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`v~~~ ·NAL PROCESSING
`I}M)·~,
`~ A PUBLICAnON OF THE IEEE SIGNAL PROCESSING SOCIETY
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`www.ieee.org/sp/index.html
`
`AUGUST 2001
`
`VOLUME 49
`
`NUMBER 8
`
`ITPRED
`
`(ISSN 1 053-587X)
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`E. '/hJ i./ I ~ I
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`I
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`'
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`PAPERS
`Methods of Sensor Array and Multichannel Processing
`A Subspace-Based Direction Finding Algorithm Using Fractional Lower Order Statistics .. T.-H. Liu and J. M. Mendel 1605
`Signal Enhancement Using Beamforming and Nonstationarity with Applications to Speech .... ..... ...... .. ... ... ...... .
`.. · .................. . .. .. ... .. . .... .. . .. ... . .. . ... ...... .... . .... ·. S. Gannot, D. Burshtein, and E. Weinstein 1614
`Signal and System Modeling
`Extensions of the Weighted-Sample Method for Digitizing Continuous-Time Filters .. ..... C. Wan and A. M. Schneider 1627
`Relations between Fractional Operations and Time-Frequency Distributions, and Their Applications .. ... .. ......... · · ·.
`· . .. .... .. ..... . . ............ ... ....... . .. ... . ... ... . . . ... . . . .... . . . ..... .. . ..... . . S.-C. Pei and J.-J. Ding 1638
`Multi window Tl!De-Varying Spectrum with Instantaneous Bandwidth and Frequency Constraints ... ... · · · · · · · · · · · · · · · · · ·
`· · . . .. . ....... . .. ... ... ... ..... . .. ... . ... .. ...... ......... . .. ............ F. (:akrak and P. J. Loughlin 1656
`Signal Detection and Estimation
`Adaptive Volterra Filters for Active Control of Nonlinear Noise Processes ...... .... ... .... ... .... ··· L Tan and J. Jiang 1667
`
`I Frequenc~. ~~~~~. ~.1~~~ .~~~. ~~~~~~. ~~~~~i~·c·a·t~~~ .~~~~~. ~~. ~~~~~~~. ~~~ .~~~~~~ .~~~~. ~~~~t~~d A: ·p: ·P;t~~;;,i;, 1677
`I On ~e Behavior of Information Theoretic Criteria for Model Order Selection ...... .... . A. P. LiavaS. and P. A. Regalia 1689
`. Ftlter Design and Theory
`Robust 'H.oo Filtering for Uncertain Discrete-Time State-Delayed Systems ..... · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
`M B " " .. · .................................................. R. M. Pal hares, C. E. de Souza, and P. L. D. Peres 1696
`- and Compactly Supported Orthogonal Symmetric Interpolating Scaling Functions · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
`·" · .. .. .. ·.. .. .. . ...
`.. .. P.-L. Shui, Z. Baa, and X.-D. Zhang 1704
`AN
`···············································
`Bou o~el D~sign of Lifting Scheme from General Wavelet .. ..... .. .. .. .. ... ... ...... ..... · · · · H. Li, Q. Wang, and L ~11 1714
`n ary Filter Optimization for Segmentation-Based Subband Coding .............. ... · · · · · · · · · · · · · · · · · · · · · A. Mertms 1718
`Nonline s·
`.
`l
`-
`ar
`tgna Processmg
`F
`d~~ency Domain Computations for Nonlinear Steady-State Solutions ..... .. .... .. ·. · · · · · · N. P. Telang and L R. Hunt 1728
`e Convergence of Volterra Filter Equalizers Using a Pth-Order Inverse Approach · · · · · · · · · ·: · · · · · · · · · · · · · · · · · · · · · ·
`Reft~~~j~~ .. ·" · ..... .......... .. .... ...... ...... ....... .......... . .. Y. Fang, L.-C. Jiao, X.-D. Zhang, and J. Pan 1734
`~oeffic1ents Counterpart of Cardan-Viete Formulas ....... . :. : .. .. .. .. . J. L. Dfaz-Barrero and J. J. Egozcue 1745
`1 Eavesd
`roppmg in the Synchronous CDMA Channel: An EM-Based Approach .... .... .......... . Y. Yao t:ind H. V. Poor 1748
`
`r
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`L
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`(Contents Continued on Back Cover)
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`- i -
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`Sony v. Jawbone
`
`U.S. Patent No. 11,122,357
`
`Sony Ex. 1004
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`JMB·ER 8
`
`ITPRED
`
`(ISSN 1 053-5
`
`••
`
`. nng
`; Fractional Lower Order Statistics .. T.-H. Liu and J. M Mej
`ionarity with Applications to Speech ................. · · · .· · · · · ·
`· · · · · · · · ..... ~ , ..... ·. S . . Gannot, D. Burshtein, and E. Weins~
`
`.
`
`.
`
`lzing Continuous-Time Filters ... ; . . . C. Wan and A. M Schneit
`equency Distributions, and Their Applications ......... .-...... · .
`· · · · · · · ............... ~ ..... · ......... S.-C. Pei and J. -J. Db
`~ous Bandwidth and Frequency-Constraints .... .-............ · · · · · · ·
`· · · · · · .. ~ . ................. ; . . . . . . . F. r;akrak and P~ J. Loughli
`
`- ii -
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`

`

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`1614
`
`IEEE TRANSACI10NS ON SIGNAL PROCESSING. VOL. 49, NO. 8, AUGUST 2001
`
`Signal Enhancement Using Beamforming and
`Nonstationarity with Applications to Speech
`
`Sharon Gannot, Student Member, IEEE, David Burshtein, Senior Member, IEEE, and Ehud Weinstein, Fellow, IEEE
`
`Abstract-We consider a sensor array located in an enclo-
`sure, where arbitrary transfer functions (TFs) relate the source
`signal and the sensors. The array is used for enhancing a signal
`contaminated by interference. Constrained minimum power
`adaptive beamforming, which has been suggested by Frost and,
`in particular, the generalized sidelobe canceler (GSC) version.
`which has been developed by Griffiths and Jim, are the most
`widely used beamforming techniques. These methods rely on the
`assumption that the received signals are simple delayed versions
`of the source signal. The good interference suppression attained
`under this assumption is severely impaired in complicated acoustic
`environments, where arbitrary TFs may be encountered. In this
`paper, we consider the arbitrary TF case. We propose a GSC
`solution, which is adapted to the general TF case. We derive a
`suboptimal algorithm that can be implemented by estimating
`the TFs ratios, instead of estimating the TFs. The TF ratios are
`estimated by exploiting the nonstationarity characteristics of the
`desired signal. The algorithm is applied to the problem of speech
`enhancement in a reverberating room. The discussion is supported
`by an experimental study using speech and noise signals recorded
`in an actual room acoustics environment.
`Index Terms-Beamforming, nonstationarity, speech enhance-
`ment.
`
`I. INTRooucnoN
`
`S IGNAL quality might significantly deteriorate in the
`
`presence of interference, especially when the signal is
`also subject to reverberation. Multisensor-based enhancement
`algorithms typically incorporate both spatial and spectral
`information. Hence, they have the potential to improve on
`single sensor solutions that utilize only spectral information.
`In particular, when the desired signal is speech, single micro-
`phone solutions are known to be limited in their performance.
`Bearnforming methods have therefore attracted a great deal of
`interest in the past three decades. Applications of bearnforming
`to the speech enhancement problem have also emerged recently.
`Constrained minimum power adaptive bearnforming, which
`has been suggested by Frost [ 1 ], deals with the problem of a
`broadband signal received by an array, where pure delay re-
`lates each pair of source and sensor. Each sensor signal is pro-
`cessed by a tap delay line after applying a proper time delay
`
`Manuscript received Marcb 28, 2000; revised April 30, 2001. The associate
`editor coordinating the review of this paper and approving il for publication was
`Dr. Alex C. Kot.
`S. Gannot
`is with
`tbe Department of Electrical Engineering
`(SISTA), Katholielr.e Universiteit Leuven, Leuven, Belgium
`(e-mail:
`Sbaron.Gannot@esaLkuleuven.ac.be).
`D. Burshtein and E. Weinstein arc with the Department of Electrical
`Engin~ring-Systems, Tei·Aviv University, Tel-Aviv.
`Israel
`(e-mail:
`bur~lyn@c:ng.tau.ac.il ; udi@eng.tau.ac.il).
`Publisher Item IdentifierS IOS3-S87X(OI)OS874-3.
`
`compensation. The algorithm is capable of satisfying some de-
`sired frequency response in the look direction while minimizing
`the output noise power by using constrained minimization of
`the total output power. This minimization is realized by ad-
`justing the taps of the filters under the desired constraint. Frost
`suggested a constrained LMS-type algorithm. Griffiths and Jim
`[2] reconsidered Frost's algorithm and introduced the general-
`ized sidelobe canceler (GSC) solution. The GSC algorithm is
`comprised of three building blocks. The first is a fixed beam-
`former, which satisfies the desired constraint. The second is
`a blocking matrix, which produces noise-only reference sig-
`nals by blocking the desired signal (e.g., by subtracting pairs of
`time-aligned signals). The third is an unconstrained LMS-type
`algorithm that attempts to cancel the noise in the fixed beam-
`former output. In [2], it is shown that Frost algorithm can be
`viewed as a special case of the GSC. The main drawback of the
`GSC algorithm is its delay-only propagation assumption.
`Van Veen and Buckley [3] summarized various methods for
`spatial filtering, including the GSC, and introduced a wider
`range of possible constraints on the beam pattern. Cox e_t al.
`[ 4] suggested constraint of tile norm of the adaptive canceler
`coefficients in order to solve the superdirectivity problem,
`i.e., its sensitivity to steering errors. In particular, they have
`suggested to update Frost's (or the Griffiths and Jim) algorithm
`by applying a quadratic constraint on the norm of the noise
`canceler coefficients. This constraint, which can limit the
`superdirectivity, is added to the usual linear constraints.
`Some authors have recently suggested using the GSC for speech
`enhancement in a reverberating environment. Hoshuyama et al.
`[5]-[7] used a three-block structure similar to the GSC. However,
`the blocking matrix has been modified to operate adaptively. In
`order to limit the leakage ofthedesiredsignal, which is responsible
`for distortion in the output signal, a quadratic constraint is imposed
`on the norm of the noise canceler coefficients. Alternatively, use of
`the leaky LMS algorithm has been suggested.
`Nordholm et a/. [8) used a GSC solution in which the
`blocking matrix is realized by spatial highpass filtering, thus
`yielding improved noise-only reference signals. Meyer· and
`Sydow [9] have suggested to construct the noise reference
`signals by steering the lobes of a multibeam bearnformer
`toward the noise and desired signal directions separately.
`Widrow and Stearns [10] have proposed a dual structure
`bearnformer. The 'master bearnformer adapts its coefficients to
`minimize the output power while maintaining the beam-pattern
`toward a predetermined pilot signal from the desired direc-
`tion. Those coefficients are continuously copied to a slave
`beamformer that . is used to enhance the speech signal. Dahl
`et a/. [11] have extended this solution by proposing a dual
`
`JOS3-S87X/Ol$10.00 C 2001 IEEE
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`- 1614 -
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`

`

`oANNOT t1 a/.: SIGNAL ENHANCEMENT USING BEAMFORMING AND NONSTATIONARITY
`
`1615
`
`beamformer that attempts to cancel both noise and jammer
`signals (e.g., loudspeaker). The pilot signal is constructed by
`offline recordings of the jammer and desired signal in the
`actual acoustic environment during a calibration phase. Thus,
`both echo cancellation and noise suppression are achieved
`simultaneously.
`Other solutions utilize a beamformer type algorithm, followed
`by a postprocessor. Zelinski [ 12] suggested a Wiener filter, fol-
`lowed by further noise reduction in a postprocessing configura-
`tion. Meyer and Simmer [ 13] addressed the problem of high co-
`herence between the microphone signals at low frequencies, as in-
`dicated by Dal-Degan and Prati [14]. They have suggested the use
`of a spectral subtraction algorithm in the low-frequency band and
`Wiener filtering in the high-frequency band. Fischer and Kam-
`meyer [15] suggested to further split the microphone array into
`differentially equispaced subarrays. This structure has been fur-
`ther analyzed by Marro eta/. [16]. Bitzer et al. [17] analyzed the
`performance of the GSC solution and showed its dependence on
`the noise field. They showed that the noise reduction might be in-
`finitely large when the noise source is directional. However, in
`the more practical situation of a reverberant enclosure, when the
`noise field can be regarded as diffused, the performance degrades
`severely. Bitzer eta/. [18] suggested aGSC with fixed Wiener fil-
`ters in the noise canceling block and further postfilters at the GSC
`output. An improved performance in the lower frequency range
`is achieved. In [ 19], it is shown that the Wiener filters can be com-
`puted in advance by utilizing prior knowledge of the noise field.
`Jan and Flanagan [20] suggested a matched filter beam-
`forming (MFBF) instead of the conventional delay and sum
`beamformer (DSBF). The MFBF configuration realizes signal
`alignment by convolving the microphone signals with the
`(estimated) acoustic transfer function (TF). Rabink.in et al. [21]
`proved that the performance of MFBF is superior to ~SBF,
`provided that the room acoustics TF is not too compltcated.
`They have also suggested truncation of the estimated acoustic
`TFs to ensure reliable estimates.
`Grenier et al. [22]-[29] have proposed GSC-based enhance-
`ment algorithms. In [29], the case where general TFs relate ~e
`source and microphones was considered. A subspace tracking
`solution [30] has been proposed. The resulting TFs are con-
`strained to the array manifold under the assumption of an FIR
`model and small displacements of the talker. The fixed beam-
`former block of the GSC is realized using MFBF.
`In this paper, we consider a sensor array located in an enclo-
`sure, where general TFs relate the source signal and the sensors.
`The array is used for enhancing a signal contaminated by inter-
`ference. We propose a GSC solution, which is adapted to the
`general TF case. The TFs are estimated by exploiting the non-
`stationarity characteristics of the desired signal. The algorithm
`is applied to the problem of speech enhancement in a rever-
`berating room. The discussion is supported by an experimental
`study using speech and noise signals recorded in an actual room
`acoustics environment. the outcome consists of the assessment
`of sound sonograms, signal-to-noise ratio (SNR) enhancement,
`and informal subjective listening tests. The paper is organized
`as follows. In Section n, we formulate the problem of beam-
`forming in a general TF environment in the frequency domain.
`The constrained power minimization is presented in Section lll,
`
`where both Frost's algorithm [I] and the Griffiths and Jim [2]
`interpretation are derived in the frequency domain. This deriva-
`tion motivates the intuitive structure suggested by other authors
`for the beamforming problem in reverberant environments. We
`then show that a suboptimal algorithm can be implemented by
`estimating the TF ratios instead of estimating the actual TFs. In
`Section IV, we address the problem of estimating the TF ratios
`by extending the nonstationarity principle, which was suggested
`by Shalvi and Weinstein (31 ]. An application of the suggested
`algorithm to the speech enhancement problem is presented in
`Section V. Section VI concludes the paper.
`
`ll. PROBLEM FORMULATION
`Consider an array of sensors in a noisy and reverberant envi-
`ronment. The received signal is comprised of two components.
`The first is some nonstationary (e.g., speech) signal. The second
`is some stationary interference signal. Our goal is to reconstruct
`the nonstationary signal component from the received signals.
`We use the following notation.
`zm(t)
`mth sensor signal;
`s( t)
`desired signal source;
`n rn ( t)
`interference signal of the mth sensor comprised of
`some directional noise component and some am-
`bient noise component;
`time-varying TFs from the desired speech source to
`the mth sensor.
`
`am(t)
`
`We have
`m = 1, ... , M
`Zm(t) = am(t) * s(t) + nm(t);
`(l)
`where * denotes convolution. Suppose that the analysis frame
`duration T is chosen such that the signal may be considered
`stationary over the analysis frame. Typically, the TFs are
`changing slowly in time so that they may also be considered
`stationary over the analysis frame. Multiplying both sides of
`(1) by a rectangular window function w(t) [w(t) = 1 over .the
`analysis frame w(t) = 0 otherwise) and applying the discrete
`time Fourier transform (DTFT) operator yields
`
`rn = 1, ... , M .
`The approximation is justified for T sufficiently large.
`Zm(t, ei"' ), S(t, ei"') and Nm(t, ei"') are the short ti~e
`Fourier transforms (STFTs) of the respective signals. Am ( e1"' )
`is the TF of the mth sensor. Note that we have assumed that
`the TFs are time invariant.
`The vector formulation of the equation set (2) is
`
`(2)
`
`where
`zT(t, ei"') = (Z1(t, ei"' ) Z2(t, ei"') · · · ZM(t, ei"'))
`AT(ei"' ) = (A1(ei"') A2(ei"' ) · .. AM(ei"')]
`N T(t, ei"') = (N1(t, ei"') N2(t, ei"')
`.. · NM(t, ei"' )] .
`
`- 1615 -
`
`

`

`1616
`
`IEEE TRANSACI10NS ON SIGNAL PROCESSING, VOL. 49, NO. 8, AUGUST 2001
`
`Fig. I. Constrained minimization.
`
`ill. CONSTRAINED OtiTPUT POWER MINIMizATION
`In [1], a beamforming algorithm was proposed under the
`assumption that the TF from the desired signal source to each
`sensor includes only gain and delay values. In this section,
`we consider the general case of arbitrary TFs. By following
`the derivation of [1] in the frequency domain, we derive a
`beamforming algorithm for the general TF case. First, we
`obtain a closed-form, linearly constrained, minimum variance
`beamformer. Then, we derive an adaptive solution. The out-
`come will be a constrained LMS-type algorithm. We proceed,
`following the footsteps of Griffiths and Jim [2], and formulate
`an unconstrained adaptive solution. We will initially assume
`that the TFs are known. Later, in Section IV, we deal with the
`problem of estimating the TFs.
`
`A. Frequency Domain Frost Algorithm
`1) Optimal Solution: Let W*(t, ei"'); m = 1, . . . , M be a
`set of M filters
`
`where • denotes conjugation, and t denotes conjugation trans-
`pose. A bearnformer is realized by flltering each sensor output
`by w•(t, ei'"') m = 1, . .. , M and summing the outputs
`Y(t, ei'"') = wt(t, ei"')Z(t, ei"')
`= wt(t, ei"')A(ei"')S(t, ei'"')
`+ wt(t, ei"')N(t, ei"')
`~Y.(t, ei"') + Yn(t, ei'"')
`where Y.(t, ei"') is the desired signal part, and Yn(t, ei'"') is the
`noise part. The output power of the beamformer is
`
`(4)
`
`E {Y(t, d"')Y*(t, d'"')}
`= E {Wt(t, d"')Z(t, ei"')Zt(t, ei"')W(t, ei"')}
`= wt(t, ei"')~zz(t, ei"')W(t, ei"')
`
`where ~zz(t , ei"') ~ E{Z(t, ei"')Zt(t, ei"')}. We want to
`minimize the output power subject to the following constraint
`on Y.(t, ei"')
`Y.(t, ei"') = wt(t, ei"')A(ei"')S(t, d"')
`= :F* ( t, ei"')S( t, d'"')
`where .r•(t, ei"') is some prespecified filter (usually a simple
`delay). We thus have the following minimization problem:
`min {Wt(t, ei"')~zz(t, ei"')W(t, ei"')}
`w
`subject to wt(t, d'"')A(d"') = :F*(t, d"').
`The minimization (5) is demonstrated in Fig. 1. The point where
`the equipower contours are tangent to the constraint plane is
`the optimum vector of beamforrning filters. The perpendicular
`F( ei"') from the origin to the constraint plane will be calculated
`in Section ill-A2.
`To solve (5), we first define the following complex Lagrange
`functional:
`.C(W) = wt(t, d"')~zz(t, d"')W(t, d"')
`+A [wt(t, ei"')A(d"')- :F*(t, ei"')]
`+A* [At(t, d"')W(d"')- :F(t, ei"')]
`
`(5)
`
`where A is a Lagrange multiplier. Setting the derivative with
`respect to w· to 0 (e.g., [32]) yields
`Y'w· .C(W) = ~zz(t, ei"')W(t, ei"' ) + AA(ei"') = 0.
`
`Now, recalling the constraint in (5), we obtain the following set
`of optimal filters:
`wopt(t, ei'"') = [At(ei"')~zi(t, ei"')A(d'"')r1
`· ~zi(t, ei"')A(ei'"'):F(ei'"').
`This closed-form solution is difficult to implement and does not
`have the ability to track changes in the environment. Therefore,
`an adaptive solution should be more useful.
`
`- 1616 -
`
`

`

`GANNOT" a/.: SIGNAL ENHANCEMENT USING BEAMFORMING AND NONSTATIONARITY
`
`1617
`
`W(t = O,elw) = F(e/W)
`W(t + 1, eJw) =
`P(e/W) [W(t,e/W)- ,Z(t,elw)Y•(t,eiW)] + F(eiW)
`t =0,1, . . .
`(P(eiw) and F(elw) are defined by (6) and (7)).
`
`Fig. 2. Frequency domain frost algorithm.
`
`2) Adaptive Solution: Consider the following steepest de-
`scent, adaptive algorithm:
`W(t + 1, d"')
`= W(t, d"') -JJ\lw.£(d"')
`= W(t, d"')- 1-' [~zz(t, d"'}W(t, d"') + AA(d"'}) .
`Imposing our constraint on W ( t + 1, ei"') yields
`F(d"') =At(d"')W(t + 1, d"')
`=At(d"')W(t, ei"')
`-JJAt(d"')~zz(t, ei"')W(t, ei"')
`-JJAt(d"')A(d"')A.
`
`Solving for the Lagrange multiplier and applying further re-
`arrangement of terms yields
`W(t+1, d"') =P(ei"')W(t, d"')
`- JJP(d"')~zz(t, d"')W(t, d"')+F(ei"')
`
`where
`
`and
`
`(6)
`
`(7)
`
`replacing
`Further simplification can be achieved by
`~zz(t, ei"') byitsinstantaneousestimatorZ(t, ei"')Zt(t, ei"')
`and recalling (4). We thus obtain
`W(t + 1, d"'}
`= P(d"') (W(t, d"'} - JJZ(t, ei"')Y*(t, d"'}) + F(d"').
`
`The algorithm is summarized in Fig. 2.
`
`B. Generalized Side/abe Canceler (GSC) Interpretation
`In [2], Griffiths and Jim considered the case where each TF
`is a delay element (with some gain). Griffiths and Jim obtained
`an unconstrained adaptive enhancement algorithm, using the
`same constrained, minimum output power criterion used by
`Frost [1]. The unconstrained algorithm is computationally
`more efficient than the constrained algorithm. Furthermore, the
`unconstrained algorithm is based on the well behaved NLMS
`scheme. In Section ill-A2, we obtained an adaptive algorithm
`for the case where each TF is represented by an arbitrary linear
`time-invariant system by tracing the derivation of Frost in the
`frequency domain. We now repeat the arguments of Griffiths
`
`and Jim for our case (arbitrary TFs) and derive an unconstrained
`adaptive enhancement algorithm.
`Consider the null space of A(ei"'}, which is defined by
`
`The constraint hyperplane
`
`is parallel to N ( eiw). In addition to that, let
`'R(d"') ~ { ~~:A(ei"') I for any real 11:}
`be the column space. By the fundamental theorem of linear al-
`gebra (e.g., [33]) 'R(ei"') 1. N(ei"') . In particular, F(ei"') is
`perpendicular to N(ei"') since
`F(d"') = ~~~~;:;ll2 A(ei"') E 'R(ei"').
`Furthermore
`Af(ei"'}F(d "')
`= At(ei"')A(d"') (At(ei"')A(ei"')) - 1 F(ei"' )= F(d "').
`Thus, F(ei"') E A(ei"') and F(ei"') 1. A(ei"'). Hence, F(ei"')
`is the perpendicular from the origin to the constraint hyperplane
`A( ei"'). The matrix P( ei"'), which is defined in (6), is the pro-
`jection matrix to the null space of A(d"'), N(ei"').
`Now, a vector in linear space can be uniquely split into a sum
`of two vectors in mutually orthogonal subspaces (e.g., [33]).
`Hence
`
`W(t, ei"') = W 0(.t, d"')- V(t, d"')
`(8)
`where W 0(t, ei"') E 'R(ei"'), and -V(t, ei"') E N(ei"'). By
`the definition of N(d"' )
`
`(9)
`where 'H(ei"') is some M x (M - 1) matrix, such that the
`columns of 'H(ei"') span the null space of A(ei"'). i.e.,
`rank {'H(ei"')} = M - 1.
`(10)
`The vector G(t, ei"') is an (M - 1) x 1 vector of adjustable
`filters. By the geometrical interpretation of Frost's algorithm
`
`W ( _;w)- F(-;"') - A(ei"') F( jw}
`-
`-
`o t, ~'""
`IIA(ei"')ll2
`e
`.
`~'""
`
`(11)
`
`[Recall that F(ei"') is the perpendicular from the origin to the
`constraint hyperplane A(ei"').] Now, using (4), (8), and (9) we
`get
`
`where
`
`.
`t
`.
`.
`YFBF(t, el"') = W 0(t, e1"')Z(t, el"')
`YNc(t, d"') = Gf(t, ei"')'Ht(ei"')Z(t, ei"').
`
`(13)
`
`- 1617 -
`
`

`

`1618
`
`IEEE TRANSAcnONS ON SIGNAL PROCESSING. VOL. 49. NO.8. AUGUST 200I
`
`The output of the constrained beamformer is a difference of two
`terms, both operating on the input signal Z(t, eiw). The first
`term YFsF(t, eiw) utilizes only fixed components (which de-
`pend on the TFs); therefore, it can be viewed as a fixed beam-
`former (FBF). We now examine the second term YNc(t, eiw).
`Note that
`
`U(t, eiw) =1lt(ei"')Z(t, eJw}
`= 1£f(eJw} (A(eJw}S(t, eJw} + N(t, eJw}]
`(14)
`
`The last transition is due to (10). U(t, eiw) are reference
`noise signals. Hence, the signal dependent component of
`YNc(t, eiw) is completely eliminated (blocked) by 1[f(eiw)
`so that YNc(t, eiw) is a pure noise term. The noise term of
`YFaF(t, eiw) can be reduced by properly adjusting the filters
`G(t, eiw), using the minimum output power criterion. This
`adjustment problem is in fact the·classical multichannel noise
`cancellation problem. An adaptive LMS solution to the problem
`was proposed by Widrow [34].
`The GSC solution is comprised of three components:
`I) fixed beamformer (FBF);
`2) blocking matrix (BM) that constructs the noise reference
`signals;
`3) multichannel noise canceler (NC).
`We now discuss each of these components in details.
`1) Fixed Beamformer (FBF): By (3), (11), and (13), we
`have
`
`The first term on the right-hand side is the signal term. The
`second is the noise term. Note that by setting .r•(eiw) = e-iwr
`(i.e., a delay), the signal component ofYFsF( t, eiw) is an undis-
`torted, delayed version of the desired signal.
`Unfortunately, we usually do not have access to the actual
`TFs (Am(eiw); m = 1, .. . , M). Later, we show how we can
`estimate the TFs ratio
`
`m=l, ... ,M.
`
`(15)
`
`Let
`
`HT(ei"') = [1 A2(ei"')
`
`AT(ei"')
`AM(ei"')]
`· · · A1(ei"') = A1(ei"') ·
`If in (11), the actual TFs are replaced by the TFs ratios, then
`
`A1(ei"')
`
`jw} -
`W (
`o t, e
`-
`
`""( jw}
`H(ei"')
`IIH(ei"')ll2 .r e
`.
`
`(16)
`
`By (3) and (13), we have
`
`YFaF(t, ei"') =Al(ei"')F*(ei"')S(t, ei"')
`
`:F•(ei"') Hf( i"')N(t
`jw)
`+ IIH(ei"')i12
`e
`.
`'e
`
`Thus, when W 0(t, eiw) is given by (16), the signal term of
`YFaF(t, eiw) is the desired signal distorted only by the first TF
`A1(ei"'). Now, suppose that
`
`In this case, W 0(t, ei"') iscomprisedofthecascadeofH(eiw),
`which is a filter matched to the TFs ratio, and F( e1w). The new
`W 0 (t, eiw) can be derived from (16) under the assumption that
`IIH(eiw)ll2 is constant. In fact, Grenier eta/. [29] argue that
`this assumption can be verified empirically. The FBF term of
`the output is now given by
`
`(t ei"') = IIA(ei~)ll2 F(eiw)S(t ei"')
`Yi
`'
`Ai(e'"'