Jacob Benesty Arden (Yiteng) Huang Adaptive Signal Processing

39 Pages·2012·0.42 MB·English

Jacob Benesty
Arden (Yiteng) Huang
Adaptive Signal Processing:
Application to Real-World Problems
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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