Jacob Benesty Arden (Yiteng) Huang Adaptive Signal Processing: Application to Real-World Problems SPIN Springer’s internal project number, if known Springer Berlin Heidelberg NewYork Barcelona Budapest HongKong London Milan Paris SantaClara Singapore Tokyo Contents 4 Multichannel Frequency-Domain Adaptive Filtering with Application to Acoustic Echo Cancellation :::::::::::::: 95 Herbert Buchner, Jacob Benesty, Walter Kellermann 4.1Introduction................................................. 95 4.2General Derivation of Multichannel Frequency-Domain Algorithms . 98 4.2.1Optimization Criterion................................... 99 4.2.2Normal Equation........................................ 102 4.2.3Adaptation Algorithm ................................... 103 4.3ConvergenceAnalysis......................................... 106 4.3.1Analysis Model ......................................... 107 4.3.2Convergencein the Mean................................. 107 4.3.3Convergenceof the Mean-Squared Error.................... 108 4.4Generalized Frequency-Domain Adaptive MIMO Filtering......... 110 4.5Approximation and Special Cases .............................. 113 4.5.1Approximation of the Frequency-Domain Kalman Gain ...... 113 4.5.2Special Cases ........................................... 114 4.6A Dynamical Regularization Strategy........................... 116 4.7E(cid:14)cient Multichannel Realization.............................. 117 4.7.1E(cid:14)cient Calculation of the Frequency-Domain Kalman Gain.. 117 4.7.2DynamicalRegularizationfor ProposedKalmanGainApproach119 4.7.3E(cid:14)cient DFT calculation of overlapping data blocks ......... 120 4.8Simulations and Real-World Applications ....................... 122 4.8.1Multichannel Acoustic Echo Cancellation................... 123 4.8.2Adaptive MIMO Filtering for Hands-Free Speech Communi- cation.................................................. 125 4.9Conclusions ................................................. 127 References ..................................................... 127 4 Multichannel Frequency-Domain Adaptive Filtering with Application to Acoustic Echo Cancellation Herbert Buchner1, Jacob Benesty2, and Walter Kellermann1 1 Univerityof Erlangen-Nuremberg, Telecommunications Laboratory, Cauerstr. 7, D-91058 Erlangen, Germany E-mail: fbuchner,[email protected] 2 Bell Laboratories, LucentTechnologies, Murray Hill, NJ 07974, USA E-mail: [email protected] Abstract. In unknown environments where we need to identify, model, or track unknown and time-varying channels, adaptive (cid:12)ltering has been proven to be an e(cid:11)ectivetool.Inthischapter,wefocusonmultichannelalgorithmsinthefrequency domain that are especially well suited for input signals which are not only auto- correlatedbutalsohighlycross-correlatedamongthechannels.Thesepropertiesare particularlyimportantforapplicationslikemultichannelacousticechocancellation. Mostfrequency-domainalgorithms,astheyarewellknownfromthesingle-channel case, are derived from existing time-domain algorithms and are based on di(cid:11)erent heuristic strategies. Here, we present a new rigorous derivation of a whole class of multichannel adaptive (cid:12)ltering algorithms in the frequency domain based on a recursive least-squares criterion. Then, from the so-called normal equation, we de- rive a generic adaptive algorithm in the frequency domain that we formulate in di(cid:11)erent ways. An analysis of this multichannel algorithm shows that the mean- squared error convergence is independent of the input signal statistics. A useful approximation provides interesting links between some well-known algorithms for the single-channel case and the general framework. We also give design rules for important parameters to optimize the performance in practice. Due to the rigor- ous approach, the proposed framework inherently takes the coherence between all inputsignalchannelsintoaccount,whilethecomputationalcomplexityiskeptlow by introducing several new techniques, such as a robust recursive Kalman gain computation in the frequency domain and e(cid:14)cient fast Fourier transform (FFT) computation tailored to overlapping data blocks. Simulation results and real-time performanceformultichannelacousticechocancellationshowthehighe(cid:14)ciencyof the approach. 4.1 Introduction Theabilityofadaptive(cid:12)lterstooperatesatisfactorilyinanunknownenviron- ment and to track time variations of input statistics make it a powerful tool in such diverse (cid:12)elds as communications,acoustics,radar,sonar,seismology, 96 Herbert Buchneret al. and biomedical engineering.Despite of the largevariety of applications,four basic classes of adaptive (cid:12)ltering applications may be distinguished [1]: sys- tem identi(cid:12)cation, inverse modeling, prediction, and interference cancelling. In speech and acoustics, where all those basic types of adaptive (cid:12)ltering can be found, weoften havetodeal with verylong(cid:12)lters (sometimes several thousand taps), unpredictably time-variant environments, and highly non- stationary and auto-correlated signals. In addition, the simultaneous processing of multiple input streams, i.e., multichannel adaptive (cid:12)ltering (MC-ADF) is becoming more and more de- sirable for future products. Typical examples are multichannel acoustic echo cancellation(systemidenti(cid:12)cation) oradaptivebeamformingmicrophonear- rays (interference cancelling). Inthischapter,weinvestigateadaptiveMIMO(multipleinputandmulti- pleoutput)systemsthatareupdatedinthefrequencydomain.Theresulting generalizedmultichannelfrequency-domainadaptive(cid:12)lteringhasalreadyled toe(cid:14)cientreal-timeimplementationsofmultichannelacousticechocancellers on standard personal computers [2,3]. Generally, we distinguish two classes of adaptive algorithms. One class includes (cid:12)lters that are updated in the time domain, usually on a sample- by-sample basis,like the classicalleast-mean-square(LMS) [4] and recursive least-squares (RLS) [5] algorithms. The other class may be de(cid:12)ned as (cid:12)lters thatareupdatedinthediscreteFouriertransform(DFT)domain(‘frequency domain’), block-by-block in general, using the fast Fourier transform (FFT) as a powerful vehicle. As a result of this block processing, the arithmetic complexity of the latter category is signi(cid:12)cantly reduced compared to time- domain adaptive algorithms. The possibility to exploit the e(cid:14)ciency of FFT algorithms is due to the Toeplitz structure of the matrices involved, which results from the transversal structure of the adaptive (cid:12)lters. The Toeplitz matrices can be expressed by circulant matrices which are diagonalizableby the DFT. Consequently, the key for deriving the frequency-domain adaptive algorithms is to formulate the time-domain error criterion so that Toeplitz and circulant matrices are explicitly shown. In addition to the low complexity, another advantage resulting from this diagonalization in frequency-domain adaptive (cid:12)ltering is that the adapta- tion stepsize can be normalized independently for each frequency bin, which results in a more uniform convergence overthe entire frequency range. Single-channelfrequency-domainadaptive(cid:12)lteringwas(cid:12)rstintroducedby Dentinoetal.,basedontheleast-mean-squares(LMS)algorithminthetime- domain [6]. Ferrara[7] wasthe (cid:12)rst to present an e(cid:14)cient frequency-domain adaptive (cid:12)lter algorithm (FLMS) that converges to the optimum (Wiener) solution.MansourandGray[8]derivedanevenmoree(cid:14)cientalgorithm,the unconstrained FLMS (UFLMS), using only three FFT operations per block instead of (cid:12)ve for the FLMS, with comparable performance [9]. However, in some applications, a major handicap with these structures is the delay in- 4 Multichannel AdaptiveFiltering, Acoustic Echo Cancellation 97 troduced between input and output. Indeed, for e(cid:14)cient implementations, this delay is equal to the length L of the adaptive (cid:12)lter, which is consider- able for applications like acoustic echo cancellation. A new structure called multi-delay (cid:12)lter(MDF), usingthe classicaloverlap-save(OLS)method,was proposed in [10,11] and generalized in [12] where the new block size N was made independent of the (cid:12)lter length L; N can be chosen as small as de- sired, with a delay equal to N. Although from a complexity point of view, the optimum choice is N = L, using smaller block sizes (N < L) in order to reduce the delay is still more e(cid:14)cient than time-domain algorithms. A more general scheme based on weighted overlapand add (WOLA) methods, the generalized multidelay (cid:12)lter (GMDF(cid:11)) was proposed in [13,14], where (cid:11) is the overlap factor. The settings (cid:11) > 1 appear to be very useful in the contextof adaptive(cid:12)ltering,sincethe (cid:12)lter coe(cid:14)cientscanbe adaptedmore frequently (every N=(cid:11) samples instead of every N samples in the conven- tional OLS scheme) and the delay can be (further) reduced as well. Thus, this structure introduces one more degree of freedom, but the complexity is increasedroughlybyafactor(cid:11).TakingtheblocksizeintheMDFaslargeas the delay permits will increase the convergence rate of the algorithm, while choosing the overlapfactor greaterthan 1 will increase the tracking abilities of the algorithm. The case of multichannel adaptive (cid:12)ltering, as shown in Fig. 4.1, has been found to be structurally more di(cid:14)cult in general. In typical scenarios, the input signals x (n), p = 1;:::;P, to the adaptive (cid:12)lter are not only p auto-correlated but also highly cross-correlated which often results in very slow convergence of the LP (cid:12)lter coe(cid:14)cients h^ (n), where (cid:20) = 0;:::;L(cid:0) p;(cid:20) 1. This problem becomes particularly severe in multichannel acoustic echo cancellation [15{17], where the signals x (n) represent loudspeaker signals p that may originate from common sources. Signal y(n) represents an echo received by a microphone. y(n) x(n) 1 ^ h(n) . 1 y^(n) e(n) .. ... ... + - + x(n) P h^ (n) P Fig.4.1. Multichannel adaptive(cid:12)ltering. Direct application of commonly used low-complexity algorithms, such as the LMS algorithm or conventional frequency-domain adaptive (cid:12)ltering, to the multichannel case usually leads to disappointing results as the cross- 98 Herbert Buchneret al. correlations between the input channels are not taken into account [18]. In contrast to this, high-ordera(cid:14)ne projection algorithms and RLS algorithms do take the cross-correlations into account. Indeed, it can be shown that the RLS providesoptimum convergencespeed even in the multichannel case [18], but its complexity is prohibitively high and, e.g., will not allow real- time implementation of multichannel acoustic echo cancellation on standard hardware any time soon. Two-channel frequency-domain adaptive (cid:12)ltering was (cid:12)rst introduced in [19]inthecontextofstereophonicacousticechocancellationandderivedfrom the extended least-mean-squares(ELMS) algorithm [20] in the time domain using similar considerations as for the single-channel case outlined above. The rigorousderivation of frequency-domain adaptive (cid:12)ltering presented in the next section leads to a generic algorithm with RLS-like properties. We will also see that there is an e(cid:14)cient approximation of this algorithm taking the cross-correlationsinto account. The single-channel version of this algorithm provides a direct link to existing frequency-domain algorithms. Theorganizationofthischapterisasfollows.InSection4.2,weintroduce afrequency-domainrecursiveleast-squarescriterionfromwhichtheso-called normal equation is derived. Then, from the normal equation, we deduce a generic multichannel adaptive algorithm that we can formulate in di(cid:11)erent ways,andweintroducethe so-calledfrequency-domainKalmangain.InSec- tion4.3,westudythe convergenceofthismultichannelalgorithm.InSection 4.4, we consider the general MIMO case and, in Section 4.5, we give a very useful approximation, deduce some well-known single-channel algorithms as special cases, and explicitly show how the cross-correlations are taken into account in the multichannel case. We also give design rules for some impor- tantparameterssuchastheexponentialwindow,regularization,andadapta- tion stepsize. A useful dynamical regularizationmethod is discussed in more detail in Section 4.6. Section 4.7 introduces several methods for increasing computationale(cid:14)ciencyinthemulti-inputandMIMOcases,suchasarobust recursiveKalmangaincomputation and FFT computation tailoredfor over- lapping datablocks. Section 4.8presentssome simulationsand multichannel real-world implementations for hands-free speech communications. Finally, our results are summarized in Section 4.9. 4.2 General Derivation of Multichannel Frequency-Domain Algorithms In the (cid:12)rst part of this section we formulate a block recursive least-squares criterion in the frequency domain. Once the criterion is rigorously de(cid:12)ned, the adaptive multichannel algorithm follows immediately. 4 Multichannel AdaptiveFiltering, Acoustic Echo Cancellation 99 4.2.1 Optimization Criterion To obtain an optimization criterion for block adaptation, we assume the (generally time-variant) adaptive (cid:12)lter coe(cid:14)cients h^p;0;:::;^hp;L(cid:0)1 for the input channels 1;:::;P to be (cid:12)xed within the block intervals of length N. Forconvenienceofnotation,thisallowsustoomitthetimeindexofthe(cid:12)lter coe(cid:14)cients during the following derivation of the block error signal. From Fig. 4.1, it can be seen that the errorsignal at time n between the output of the multichanneladaptive(cid:12)ltery^(n) andthe desiredoutput signal y(n) is given by e(n)=y(n)(cid:0)y^(n); (4.1) with P L(cid:0)1 y^(n)= x (n(cid:0)(cid:20))h^ : (4.2) p p;(cid:20) p=1(cid:20)=0 XX Bypartitioningthe impulse responses h^ into segmentsof length N asin p [10,11], (4.2) can be written as P K(cid:0)1N(cid:0)1 y^(n)= x (n(cid:0)Nk(cid:0)(cid:20))h^ ; (4.3) p p;Nk+(cid:20) p=1 k=0 (cid:20)=0 X X X where we assume that the total (cid:12)lter length L is an integer multiple of N (N (cid:20)L), so that L=KN. For convenient notation of the multichannel algorithms, we rewrite (4.3) in vectorized form P K(cid:0)1 P y^(n)= xT (n)h^ = xT(n)h^ =xT(n)h^; (4.4) p;k p;k p p p=1 k=0 p=1 X X X where x (n) =[x (n(cid:0)Nk);x (n(cid:0)Nk(cid:0)1);(cid:1)(cid:1)(cid:1);x (n(cid:0)Nk(cid:0)N +1)]T;(4.5) p;k p p p h^p;k =[h^p;Nk;^hp;Nk+1;(cid:1)(cid:1)(cid:1);^hp;Nk+N(cid:0)1]T; (4.6) x (n) =[xT (n);xT (n);(cid:1)(cid:1)(cid:1);xT (n)]T; (4.7) p p;0 p;1 p;K(cid:0)1 h^ =[h^T ;h^T ;(cid:1)(cid:1)(cid:1);h^T ]T; (4.8) p p;0 p;1 p;K(cid:0)1 x(n) =[xT(n);xT(n);(cid:1)(cid:1)(cid:1);xT(n)]T; (4.9) 1 2 P h^ =[h^T;h^T;(cid:1)(cid:1)(cid:1);h^T]T: (4.10) 1 2 P Superscript T denotes transposition of a vector or a matrix. The length-N vectors h^ , k = 0;:::;K (cid:0)1, represent sub-(cid:12)lters of the partitioned tap- p;k weight vector h^ of channel p. p 100 Herbert Buchneret al. Wenowde(cid:12)netheblockerrorsignaloflengthN.Basedon(4.1) and (4.4) we write e(m)=y(m)(cid:0)y^(m); (4.11) with m being the block time index, and P K(cid:0)1 P y^(m)= UT (m)h^ = UT(m)h^ =UT(m)h^; (4.12) p;k p;k p p p=1 k=0 p=1 X X X where e(m)=[e(mN);(cid:1)(cid:1)(cid:1);e(mN +N (cid:0)1)]T; (4.13) y(m)=[y(mN);(cid:1)(cid:1)(cid:1);y(mN +N (cid:0)1)]T; (4.14) y^(m)=[y^(mN);(cid:1)(cid:1)(cid:1);y^(mN +N (cid:0)1)]T; (4.15) U (m)=[x (mN);(cid:1)(cid:1)(cid:1);x (mN +N (cid:0)1)]; (4.16) p;k p;k p;k U (m)=[UT (m);(cid:1)(cid:1)(cid:1);UT (m)]T; (4.17) p p;0 p;K(cid:0)1 U(m)=[UT(m);(cid:1)(cid:1)(cid:1);UT(m)]T: (4.18) 1 P Itcaneasilybeveri(cid:12)edthatU ,p=1;:::;P,k =0;:::;K(cid:0)1areToeplitz p;k matrices of size (N (cid:2)N): x (mN (cid:0)Nk) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) x (mN (cid:0)Nk(cid:0)N +1) p p UT (m)=2 xp(mN (cid:0)Nk+1) ... ... 3 p;k .. .. .. .. 6 . . . . 7 6 7 6x (mN (cid:0)Nk+N (cid:0)1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) x (mN (cid:0)Nk) 7 6 p p 7 4 5 These Toeplitz matrices are now diagonalized in two steps: Step 1: Transformation of Toeplitz matrices into circulant matrices. Any Toeplitz matrix U can be transformed, by doubling its size, to a p;k circulant matrix U0T (m)UT (m) Cp;k(m)= UpT;k(m)Up0T;k(m) ; (4.19) (cid:20) p;k p;k (cid:21) where the U0 are also Toeplitz matrices and can be expressed in terms of p;k the elements of UT (m), except for an arbitrary diagonal, e.g., p;k x (mN (cid:0)Nk(cid:0)N) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) x (mN (cid:0)Nk+1) p p U0T (m)=2xp(mN (cid:0)Nk(cid:0)N +1) ... ... 3: p;k 6 ... ... ... ... 7 6 7 6 x (mN (cid:0)Nk(cid:0)1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) x (mN (cid:0)Nk(cid:0)N)7 6 p p 7 4 5
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