New Insights Into the Stereophonic Acoustic Echo Cancellation Problem and an Adaptive Nonlinearity Solution

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1 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL 10, NO 5, JULY New Insights Into the Stereophonic Acoustic Echo Cancellation Problem and an Adaptive Nonlinearity Solution Tomas Gänsler, Member, IEEE, and Jacob Benesty, Member, IEEE Abstract In this paper, we expand the knowledge regarding the problems of two-channel (or stereophonic) echo cancellation The major difference between two-channel and the single-channel echo cancellation is the problem of nonunique solutions in the twochannel case In previous work, this nonuniqueness problem has been linked to the coherence between the two incoming audio channels One proven solution to this problem is to distort the signals with a nonlinear device In this work, we present new theory that gives insight to the existing links between: i) Coherence and level of distortion, and ii) coherence and achievable misalignment of the stereophonic echo canceler Furthermore, we present an adaptive nonlinear device that incorporates this new knowledge in such a way that a pre-specified maximum misalignment is maintained while improving the perceived quality by minimizing the introduced distortion Moreover, all the ideas presented can be generalized to the multichannel ( 2) case Index Terms Adaptive filter, coherence, frequency-domain, misalignment, nonuniqueness, stereo I INTRODUCTION IT HAS BEEN shown that a two-channel echo cancellation (EC) system needs special attention because of its inherent nonuniqueness problem, ie, if the multiple audio streams (in, eg, teleconferencing) originate from the same source, the echo canceler cannot provide a unique echo path solution [1], [2] The only way to mitigate this nonuniqueness problem is to diminish the linear relation between the audio channels, ie, to decorrelate them [2] This decorrelation must of course be done carefully so that the stereo effect is not degraded, and the introduced distortion is (almost) inaudible A successful method for decorrelating the signals, by means of a static nonlinearity, was proposed in [2] Since these results were published, many other proposals have been presented for solving the nonuniqueness problem [3] [5] However, when multiple talkers are active, or when there is background music playing, there is no need for decorrelation (in theory), because the normal equation to be solved by the echo canceler in this case is indeed nonsingular Recently, a software stereo echo cancellation system [6] has been developed from the ideas presented in [7] [9] Experi- Manuscript received June 25, 2001; revised April 10, 2002 The associate editor coordinating the review of this manuscript and approving it for publication was Dr Dirk van Compernolle T Gänsler is with the Media Signal Processing Research Department, Agere Systems, Murray Hill, NJ USA ( gaensler@agerecom) J Benesty is with Lucent Technologies, Bell Labs, Murray Hill, NJ USA ( jbenesty@bell-labscom) Publisher Item Identifier /TSA ments with this system, which is the first ever built, have shown that in a high quality speech transmission, the necessary distortion introduced may be audible These observations inspired the idea of using an adaptive nonlinearity in order to reduce the channel coherence (ie, the channel correlation) in a twochannel EC system Assuming that we can measure the channel coherence and this coherence is high (close to 1), we add distortion On the other hand, if the coherence is low ( 09), eg, when music is playing in the background of a conversation, we do not add any distortion This would certainly improve the overall perceived quality of the two-channel system To properly control the level of nonlinearity, from the view of both the performance of the EC and the allowed perceived distortion, more knowledge is needed of how the nonlinearity influences these properties Investigations of perception of different nonlinearities and a framework for analyzing their influence on channel coherence have been made in [10] Results from those investigations will be used in this paper to set bounds on the level of nonlinearity in order to achieve a certain perceptual quality Moreover, new important links between the level of nonlinearity, coherence, and misalignment will be derived for performance control of the EC This paper is organized as follows Section II introduces the definitions, notation, and the established theory needed for describing the two-channel echo cancellation problem New insights into the misalignment problem and its links to coherence and level of nonlinearity will be presented in Section III In Section IV, we propose an adaptive nonlinearity for mitigating the nonuniqueness problem and controlling the misalignment according to specifications, while maintaining a reasonable perceptual quality of the transmitted signals Section V verifies our theoretical concepts of the adaptive nonlinearity with simulation results and compares these with existing techniques for decorrelation Section VI summarizes the results and presents our conclusions II BACKGROUND TO THE NONUNIQUENESS PROBLEM The two-channel teleconference situation is shown in Fig 1 Communication is hands-free between a transmission room and a receiving room we denote the signals picked up by the microphones in the transmission room by, and the return signal picked up by one of the microphones in the receiving room by The receiving room signal is in general composed of echo, ambient noise, and possibly re /02$ IEEE

2 258 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL 10, NO 5, JULY 2002 The minimization of this criterion leads to the normal equation (8) is an estimate of the input signal covariance matrix and (9) Fig 1 Block diagram of a generic two-channel acoustic echo canceler ceiving room speech, which in this scenario is referred to as double-talk Thus, we have the receiving room signal model is the echo, denotes convolution and are the receiving room echo paths Define the two input excitation vectors as the superscript denotes transposition The error signal at time between one arbitrary microphone output in the receiving room and its estimate is (1) (2) (3) (10) is an estimate of the cross-correlation vector between the input and output signals (in the receiving room) If there is a single audio source as shown in Fig 1, the transmission room signals, will be linearly related through the transmission room paths, Under this assumption, it can easily be shown that the normal equation (8) is singular [2], [9] Therefore, the echo canceler has an infinite set of solutions, and all except the true receiving room response ( ), are dependent on the transmission room paths and This forces the echo canceler to reconverge whenever there is an echo path change in the transmission room (the change may be caused by, eg, a change of talkers or a moving talker) as well as the receiving room (normally caused by talkers moving) This is in contrast to single-channel echo cancelers only echo path variation in the receiving room has to be considered The singularity, or closeness to singularity, of the normal equation (8) is revealed by the condition number of the covariance matrix (9) Assuming that, it was shown in [2] that the eigenvalues and hence the condition number of the covariance matrix, can be obtained by trivially finding the roots of a product of second order polynomials Assuming that, ( ), we find the eigenvalues are given by (4) and are the two modeling filters In general, we have two microphones in the receiving room, ie, two return channels Hence, we need four adaptive filters for the general two-channel case However, we will just consider one return channel, ie, two filters, because the derivation is similar for the other channel Let us look at the recursive least-squares error criterion with respect to the modeling filters is a forgetting factor (5) (6) (7) (11), are the auto-spectra of the signals and at frequency respectively, is the cross-spectrum between the signals, denotes the eigenvalues of the matrix (9), and (12) is the coherence function between the two signals Expression (11) shows that the minimum eigenvalue is lower bounded by the factor and that if any, then the covariance matrix (9) is singular Hence, a practical method to monitor the closeness to singularity of the normal equation is to use the coherence, or magnitude coherence, between the transmission signals

3 GÄNSLER AND BENESTY: STEREOPHONIC ACOUSTIC ECHO CANCELLATION PROBLEM 259 To solve the nonuniqueness problem it has been proposed [2] that the transmission room signals, and should be distorted in such a way that they no longer are linearly related, then (8) becomes positive definite and only one solution to this equation exists [ie, ] for the echo canceler to identify An effective method for distorting the signals is to pass one channel through a positive and the other channel through a negative half-wave rectifier [11], ie, (13a) (13b) quantifies the level of introduced nonlinearity (distortion) This method has been shown (by listening tests) not to degrade the stereo effect and it is virtually inaudible (for speech signals) for smaller than 05 For music it may be objectionable Using for high quality speech (ie, wideband, 8 khz bandwidth), may also be objectionable for some people This is most likely due to the fact that the rectifier boosts higher frequencies which becomes audible because of poor masking from the original speech at these frequencies Moreover, it is customary to normalize (13) with the factor in order to have the same average signal power before and after processing Ideally, we would like to pass uncorrelated audio streams without distorting them so that a high audio quality is maintained in the receiving room However, when the signals originate from the same source, ie, when they are linearly related or highly coherent, we would like to introduce some distortion and thereby avoid the problem of nonuniqueness for the echo canceler This can actually be done by monitoring the magnitude coherence function of the transmission room signals as described above, and using its value to control the level of nonlinearity ( ) in (13) Furthermore, by using a certain adaptive algorithm in the echo canceler, we can obtain an estimate of the magnitude coherence with very low computational complexity This algorithm, the two-channel frequency-domain algorithm [7], [9], [12], computes the magnitude coherence explicitly in order to update the estimate of the echo path In the following section, we will investigate the links that exist between the misalignment and coherence, and between the level of nonlinearity and coherence When these links have been established, we can propose a solution that adaptively adjusts the level of nonlinearity according to performance specifications of the echo canceler III USEFUL LINKS BETWEEN NONLINEARITY, COHERENCE, AND MISALIGNMENT To find a good way of adjusting the level of nonlinearity, we need answers to the following questions i) How does the level of nonlinearity ( ) influence the level of coherence ( )? ii) How does the level of coherence influence the misalignment ( )? (The misalignment is defined as the norm of the estimation error of the echo path: ) A Link Between Nonlinearity and Coherence The first question i) can be answered by looking at a theoretical model (in the same fashion that was done in [10]) for the coherence between the two signals that have passed through a nonlinearity However, a modification of the ideas in [10] is necessary for the positive and negative half-wave rectifier Starting from (13), we would like to find the coherence between the signals processed by the nonlinearity as a function of the spectra and cross-spectra of, and level of nonlinearity ( ) In [10], by assuming Gaussian distributed signals, it was shown that the coherence between and (denoted by ) is given by (14) is a constant depending on the nonlinear function (in our case, the half-wave rectifier) and are the cross- and auto-spectra (of the corresponding signal, in this case and ) as discussed in the previous section We can compute these spectra from their corresponding (cross-)correlation functions using the Fourier transform (15) Furthermore, for the positive and negative half-wave rectifier (13), we have [10] (16) We now want to evaluate (14) for our situation Hence,, as well as, have to be computed for an assumed model of the signals In this section, we choose to model the transmission signals and as constant spectrum (white) Gaussian signals The coherence between the signals is also constant, and the signals are band-limited in frequency between with variance (The sampling frequency is denoted by ) We refer to this model as the anechoic model After sampling, we get (17a) (17b) is the unity impulse function and the time lag variable, and is a possible time shift between the channels 1 Applying (15) to this model results in and (18a) (18b) (19) 1 For simple time shifts, identical nonlinearities in each path, as in [2] does not decorrelate the signals However, the positive and negative half-wave rectifier of (13) resolves this problem

4 260 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL 10, NO 5, JULY 2002 Hence, the magnitude of the coherence between the channels before passing the nonlinearity (positive and negative half-waverectifier) is constant and equal to for this model This may not look like an appropriate model for a real-life situation However, for speech in an office environment, it works well as will be shown in our simulations We can motivate the flat magnitude coherence since the average speech spectrum rolls off with about 6 db/octave (above 1 khz) and room noise can be modeled as noise (ie, it rolls off with 3 db/octave) Hence, this speech-to-noise ratio results in a fairly flat magnitude coherence between the channels Computing the spectra, is somewhat more complicated Expressions for, can be found as a function of, by using the methods outlined in [13] In [10], we find expressions for auto correlation of the signals and (25b) which together with (14) gives us the desired link between the nonlinearity variable [through, (16)] and the coherence between the transmission signal after they have passed through the positive and negative half-wave rectifier Furthermore, for, wehave (26) (20) is the normalized correlation function given by (21) The normalized cross-correlation function is analogously defined In [10], the result for the negative half-wave rectifier it is not explicitly given However, the sign difference between positive and negative rectifiers disappears in the autocorrelation function The cross-correlation between a positive half of a signal ( ) and a negative half ( ), on the other hand, needs some special attention The simplest way to find this function is to observe that (27) Hence, we can now conclude that the answer to question i) is given by (27) with the assumption of the anechoic model In Section IV we will present a method for adjusting the nonlinearity level properly, For that method, it is useful to rewrite the function in (27) as follows: (28) (29) is in closed form as shown in (30) at the bottom of the page, and is found as a simple recursion (22) The function is known from (20) and using the fact that for signals with symmetric probability density [10] (23) we finally conclude by using (20), (22), and (23) that (24) For the model we have adopted here (17), we can now compute the corresponding spectra of (20) and (24) as (25a) (31) B Link Between Coherence and Misalignment The second question ii) is given by the misalignment formula for the two-channel frequency-domain algorithm 2 A simple closed form expression for the misalignment is given in [9, Ch 2 The resulting formula can also be derived for the two-channel recursive least-squares (RLS) algorithm However, we prefer to use the frequency-domain approach because of the usefulness of the frequency-domain adaptive algorithm in practice (30)

5 GÄNSLER AND BENESTY: STEREOPHONIC ACOUSTIC ECHO CANCELLATION PROBLEM 261 9] In this paper, however, we will improve the misalignment formula so that it includes the coherence between the transmission signals For the two-channel frequency-domain adaptive algorithm [9], it can be shown that the expected normalized misalignment energy is given by (32) is the near-end speech and ambient noise power, is the block time index, and is a spectral matrix of the transmission signals This matrix is defined as (33) denotes expectation and, which is the spectral density matrix of the far-end signals, is defined in detail in the Appendix Equation (32) is found under the assumption that varies slowly, ie, As shown in the Appendix, we can express this spectral matrix as (34) the Fourier matrix,, is defined in the Appendix and the Toeplitz correlation matrix is defined as is still constant with frequency, ie,, We then find by combining (32), (37), and (38) EBR is the echo-to-background ratio (39) (40) The excess misalignment (abbreviated, ex mis ) is the term in (39) that is solely dependent on the channel coherence (41) Fig 2 shows the excess misalignment as a function of magnitude-squared coherence it is assumed that the coherence is equal for all frequencies The lower bound of (41) is achieved when the channels are uncorrelated ( ) We can now make an interesting connection between misalignment as it is defined in (32) and the (normalized) excess mean-squared error (MSE) defined as (42) Using the analysis results for convergence in the mean-square in [9, Ch 8], we find that (35) If we let, the Toeplitz correlation matrices, are diagonalized by the Fourier matrices and, hence we find (36),, are spectra and cross-spectrum vectors for the normalized frequencies, Provided that the coherence is not equal to 1 for any frequency, as described in Section II, the inverse of this 2 2 block-diagonal is (43) ie, the mean-square error is independent of the channel coherence In the case of a fully excited signal space [ie, ] the misalignment coincides with the mean-square error However, for the case of coherent channels, the misalignment is always greater than the MSE What is even more interesting is the following: If we have an under-excited two-channel system ( ), and there is an (instantaneous) echo path change in the transmission room, the (normalized) power that will go through the system is This means that the difference in MSE before and just after a transmission room echo path change ( )is is the diagonal coherence matrix (37) (38) Now, to be somewhat more general than (17), assume a model,,isnot constant with frequency, but (44) To conclude, by looking at (39), the (total) misalignment can be controlled by changing the forgetting factor of the adaptive algorithm Furthermore, what is also evident from (39) is that we can lower the excess misalignment by manipulating the channel coherence For a certain desired maximum excess misalignment, we can compute a required using (41) and finally the required level of nonlinearity using (29) and (28) A more detailed description of this method will be given next

6 262 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL 10, NO 5, JULY 2002 TABLE I TWO-CHANNEL FREQUENCY-DOMAIN ECHO CANCELER WITH MODIFICATIONS USED IN ALL SIMULATIONS SEE [9] FOR MORE DETAILS Fig 2 Excess misalignment due to channel coherence Under the assumption that the coherence is equal for all frequencies, this result is valid for an arbitrary L IV PROPOSED SOLUTION: THE ADAPTIVE NONLINEARITY In this section, we propose a method for adjusting the level of the nonlinearity in such away that performance specifications of the echo canceler are met and at the same time, the perceived distortion introduced by the nonlinearity is minimized The basis for this adaptive nonlinearity is the ability to estimate channel coherence A two-channel frequency-domain adaptive algorithm for echo cancellation is given in Table I For each iteration, this algorithm uses a block of samples to update the estimated echo path,, (45) and is given in (6) Looking at the frequency-domain algorithm in Table I, we find that the magnitude-squared coherence (for the transmission signals after the nonlinearity) is estimated as (46) is the estimated coherence at frequency for time block We will exploit this intermediate step from the adaptive echo cancellation algorithm in our adaptive nonlinearity Assume we have the following situation We would like to keep the excess misalignment below a certain desired level and furthermore, we assume that the coherence is constant with frequency, then we have This means that we have a desired magnitude coherence given by (47) Equation (41) and the main diagonal of (46) can be used for estimation of the excess misalignment according to From this estimate, an average magnitude coherence can be calculated that results in an equivalent amount of excess misalignment, ie, (48) We now adjust the nonlinearity that is applied to the next block of data in such a way that This is done by first computing an estimate of the coherence of the unprocessed signals (before the half-wave rectifiers) using (29) (49)

7 GÄNSLER AND BENESTY: STEREOPHONIC ACOUSTIC ECHO CANCELLATION PROBLEM 263 then applying (28) together with (49) and the desired coherence of (47) In practice, we bound this estimate according to (50) (51) in order to preserve the perceived quality of the signals 3 A suitable level for is 05 V SIMULATIONS In this section, we will verify our derived expressions for the link between the nonlinearity and the coherence (26) and the link between the coherence and the misalignment (39) Furthermore, we will present an example of the performance of the proposed adaptive nonlinearity (51) and compare this result with what is achieved using a constant level of nonlinearity in (13) A Evaluation of Theoretical Links Between Nonlinearity, Coherence, and Misalignment For these experiments, we use signals generated such that they obey the model given in (17) Let,, and be white, mutually uncorrelated Gaussian distributed signals with equal variance The transmission signals are then generated by (52a) Fig 3 Comparison of theoretical (solid) and estimated (3) magnitude coherence after processing the transmission signals with a positive and a negative half-wave rectifier (13) The magnitude coherence of the signal before the rectifiers are found at =0 (52b) is defined in (19) We start by comparing (26), ie,, with estimates of the coherence function of the signals (52) after they have passed through the nonlinear function given in (13) This comparison is done for different nonlinearity levels ( ) and coherence ( ) of the unprocessed signals The magnitude coherence of our signal model is equivalent to the normalized correlation coefficient when, ie, the cross-correlation normalized by the individual standard deviation of the signals (all mean values are removed) Hence, the coherence values are estimated according to Fig 4 Echo path response used in simulations is the length of the data Here, we have chosen The results from this comparison are shown in Fig 3 we find that theory and simulation completely agree over our choice of range of the parameters: and Next, we evaluate the misalignment formula (39) and compare it with estimates The receiving room echo paths ( ) 3 A simple alternative to (49) and (50) is: ^ (m +1) = ^ (m) + C(j^ (m)j 0j j) C is a feedback constant Fig 5 Comparison of theoretical (solid) and estimated misalignment (3)asa function of magnitude-squared coherence

8 264 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL 10, NO 5, JULY 2002 TABLE II ADAPTIVE NONLINEARITY USED IN THE SIMULATIONS THE MAGNITUDE OPERATOR j 1 jis APPLIED ELEMENTWISE Fig 6 Example of estimated magnitude-squared coherence for measured of the algorithm speech signals: (a) with small regularization = in Table I and (b) with normal regularization =5 we use are shown in Fig 4 Fig 5 shows the theoretical misalignment, given by expression (39), and the estimated misalignment that results from using the frequency-domain adaptive algorithm in Table I Here, the magnitude coherence ( ) is varied between 0 and 1 The final estimate of the misalignment is found by averaging instantaneous normalized misalignment estimates over samples after the echo canceler has converged The ambient noise level, (30 db) For the adaptive filter we have chosen (corresponds to 64 ms at 16 khz sample rate), and,, Hence, these settings results in A comparison of this theoretical curve with our estimates shows that theoretical and simulation results match well (see Fig 5) Having verified the theoretical formulae, we will use these to control the proposed adaptive nonlinearity described in Section IV in a more realistic scenario using recorded speech B Adaptive Nonlinearity In this section, the coherence estimate of the algorithm in Table I, the link between MSE and excess misalignment, and the adaptive nonlinearity of Table II are studied We use real-life speech data described in [14] Data and general algorithm settings are as follows Transmission Room Speech: Stereo recordings with a male talker At times 309, 618, 669, 721, 772 s, there are talker position changes; and from 40 s to 50 s there is some background music playing which is somewhat shifted in the stereo image plane toward the left channel Receiving Room Speech: The receiving room speech is generated by filtering the (nonlinearly) processed transmission room speech through an echo path model This model is a measured acoustic response between the left loudspeaker and a standard cardioid microphone positioned on top of a workstation The original impulse response has a length of 256 ms, consisting of 4096 coefficients at 16 khz sampling rate In our simulations, however, we restrict the echo path to 1024 coefficients (see Fig 4) The other parameter settings are the same as in the previous section except for regularization of in Table I (see [9] and [15] for more details on regularization) In this simulation we have, Fig 6(a);, Figs 6(b), 7, and 9 First, the magnitude-squared coherence of the above described transmission room speech as a function of frequency is studied These estimates, obtained with a small regularization parameter or a large regularization parameter are shown in Fig 6(a) and (b), respectively (These are estimates obtained when there is no talker position change or background music) Not surprisingly, we find that regularization severely biases the coherence estimate at higher frequencies the speech level is lower It is therefore advantageous to use only lower frequencies when averaging the squared coherence function of (48), and we from now on will modify our estimates so that only coherence values over the interval 4, ie, 1000 to 4000 Hz are used Second, we show that the link between excess MSE before and after echo path change and the coherence (44) in fact is fairly accurate for a real-life speech situation and not only restricted 4 Note that the number of FFT bins in the theoretical derivation of Section III is L while in the algorithm (Table I) it is 2L

9 GÄNSLER AND BENESTY: STEREOPHONIC ACOUSTIC ECHO CANCELLATION PROBLEM 265 Fig 7 Examples of mean-squared-error-change when transmission echo paths changes occur at 309 s and 618 s (a), (d) Estimated magnitude coherence (b), (e) Predicted MSE difference change (44) Note that the level is about 11 db just before the echo path change (c), (f) MSE curve that shows the actual MSE jump at 309 and 618 s, respectively Fig 9 Performance of adaptive nonlinearity (a) Left channel transmission room speech (upper), residual error (lower) Background music starts at about 40 s and ends at 50 s (b) Estimated magnitude coherence (solid) and applied nonlinearity (dashed) (c) Mean-squared error (solid) and misalignment (dashed) Fig 8 Level of introduced nonlinearity (50) (solid) and (m+1) (51) (dotted) as a function of ^, ^ is given in (49) to our white source excited anechoic model Fig 7 shows the estimated coherence, predicted change of MSE given by (53) and actual excess MSE as a function of time for two different occasions calculated as (54) LPF denotes a first-order lowpass filter with a pole at 0999 (time constant 1/16 s at 16 khz sampling rate) The predicted MSE change is about 11 db just before echo path changes occurs at 309 and 618 s, respectively Looking at Fig 7(c) and (f) we find that the MSE difference from these points in time to the peaks are within 1 or 2 db from the predicted value Third, we give an example of how the adaptive nonlinearity operates Table II shows the whole algorithm for the adaptive nonlinearity that is used in this simulation Fig 8 shows the applied as a function of magnitude coherence of the unprocessed transmission signals In this figure, we have chosen the desired processed coherence ( ) to be 09 The solid line in the figure presents the level of nonlinearity The dashed line presents the function we are going to use because of the restriction to the allowed distortion that can be introduced Fig 9 shows the performance results The coherence between the channels is high so the adaptive nonlinearity adjusts the level to maximum (05) [Fig 9(b)] except for the parts there are talker position changes or background music The result is a fairly good misalignment performance and informal listening tests have shown that we achieve a better perceived quality, since no distortion is introduced, of the background music sequence VI CONCLUSIONS Previously, it was shown that there is a link between the nonuniqueness problem and coherence between the audio channels in the two-channel echo cancellation problem [2] In this paper, however, we have expanded the knowledge of

10 266 IEEE TRANSACTIONS ON SPEECH AND AUDIO PROCESSING, VOL 10, NO 5, JULY 2002 two-channel echo cancellation problem by presenting new theory that gives insight to how to handle the misalignment problem (a unique solution exists but there is still poor misalignment because of coherence between the channels) inherent in two-channel echo cancellation These new ideas involve the following: theory for predicting the required level of nonlinearity in order to achieve a specified level of channel coherence; an expression (39) of misalignment for the two-channel case that accounts for the inter-channel coherence; an expression (44) that predicts the level of mean-squarederror shift after the echo paths changes in the transmission room; a proposed adaptive nonlinearity that results in a better average perceptual quality of the transmitted audio signals All these ideas have been derived by using a simple model for the transmission room (the anechoic model) primarily the coherence is assumed constant for all frequencies However, one should be aware of the model s limitations when used in practice, eg, the prediction of misalignment can be poor because of mismatch at high frequencies APPENDIX DETAILED DESCRIPTION OF THE SPECTRAL MATRIX is an Toeplitz matrix and The definition of is [9, Eq (865)] (55) Assuming stationary input signals ( ) and, we find that For simplicity, we choose to remove the subscript in the equations of this appendix, eg, The following definitions are useful in the derivation of the spectral matrix as defined in (33): (56) If we look at a (general) element we find is a positive integer It can now be determined that (57)

11 GÄNSLER AND BENESTY: STEREOPHONIC ACOUSTIC ECHO CANCELLATION PROBLEM 267 is the (cross-)correlation between and for lag That is, we have (58) Note that is a circulant correlation matrix Substituting the elements in (56) with the corresponding ones given in (58) results in [6] V Fischer, T Gänsler, J Benesty, and E J Diethorn, A software stereo acoustic echo canceler under Microsoft Windows, in Proc IWAENC, 2001 [7] J Benesty and D R Morgan, Multi-channel frequency-domain adaptive filtering, in Acoustic Signal Processing for Telecommunication,S L Gay and J Benesty, Eds Norwell, MA: Kluwer, 2000, ch 7, pp [8] T Gänsler and J Benesty, A frequency-domain double-talk detector based on a normalized cross-correlation vector, Signal Process, vol 81, pp , Aug 2001 [9] J Benesty, T Gänsler, D R Morgan, M M Sondhi, and S L Gay, Advances in Network and Acoustic Echo Cancellation Berlin, Germany: Springer-Verlag, 2001 [10] D R Morgan, J L Hall, and J Benesty, Investigation of several types of nonlinearities for use in stereo acoustic echo cancellation, IEEE Trans Speech Audio Processing, vol 9, pp , Sept 2001 [11] J Benesty, D R Morgan, J L Hall, and M M Sondhi, Synthesized stereo combined with acoustic echo cancellation for desktop conferencing, Bell Labs Tech J, vol 3, pp , July-Sept 1998 [12] J Benesty and D R Morgan, Frequency-domain adaptive filtering revisited, generalization to the multi-channel case, and application to acoustic echo cancellation, in Proc IEEE ICASSP, 2000, pp [13] R F Baum, The correlation function of Gaussian noise passed through nonlinear devices, IEEE Trans Inform Theory, vol IT-15, pp , July 1969 [14] P Eneroth, S L Gay, T Gänsler, and J Benesty, A real-time stereophonic acoustic subband echo canceler, in Acoustic Signal Processing for Telecommunications, S L Gay and J Benesty, Eds Norwell, MA: Kluwer, 2000, ch 8, pp [15] H Buchner and W Kellermann, Acoustic echo cancellation for two and more reproduction channels, in Proc IWAENC, 2001 since That is, (59) is our desired expression for the spectral matrix (59) (60) ACKNOWLEDGMENT The authors would especially like to thank D Morgan for sharing his knowledge of statistical analysis of nonlinear processing of random signals They are also very grateful to G Elko for proofreading this manuscript REFERENCES [1] M M Sondhi, D R Morgan, and J L Hall, Stereophonic acoustic echo cancellation An overview of the fundamental problem, IEEE Signal Processing Lett, pp , Aug 1995 [2] J Benesty, D R Morgan, and M M Sondhi, A better understanding and an improved solution to the specific problems of stereophonic acoustic echo cancellation, IEEE Trans Speech Audio Processing, vol 6, pp , Mar 1998 [3] S Shimauchi, Y Haneda, S Makino, and Y Kaneda, New configuration for a stereo echo canceller with nonlinear pre-processing, in Proc IEEE ICASSP, 1998, pp [4] Y Joncour and A Sugiyama, A stereo echo canceller with pre-processing for correct echo path identification, in Proc IEEE ICASSP, 1998, pp [5] A Gilloire and V Turbin, Using auditory properties to improve the behavior of stereophonic acoustic echo cancellers, in Proc IEEE ICASSP, 1998, pp Tomas Gänsler (M 97) was born in Sweden in 1966 He received the MS degree in electrical engineering and the PhD degree in signal processing from Lund University, Lund, Sweden, in 1990 and 1996, respectively From 1997 to September 1999, he was an Assistant Professor at Lund University During 1998 he was a Consultant with Bell Labs, Lucent Technologies; in October 1999, he joined as a Member of Technical Staff Since February 2001, he has been with Agere Systems, a spinoff from Lucent Technologies Microelectronics Group His research interests include robust estimation, adaptive filtering, mono/multichannel echo cancellation and subband signal processing He coauthored the book Advances in Network and Acoustic Echo Cancellation and he is also a coauthor of the book Acoustic Signal Processing for Telecommunication Jacob Benesty (M 98) was born in 1963 He received the Master degree in microwaves from Pierre & Marie Curie University, France, in 1987, and the PhD degree in control and signal processing from Orsay University, France, in April 1991 While pursuing the PhD degree, he worked on adaptive filters and fast algorithms at the Centre National d Etudes des Telecomunications (CNET), Paris, France From January 1994 to July 1995, he was with Telecom Paris University working on multichannel adaptive filters and acoustic echo cancellation He joined Bell Labs, Lucent Technologies (formerly AT&T) in October 1995, first as a Consultant and then as a Member of Technical Staff Since then, he has been working on stereophonic acoustic echo cancellation, adaptive algorithms, source localization, robust network echo cancellation, and blind identification He was the co-chair of the 1999 International Workshop on Acoustic Echo and Noise Control He co-authored the book Advances in Network and Acoustic Echo Cancellation (Berlin, Germany: Springer-Verlag, 2001) He is also co-editor/co-author of the book Acoustic Signal Processing for Telecommunication (Norwell, MA: Kluwer, 2000) Dr Benesty is a member of the IEEE Signal Processing Society Technical Committees on Audio and Electroacoustics He is the recipient, with Morgan and Sondhi, of the IEEE Signal Processing Society 2001 Best Paper Award