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Springer Topics in Signal Processing Volume 4 SeriesEditors J.Benesty,Montreal,QC,Canada W.Kellermann,Erlangen,Germany Springer Topics in Signal Processing EditedbyJ.BenestyandW.Kellermann Vol.1:Benesty,J.;Chen,J.;Huang,Y. MicrophoneArraySignalProcessing 250p.2008[978-3-540-78611-5] Vol.2:Benesty,J.;Chen,J.;Huang,Y.;Cohen,I. NoiseReductioninSpeechProcessing 240p.2009[978-3-642-00295-3] Vol.3:Cohen,I.;Benesty,J.;Gannot,S.(Eds.) SpeechProcessinginModernCommunication 360p.2010[978-3-642-11129-7] Vol.4:Benesty,J.;Paleologu,C.;Gänsler,T.;Ciochina˘,S. APerspectiveonStereophonicAcousticEchoCancellation 139p.2011[978-3-642-22573-4] · Jacob Benesty Constantin Paleologu · Tomas Gänsler Silviu Ciochina˘ A Perspective on Stereophonic Acoustic Echo Cancellation ABC Prof.Dr.JacobBenesty Dr.TomasGänsler INRS-EMT mhacousticsLLC UniversityofQuebec Summit,NewJersey H5A1K6Montreal,QC USA Canada Email:[email protected] Email:[email protected] Prof.Dr.ConstantinPaleologu Prof.Dr.SilviuCiochina˘ UniversityPolitehnicaofBucharest PolitehnicaUniversityofBucharest TelecommunicationsDepartment TelecommunicationsDepartment 061071Bucharest 061071Bucharest Romania Romania Email:[email protected] Email:[email protected] ISBN978-3-642-22573-4 e-ISBN978-3-642-22574-1 DOI10.1007/978-3-642-22574-1 SpringerTopicsinSignalProcessing ISSN1866-2609 e-ISSN1866-2617 LibraryofCongressControlNumber:2011933378 (cid:2)c 2011Springer-VerlagBerlinHeidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks. DuplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGerman CopyrightLawofSeptember9,1965,initscurrentversion,andpermissionforusemustalways beobtainedfromSpringer.ViolationsareliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoes notimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. CoverDesign:WMXDesignGmbH,Heidelberg Printedonacid-freepaper 987654321 springer.com Contents 1 Introduction.............................................. 1 1.1 Stereophonic Acoustic Echo Cancellation (SAEC) .......... 1 1.2 Organizationof the Book................................ 3 References ................................................. 4 2 Problem Formulation ..................................... 5 2.1 Stereophonic Acoustic Echo Model........................ 5 2.2 Widely Linear (WL) Model .............................. 6 2.3 Measures .............................................. 9 References ................................................. 10 3 System Identification with the Wiener Filter ............. 13 3.1 Mean-Square Error (MSE) Criterion and Wiener Filter...... 13 3.2 Nonuniqueness Problem ................................. 16 3.3 Distortion for a Unique Solution.......................... 17 3.4 Deterministic Algorithm................................. 20 3.5 Regularized MSE Criterion .............................. 24 References ................................................. 26 4 A Class of Stochastic Adaptive Filters .................... 29 4.1 Least-Mean-Square (LMS) Algorithm ..................... 29 4.2 Performance of the LMS Algorithm....................... 30 4.3 Normalized LMS (NLMS) Algorithm...................... 34 4.4 Interpretation of the NLMS Algorithm .................... 35 4.5 Regularization of the NLMS Algorithm.................... 37 4.6 Variable Step-Size NLMS (VSS-NLMS) Algorithm.......... 39 4.7 Improved Proportionate NLMS (IPNLMS) Algorithm ....... 41 4.8 Regularization of the IPNLMS Algorithm.................. 44 4.9 VSS-IPNLMS Algorithm ................................ 45 4.10 Extended NLMS (ENLMS) Algorithm .................... 46 References ................................................. 47 vi Contents 5 A Class of Affine Projection Algorithms .................. 49 5.1 Affine Projection Algorithm (APA) ....................... 49 5.2 Interpretation of the APA ............................... 50 5.3 Regularization of the APA............................... 52 5.4 Variable Step-Size APA (VSS-APA) ...................... 55 5.5 Improved Proportionate APA (IPAPA).................... 59 5.6 Memory PAPA......................................... 60 References ................................................. 62 6 Recursive Least-Squares Algorithms ...................... 63 6.1 Least-Squares Error Criterion and Normal Equations ....... 63 6.2 Recursive Least-Squares (RLS) Algorithm ................. 65 6.3 Fast RLS (FRLS) Algorithm............................. 67 References ................................................. 69 7 Double-Talk Detection.................................... 71 7.1 Principles of a Double-Talk Detector (DTD) ............... 71 7.2 DTDs Based on the Holder’s Inequality ................... 73 7.3 DTD Based on Cross-Correlation......................... 75 7.4 DTD Based on Normalized Cross-Correlation .............. 76 7.5 Performance Evaluation of DTDs......................... 77 References ................................................. 78 8 Echo and Noise Suppression as a Binaural Noise Reduction Problem....................................... 81 8.1 Problem Formulation ................................... 81 8.2 WL Model............................................. 82 8.3 Performance Measures .................................. 84 8.3.1 Noise Reduction.................................. 84 8.3.2 Speech Distortion ................................ 86 8.3.3 MSE Criterion ................................... 87 8.4 Optimal Filters ........................................ 89 8.4.1 Maximum Signal-to-Noise Ratio (SNR) ............. 89 8.4.2 Wiener.......................................... 90 8.4.3 Minimum Variance Distortionless Response (MVDR) . 92 References ................................................. 93 9 Experimental Study ...................................... 95 9.1 Experimental Conditions ................................ 95 9.2 NLMS, VSS-NLMS, IPNLMS, and VSS-IPNLMS Algorithms 96 9.3 APA, VSS-APA, IPAPA, and MIPAPA.................... 114 9.4 FRLS Algorithm ....................................... 132 References ................................................. 133 Index......................................................... 137 Chapter 1 Introduction 1.1 Stereophonic Acoustic Echo Cancellation (SAEC) Research and development of stereophonic echo control systems has been a subjectof interestoverthe last20+years.Infact, one ofthe firstpapersde- scribing the characteristics of stereophonic echo cancellation was presented in 1991 [1]. In this paper, it is pointed out that the loudspeaker (input) sig- nals are linearly related through non-invertible acoustic room responses (in the case of one source, but not necessarily two or more). The consequence of this linear relationship is that the underlying normal (or Wiener-Hopf) equations to be solved by the adaptive algorithm is an ill-conditioned, or in the worst case, a singular problem. In the singular case, the adaptive filter can drift between candidates in the set of available nonunique solutions, all minimizing the varianceof the output error.However,these solutions do not necessarilyminimizefiltermisalignment.Asaresult,someunstablebehavior for certain time varying events may be expected. Even though the problem of nonuniqueness was described, analyzed, and solutions presented in early publications, e.g., [2], [3], [4], many following proposals have been confused over what has to be done to solve the problem correctly. Fundamentally, the coresolutionto the stereophonicacousticechocancellation(SAEC)problem musttackletwoissues:(a)provideapropersolutiontothe inherentill-posed problem of stereophonic echo cancellation and (b) mitigate the effect that the ill-posed problem has on the convergence rate and tracking of the adap- tive algorithm. The former problem (a), can only be solvedby manipulating the signals actually transmitted to the near-end (receiving) room, i.e., us- ing a preprocessor on the loudspeaker signals to decorrelate them (or more accuratelyto reduce the coherence)beforethe SAECas wellasbefore trans- mitting them to the far-end room. To see that this is the case, we can look at the normal equations the echo canceler has to solve, (cid:2) R(cid:2)xh =p(cid:2)xd, (1.1) J.Benestyetal.:APerspectiveonStereophonicAcousticEchoCancellation,STSP4,pp.1–4. springerlink.com (cid:2)c Springer-VerlagBerlinHeidelberg2011 2 1 Introduction where R(cid:2) is the correlation matrix of the excitation (loudspeaker) signal x (cid:2) (left and right stereo channels), h is the estimated echo path, and p(cid:2)xd is the cross-correlation between the excitation signal and the microphone signal. See Chapter 3 for more details of the problem formulation, notation, and normal equations. The estimated echo path is given by the solution to the normal equations (1.1), which is found to be (cid:3)2L (cid:2) (cid:2) h =ht+ βf,iqi, (1.2) i=R+1 (cid:2) wherehtisthetrueechopathoflength2L,qiareeigenvectors(corresponding to the nullspace of R(cid:2)x), R is the rank of the correlationmatrix, and βf,i are arbitrary factors. This solution is easily shown to be valid by using (1.1) and (1.2), (cid:3)2L (cid:2) (cid:2) R(cid:2)xh =R(cid:2)xht+ βf,iR(cid:2)xqi (1.3) i(cid:4)=R+1(cid:5)(cid:6) (cid:7) 0 =p(cid:2)xd. Note thatthe solution(1.2)is independent ofanyadaptivealgorithmwe use in our echo canceler system. Whatever adaptive algorithm used will end up with non-zero scalar-values (βf,i). It is clearly seen that we can only achieve a unique solutionif R=2L and this condition canonly be met if we modify (preprocess) the signals that actually excite the transmission room. Having concluded that preprocessing of far-end loudspeaker signals that actually are transmitted to the near-end room is the only way to achieve a uniquesolution,weturntothelatterproblem(b).Therehasbeenacommon misunderstanding in several publications that manipulating the adaptive al- gorithm to improve convergence rate solve (a) which is not true. However, usinganalgorithmtailoredtoexploitthecross-correlationbetweenthechan- nels addresses problem (b), i.e., it mitigates the effects of the ill-conditioned normal equations to be solved. Remember, even with sophisticated prepro- cessing,itis difficulttoachieveawell-conditionedsystem.Variousalgorithm choices for problem (b) have been presented in literature. For example, a natural choice of algorithmis the recursiveleast-squares (RLS) ([5], [6], [7]), which was the preferred algorithm in [1] and subsequent papers such as [8], [9]. In order to build a working echo cancellation system, it is crucial to con- troltheadaptivealgorithmproperlyduringdifferenttalkerconditions.Talker conditionsusuallyinclude; singletalkcases,i.e.,onlythefar-endornear-end talker is active, double-talk where both talkers are active simultaneously, as wellastheidleconditionwithneithersideactive.Anumberofcontrolmech- anismsarecommonlyemployedtocontrolthealgorithmsunderthesevarious 1.2 OrganizationoftheBook 3 conditionsandoneofthemostimportantisthe double-talkdetector(DTD). TheobjectiveoftheDTDistostopalgorithmdivergenceduringdouble-talk. Its functionality can either be incorporated directly into the adaptive algo- rithm,e.g.,asastep-sizecontrolmechanism,orasaseparatecontrolmodule. Becauseofitsimportanceandtheexistingwealthofpublicationsinthisarea a chapter in this book is solely devoted to this problem. Another equally important aspect of echo canceler systems is handling of residual echo, usu- ally referred to residual echo suppression or nonlinear processing (NLP) (of theresidualecho).Inarealisticacousticenvironment,linearcancellationcan neverprovidesufficientechoattenuationineverytalkercondition.Tohandle loud echoes, e.g., at initial convergence, echo path changes, or large acoustic coupling conditions, echo suppression is required to complement the linear echo canceler. Aspects of combined residual echo and noise suppression is therefore presented as a separate chapter. 1.2 Organization of the Book The objectives of this book are to recast the stereophonic echo cancellation problem using the widely linear (WL) model, as well as in this framework presentandanalyzesomeofthetypicalalgorithmsappliedtothestereophonic case.Chapter2describesthestereophonicechocancellationproblemasaWL model and redefines some of the evaluation criteria commonly used in echo cancellation. General identification of the stereophonic echo paths using the Wiener formulation in the WL stereo framework is discussed in Chapter 3. Thischapteralsoanalyzesthenonuniquenessproblemandpresentsanewap- proach to preprocessing the loudspeaker signals. Three chapters are devoted to classical as well as improved variants of adaptive filters for the SAEC problem. Stochastic gradient methods, of which the normalized least-mean- square (NLMS) algorithm belongs, is the topic of Chapter 4. This chapter alsodiscussesindetailhowtoappropriatelyregularizethealgorithms.Regu- larization is extremely important for practical implementations of echo can- celers. Moreover, variable step-size control for NLMS based algorithms are presented. For the stereophonic problem, the ability of the adaptive algo- rithm to exploit the spatial correlation between the channels is important. A family of algorithms with this ability is based on affine projections (APs). Chapter 5 goes into details of these algorithms. AP algorithms (APAs) have less degrees of freedom for spatial decorrelation compared to RLS based al- gorithms. However, the APA is less computationally complex compared to the RLS and is therefore an interesting alternative for realtime implementa- tions. RLS adaptive filters are the most flexible algorithms when it comes to handling the problems occurring in stereophonic echo cancellation systems. Hence, afull derivationofthe WL model-basedRLSas well asa fastversion are described in Chapter 6. The problems of double-talk and residual echo 4 1 Introduction and noise handling are treated in Chapters 7 and 8, respectively. Chapter 9 presents extensive simulation results from most of the algorithms described in previous chapters. References 1. M. M. Sondhi and D. R. Morgan, “Acoustic echo cancellation for stereophonic tele- conferencing,”inProc. IEEE WASPAA,1991. 2. M.M.Sondhi,D.R.Morgan,andJ.L.Hall,“Stereophonicacousticechocancellation– An overview of the fundamental problem,” IEEE Signal Process. Lett., vol. 2, pp. 148–151, Aug.1995. 3. F. Amand, A. Gilloire, and J. Benesty, “Identifying the true echo path impulse re- sponses in stereophonic acoustic echo cancellation,” in Proc. EUSIPCO, pp. 1119– 1122,1996. 4. J. Benesty, D. R. Morgan, and M. M. Sondhi, “A better understanding and an im- proved solution to the specific problems of stereophonic acoustic echo cancellation,” IEEE Trans. Speech, Audio Process., vol.6,pp.156–165, Mar.1998. 5. J. Cioffi and T. Kailath, “Fast recursive-least-squares transversal filters for adaptive filtering,” IEEE Trans. Acoust., Speech, Signal Process., vol. 34, pp. 304–337, Apr. 1984. 6. M.G.Bellanger,Adaptive Filters and Signal Analysis.NY:Dekker,1988. 7. M.G.BellangerandP.A.Regalia,“TheFRL-QRalgorithmforadaptivefiltering:the caseofmultichannel signal,”Signal Process.,vol.22,pp.115–126, Feb.1991. 8. J.Benesty, F.Amand,A.Gilloire,andY.Grenier,“Adaptive filteringalgorithmsfor stereophonicacousticechocancellation,”inProc.IEEEICASSP,pp.3099–3102,1995. 9. P. Eneroth, S. L. Gay, T. G¨ansler, and J. Benesty, “A real-time stereophic acoustic subband echo canceler,” in Acoustic Signal Processing for Telecommunication, S. L. GayandJ.Benesty,eds.,KluwerAcademicPublishers,2000,Chapter8,pp.135–153.

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