitate real-time implementation, various methods
`are employed to reduce the computational complex-
`ity, such as the partially adaptive beamforming [l],
`wavelet-based beamforming [2] and subband beamform-
`ing [3]. In the latter, the received sensor signals are first
`split into decimated subbands, then an independent
`beamformer is applied to each subband. The advan-
`tage.arises from the processing in decimated subbands,
`although at the expense of having to project constraints
`into the subband domain as well.
`
`We here focus on a linearly constrained minimum
`variance (LCMV) beamformer, which can be efficiently
`implemented as a generalized sidelobe canceller (GSC)
`[4, 51. Different from [3], instead of performing beam-
`forming in subbands by decomposing the input sen-
`sor signals, we employ subband adaptive filtering tech-
`niques for the adaptive process of the GSC structure
`only. Specifically, noting that there are in total M - S
`input tapped delay-lines for the adaptive part of the
`GSC, we decompose each of the tap-delay line signals
`and the reference signal d[n] into K subbands by a
`K-channel filter banks as shown in Fig. 3 and perform
`adaptive minimisation in each subband. Simulation re-
`sults with different blocking matrices and different or-
`der of derivative constraints show that this new method
`outperforms the fullband counterpart in addition to its
`very low computational complexity.
`
`The rest of this paper is organised as follows: Sec-
`tion 2 is a brief review of GSC-based broadband beam-
`forming based on a generalized sidelobe canceller with
`derivative constraints. In Section 3, we introduce the
`proposed subband-based GSC structure. Simulation
`and results will be given in Section 4 and conclusions
`are drawn in Section 5.
`
`0-7803-701 1-2/01/$10.00 02001 IEEE
`
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`I.E
`
`synthesis filter bank
`analysis filter bank
`Fig. 3: K channel filter banks with decimation N .
`
`B and a quiescent vector wq. Thereafter, standard un-
`constrained optimisation algorithms such as least mean
`square (LMS) or recursive least squares (RLS) algo-
`rithms can be invoked [8]. Fig. 2 shows the principle
`of a GSC, where the desired signal d[n] is obtained via
`wq 7
`d[n] = w:
`
`with w; = C(CHC)-lf . (8)
`
`. x,
`
`The input signal U, = [uo[n] u1[12] . . . U M - S - ~ [ ~ ] ]
`T
`to the following multichannel adaptive filter (MCAF)
`is obtained by U, = BH%,, whereby the M x (A4 - S)
`blocking matrix B must satisfy
`
`CHB = 0
`
`where C = [CO -..CSI] .
`
`(9)
`
`In the next section, we will focus on the multiple-input
`optimisation process and introduce our subband adap-
`tive GSC structure by employing the subband adaptive
`filtering techniques.
`
`3. SUBBAND ADAPTIVE GENERALIZED
`SIDELOBE CANCELLER
`
`Fig. 2: Structure of a generalized sidelobe canceller.
`
`2. GENERALIZED SIDELOBE
`CANCELLER
`
`An LCMV beamformer performs the minimization of
`the variance or power of the output signal with respect
`t o some given spatial and spectral constraints. For a
`beamformer with M sensors and J filter taps following
`each sensor as shown in Fig. 1, the output e [ n ] can be
`expressed as:
`
`e [ n ] = wH . x,
`
`(1)
`
`where coefficients and input sample values are defined
`as
`
`w = [WZ w:
`w1 = [WO[Z] w$]
`
`T
`
`T
`
`H
`. .. WT1]
`(2)
`. . . WM-l[q]
`(3)
`x, = [%: %Ll . . . 5&+J
`T
`(4)
`x, = [zoln] z1[4 . . . zM-l[flIl
`.
`(5)
`The data vector 2, is a time slice as given in Fig. 1.
`A coefficient wm[Z] is defined to sit at the tap position
`1 of the mth filter fm. The LCMV problem can now
`be formulated as [6]
`
`subject to
`
`CHw = f
`
`(6)
`
`minwHR,,w
`W
`where R,, is the covariance matrix of observed array
`data in x,, C E C M J x S J is a constraint matrix and f E
`CsJ is the constraining vector. The constraint matrix
`here imposes derivative constraints of order S - 1 [7],
`
`Subband decompositions for adaptive filtering applica-
`tions are commonly based on oversampled modulated
`filter banks (OSFB) as shown in Fig. 3 where the in-
`put signal is divided into K frequency bands by analysis
`filters and then decimated by a factor N . Due to over-
`sampling, i.e. N < K , a low alias level in the subband
`signals can be achieved. This is important since alias-
`ing will limit the performance of an subband adaptive
`filtering (SAF) system [9]. Due to its lower update rate
`and fewer coefficients to represent an impulse response
`of a given length, the subband implementation only
`necessitates KIN2 (KIN3) of the operations required
`for a fullband adaptive algorithm with a complexity of
`(?(La) (O(L$)), where La is the total number of coef-
`ficients in the fullband realisation [3].
`... (A4 - 1 - m ~ ) ~ ]
`with ci = [ ( - m ~ ) ~ (1 - m ~ ) ~
`
`T
`When applying SAF techniques to the MCAF in the
`and a phase origin point mo.
`GSC structure in Fig. 2, the subband setup as shown
`in Fig. 4 arises. There, the blocks labelled A perform
`The constrained optimisation of the LCMV prob-
`an OSFB analysis operations, splitting the signal into
`lem in (6) can be conveniently solved using a GSC.
`K frequency bands each running at an N times lower
`The GSC performs a projection of the data onto an
`sampling rate compared to the fullband input to the
`unconstrained subspace by means of a blocking matrix
`
`.Cs1]
`
`with Ci =
`
`[ 1
`
`:i]
`
`(7)
`
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`\
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`:
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`I
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`.
`.
`.
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`.
`.
`.
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`
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`Fig. 4: Subband adaptive GSC; an independent MCAF
`is applied to each subband.
`
`block. Within each subband, an independent MCAF
`is operated, and a synthesis filter bank, labelled S, re-
`combines the different subsystem outputs to a fullband
`beamformer output e[n].
`In addition to the lower computational complex-
`ity of this subband adaptive GSC, it promises faster
`convergence speed for LMS-type adaptive algorithms
`because of the pre-whitening effect of the input signal.
`Next, we will give some simulation results to demon-
`strate the performance of our subband adaptive GSC.
`
`4. SIMULATIONS AND RESULTS
`
`In our simulation, we use a beamformer with M = 15
`sensors and J = 60 coefficients for each attached filter.
`Each of the input signals ui[n] (i = 0,2, . . . , M - S - 1)
`and the reference signal d[n] are divided into K = 8
`subbands by an oversampled GDFT filter bank [lo]
`with decimation factor N = 6 as characterised in Fig. 5.
`This subband adaptive GSC is constrained to received
`a signal of interest from broadside, which is white Gaus-
`sian with unit variance. The beamformer should adap-
`tively suppress a broadband interference signal cov-
`ering the frequency interval 52 = [0.257r;0.75~] from
`8 = 30" and with a signal-to-interference ratio (SIR)
`of -24 dB. The sensor signals are corrupted by additive
`Gaussian noise at an SNR of 20 dB.
`
`0.2
`
`0.4
`
`o
`1
`1.2
`1.4
`0.6
`0.8
`normalised angular frequency R I n
`Fig. 5: Magnitude response of K = 8 channel filter
`bank decimated by N = 6.
`
`1.6
`
`1.8
`
`2
`
`30
`
`- Wband(SVD)
`-. Fullband ( W O )
`
`500
`
`1000
`
`2000
`
`2500
`
`1500
`Iterations n
`Fig. 6: Learning curves for simulation I (S = 2).
`
`I
`3000
`
`5m
`
`1000
`
`1500
`Iterations n
`Fig. 7: Learning curves for simulation I1 (S = 2).
`
`2000
`
`2500
`
`3M
`
`In order to compare the performance of our subband
`method with its fullband counterpart, we give four ex-
`amples based on two commonly used approaches for
`building the blocking matrix, each with two different
`orders of constraints. The first approach is based on
`the cascaded columns of difference (CCD) method [ll],
`the second on a singular value decomposition (SVD) [5].
`The four examples are: (I) SVD method with first order
`derivative constraints (S = 2), (11) CCD method with
`S = 2, (111) SVD method with zero order derivative
`constraints ( S = l), (IV) CCD method with S = 1.
`The step size in the NLMS adaptation for the first
`two examples is set to CL = 0.30, and to fi = 0.20 for ex-
`amples (111) and (IV). Simulation results for these four
`cases are shown in Fig. 6 to Fig. 9, respectively. As a
`performance criterion, these figures display the ensem-
`ble mean square value of the residual error, which is
`defined as the difference between the beamformer out-
`put e[n] and the appropriately delayed desired signal
`received from broadside.
`
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`500
`
`1wo
`
`1m
`Iterations n
`Fig. 8: Learning curves for simulation I11 (S = 1).
`
`xm
`
`2500
`
`3x0
`
`I
`0
`
`5w
`
`1wo
`15w
`Iterations n
`Fig. 9: Learning curves for simulation IV (S = 1).
`
`2wo
`
`I
`
`2500
`
`gence speed because of its pre-whitening effect. Supe-
`riority of this new method to fullband implementation
`has been demonstrated by four examples based on dif-
`ferent approaches for the blocking matrix and different
`orders of derivative constraints.
`
`6. REFERENCES
`
`[l] D. J. Chapman, “Partial Adaptivity for Large Ar-
`rays,” IEEE Trans AP, 24(9):685-696, Sept. 1976.
`[2] Y. Y. Wang, W. H. Fang, and J. T. Chen,
`“Improved Wavelet-Based Beamformers with Dy-
`namic Subband Selection,” in Proc. IEEE AP-S
`Int. Symp., 1999.
`[3] S. Weiss, R. W. Stewart, M. Schabert, I. K.
`Proudler, and M. W. Hoffman,
`“An Efficient
`Scheme for Broadband Adaptive Beamforming,”
`in Asilomar Conf. SSC, I:496-500, Monterey, CA,
`Nov. 1999.
`[4] L. J. Griffith and C. W. Jim, “An Alternative Ap-
`proach to Linearly Constrained Adaptive Beam-
`forming,” IEEE Trans AP, 30( 1):27-34, Jan. 1982.
`[5] K. M. Buckley and L. J. Griffith,
`“An Adap-
`tive Generalized Sidelobe Canceller with Deriva-
`tive Constraints,” IEEE Trans AP, 34(3):311-319,
`Mar. 1986.
`[6] 0. L. Frost, 111, “An Algorithm for Linearly Con-
`strained Adaptive Array Processing,” Proc. IEEE,
`60(8):926-935, Aug. 1972.
`
`[7] K.C. Huarng and C.C. Yeh, “Performance Analy-
`sis of Derivative Constraint Adaptive Arrays with
`Pointing Errors,” IEEE Trans AP, 40(8):975-981,
`Aug. 1992.
`[8] S. Haykin, Adaptive Filter Theory, Prentice Hall,
`Englewood Cliffs, 2nd edition, 1991.
`[9] S. Weiss, R. W. Stewart, A. Stenger, and
`R. Rabenstein, “Performance Limitations of Sub-
`in Proc. EUSIPCO,
`band Adaptive Filters,’’
`III:1245-1248, Rodos, Greece, Sep. 1998,
`[lo] S. Weiss and R. W. Stewart, On Adaptive Filtering
`in Oversampled Subbands, Shaker Verlag, Aachen,
`Germany, 1998.
`[ll] N. K. Jablon,
`“Steady State Analysis of
`the Generalized Sidelobe Canceler by Adaptive
`IEEE Trans A P,
`Noise Canceling Techniques,”
`34(3):330-337, Mar. 1986.
`
`F
`
`From these results we can see that the subband
`adaptive method always has a faster convergence speed
`because of its pre-whitening effect. Comparing Fig. 6
`with Fig. 7 and Fig. 8 with Fig. 9, we see the fullband
`performance changes according to different building of
`the blocking matrix, whereas the subband method has
`a relatively uniform performance independent of set-
`tings. With the added benefit of its low computational
`complexity due t o processing in decimated subbands,
`the presented subband method outperforms the tradi-
`tional fullband implementation.
`
`5. CONCLUSIONS
`
`A novel subband adaptive Generalized Sidelobe Can-
`celler for broadband beamforming has been proposed.
`By employing subband adaptive filtering techniques,
`the computational complexity is greatly reduced. More-
`over, the new method can also achieve a faster conver-
`
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