Convex Combination of MIMO Filters for Multichannel Acoustic Echo Cancellation

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1 Convex Combination of MIMO Filters for Multichannel Acoustic Echo Cancellation Danilo Comminiello, Simone Scardapane, Michele Scarpiniti, Raffaele Parisi, and Aurelio Uncini Department of Information Engineering, Electronics and Telecommunications (DIET) Sapienza Universit of Rome Via Eudossiana 8, 0084 Rome, Ital Abstract This paper introduces a new framework for multichannel adaptive filtering, aiming at improving performance of an overall filtering sstem The proposed architecture relies on the properties of the adaptive combination of filters which exploits the capabilities of different constituents, thus adaptivel providing at least the behaviour of the best performing filter Appling this concept to multichannel filtering sstems, we define a scheme for the combination of multiple-input multipleoutput (MIMO) filters More precisel the proposed structure involves the combination of two different multiple-input singleoutput (MISO) sstems for each MIMO output We propose such framework with application to the multichannel acoustic echo cancellation (MAEC) with the goal of giving robustness to the sstem against impulsive background noise, and thus improving overall cancelling performance Experimental results show the effectiveness of the proposed combined MAEC in the presence of adverse environmental conditions I INTRODUCTION The growing availabilit of resources for data transmission and processing is encouraging the spread of hands-free speech communication sstems which offer users an immersive experience ] 3] An immersive communication scenario is characterized b the use of multiple microphones and loudspeakers, which aim at capturing and reproducing desired signals while focusing on preserving the audio qualit perceived b users In such context, the presence of interfering sources and reverberation ma cause a qualit degradation of speech communications In order to address this problems, microphones and loudspeakers are usuall connected with signal processing sstems, thus resulting in intelligent acoustic interfaces 3], 4] One of the main problems of immersive scenarios is the multichannel acoustic echo cancellation (MAEC), which is caused b the multiple coupling between microphones and loudspeakers MAEC ma be seen as a straightforward generalization of the monophonic acoustic echo cancellation (AEC), but it entails more problems to tackle, such as nonuniqueness and slow convergence due to inter-channel correlation, and the poor abilit to react to changes in the environmental conditions 5] Such issues have roused remarkable interest over the ears 6] 9] The effectiveness of an MAEC strictl relies on the design of a multiple-input multiple-output (MIMO) filtering sstem, whose main task is to estimate several acoustic impulse responses (AIRs), depending on the number of microphones and loudspeakers It results in a large number of coefficients to adapt, therefore an appropriate choice of the adaptive algorithm becomes essential Generall, in the time-domain, first-order adaptive algorithms, such as the least mean squares (LMS), are ver attractive due to their simplicit and low computational cost However, LMS-based algorithms do not take into account the cross-correlation statistics of the input signals, thus resulting in poor convergence performance 5] Hessian-based algorithms, such as the recursive least squares (RLS), improve convergence abilities since the consider interchannel correlations Unfortunatel, the RLS entails a larger computational cost and, moreover, it ma perform worse than first-order algorithms in adverse environments 0] This is the reason wh here we consider the affine projection algorithm (APA) to adapt the MIMO filters, since it can be seen as a generalization of the normalized LMS (NLMS) algorithm, in which cross-correlations of the input signals are involved ] However, even a multichannel APA ma suffer from adverse conditions, especiall in the presence of impulsive noise or in changing environments, which ma make the adaptation process unstable and reduce performance In order to tackle this problem, we introduce a new adaptation framework which relies on the adaptive combination of MIMO filtering sstems Adaptive combination of filters is capable of automaticall switching between constituents according to the best performing filter, thus alwas providing the best possible performance 2] Combined adaptive schemes are usuall adopted with filters of the same famil and complementar properties, but also with filters of different families or using different updating rules 3] 5] Recentl, combination of filters was successfull applied to multiple-input single-output (MISO) sstems 6] 8] with application to adaptive beamforming for noise reduction In this paper we exploit the properties of adaptive combination of filters to perform a combination of MIMO filtering sstems In particular, we combine two MIMO sstems, each one having different adaptation settings We describe in detail a possible combining architecture for MIMO filters and we appl such technique to MAEC As a consequence, an improvement of the overall performance of the canceller is expected, especiall in the presence of an impulsive noise that ma cause a change in the environment This paper is organized as follows: Section II describes the MAEC framework using the combined MIMO architecture updated b using the multichannel APA The combination of adaptive MIMO filters is proposed in Section III Section IV contains the evaluation of the effectiveness of the proposed framework with application to MAEC, and some conclusion are finall drawn in Section V Multichannel Audio Algorithms: New Trends and Innovative Applications 778

2 II COMBINED MULTICHANNEL ACOUSTIC ECHO CANCELLATION The MAEC framework including the combination of MIMO filtering sstems is depicted in Fig and it is referred to as combined MAEC (CMAEC) At n-th time instant, P speech signals coming from the remote environment, also known as far-end, and denoted as x p n], with p =,, P, arrive at the other side of the full-duplex communication, or also said near-end The far-end signals are reproduced b P loudspeakers and then acquired b microphones The acoustic coupling between microphones and loudspeakers is characterized b P acoustic paths, which also contain information about environment reverberations The desired signals d q n] (q =,, ) acquired b the microphones represent the echo signals, which ma be possibl superimposed on an near-end contribution, containing the near-end speech signal s n] with the addition of background noise v n] At the same time, the far-end signals x p n] are processed b the CMAEC in order to estimate the AIRs between microphones and loudspeakers In particular, each input signal x p n] is collected in an input data matrix X n,p R K M : X n,p = x T n,p x T n,p x T n K,p x p n] x p n M ] x p n ] x p n M 2] = x p n K ] x p n M K 2] where M is the length of the adaptive filters and K denotes the number of previous entries to keep in memor, ie the projection order Data matrices can be represented b a unique input data matrix X n R K MP : () X n = X n, X n,2 X n,p ] T (2) The input matrix is processed b the CMAEC, which is composed of J MIMO filtering sstems In this paper we take into account the combination of J = 2 MIMO filters, but it can be also generalized to the case of J > 2 Each MIMO filter is represented b the following matrix W n (j) R MP, with j =, 2: W (j) n = n, n,2 n, n,2 n,22 n,2 n,p n,p 2 n,p where inter-channel paths are also taken into account Each individual adaptive filter w n,pq R M included in (3) can be expressed as: n,pq = pq,0 n] The output of each MIMO filter Y n (j) achieved: Y (j) n (3) ] T w(j) pq, n] w(j) pq,m n] (4) R K can be easil = X (j) n W (j) n (5) e n e Fig xp x Combined MAEC Combined MAEC sstem d d that can be also seen as a concatenation of individual ] T output vectors, ie Y n (j) = (j) n, (j) n,2 (j) n,, for j =, 2 Taking into account the matrix of the desired signals D n R K, ie: d n] d n] d n ] d n ] D n =, (6) d n K ] d n K ] for each MIMO filter it is possible to achieve the error signals, which are contained in the following error matrix: E (j) n = D n Y (j) n (7) Similarl to (5), E (j) n can be seen as a concatenation of individual error vectors, ie E (j) n = e (j) n, e (j) n,2 e (j) n, ] T Each MIMO filter is individuall updated according to the multichannel affine projection algorithm ]: W n (j) = W (j) n µ jx T ( n δj X n X T ) n E (j) n (8) where µ j and δ j are respectivel the step size parameter and the regularization factor, which are the same for all the filters of the j-th MIMO sstem In the next section, a possible adaptive scheme is derived for the combination of the two MIMO sstems, thus achieving the overall error signals for the CMAEC III A CONVEX COMBINATION SCHEME FOR MIMO FILTERS The trademark of the proposed sstem lies in the CMAEC, which involves the combination of the two MIMO filters These sstems can be differentiated in several was according to their update rule In this paper we distinguish the two MIMO Multichannel Audio Algorithms: New Trends and Innovative Applications 779

3 filters b choosing different values for the step size parameter, a small value for j = and a large one for j = 2, according to 2], 3] As regards the kind of adaptive combination, in this work we focus on the convex constrained combination using a sigmoid nonlinearit on the output stage, since it introduces less gradient noise compared to unconstrained and affine constrained combinations The chosen method for combining MIMO filters is based on the sstem-b-sstem (SS) combination scheme 6] and it is represented in Fig 2 An MIMO sstem can be seen as a parallel of MISO sstems, each receiving the same P inputs The idea which underpins the SS scheme is to combine the outputs of each pair of MISO sstems, for all the output channels The output of each MISO sstem for the q-th MIMO output channel, that we denote as q (j) n], is achieved b summing each filter output of the corresponding MISO sstem, ie: x w n, w n, 2 w n, 2 w n, n n 2 (j) q n] = P p= x T n,p n,pq (9) xp w np, The two MISO sstem outputs related to the q-th output channel are then convexl combined generating the q-th MIMO output: q n] = λ q n] () q n] ( λ q n]) (2) q n] (0) where λ q n] is the q-th mixing parameter, which is constrained to remain in the range 0, ] 2] It is worth noting that, in order to not further increase the computational complexit, in (0) we do not consider the last K output samples but onl the current sample The q error signal e q n] after the combination is achieved as: w n, P 2 w np, 2 w n, P n n 2 e q n] = d n] q n] () The mixing parameters in (0) are usuall updated b using an auxiliar parameter, a q n], related to λ q n] b the expression: λ q n] = sgm (a q n]) = (2) e an] The update rule of the auxiliar parameter can be defined according to 2], 9]: a q n ] = a q n] µ a r q n] e q n] e q n] λ q n] ( λ q n]) where and (3) e q n] = e (2) q n] e () q n], (4) r q n] = ηr q n ] ( η) e 2 q n] (5) is the estimated power of e q n]; η is a smoothing factor Therefore, µ a /r q n] in (3) represents a normalized step size B using the adaptation rule (3), the combination scheme is able to adaptivel recognize and select the best performing filters, thus improving the overall performance of the CMAEC The computational complexit of the CMAEC is subjected to several parameters, but most of all it depends on the adopted Fig 2 Sstem-b-sstem scheme of the combined MAEC adaptive algorithm and the number of channels In particular, the overall cost of a CMAEC is exactl the double of a MAEC with the addition of an overhead due to the convex combinations Such overhead consists of 8 multiplications, 3 additions and function evaluations The reduction of the computational complexit of the CMAEC can be taken into account in future developments, in particular in the choice of the adaptive algorithm, which represents the most burdensome component IV EXPERIMENTAL RESULTS In this section we present two sets of experiments aiming at assessing the performance of the proposed combined MIMO sstem In order to have an exhaustive view of the capabilities of the sstem, we first analse its convergence performance and then its effectiveness in an MAEC application A Convergence Performance of the Combined MIMO Filter In the first set of experiments we assess the convergence behaviour of the proposed combined MIMO filter in terms of the excess mean square error (EMSE), defined as the Multichannel Audio Algorithms: New Trends and Innovative Applications 780

4 2 4 6 Conventional MIMO, µ Conventional MIMO, µ 2 Combined MIMO EMSE db] ERLE db] Samples 5 Conventional MAEC, µ Conventional MAEC, µ 2 CMAEC Samples Fig 3 Convergence performance in terms of the EMSE Fig 4 Performance evaluation in terms of the ERLE excess over the minimum mean square error The scenario involves an unknown multichannel sstem to identif, which is composed of P = 4 input channels and = 4 output channels Each inter-channel impulse response is formed b M = 7 independent random values between and The input signal is generated b means of a first-order autoregressive model, whose transfer function is α 2 / ( αz ), with α = 08, fed with an iid Gaussian random process The length of the input signal is L = 0000 samples In order to evaluate the abilit of the proposed architecture to react to possible alterations in the environment, the impulse responses to estimate are randoml reassigned at the time instant n = L/2 An additive iid noise signal v n] with variance σ 2 v = 00 is added at the output of each channel In order to identif the unknown P = 6 impulse responses we adopt the proposed combined MIMO filter, which is composed of two MIMO filters, one using a small step size µ = 000 and the other one using a larger value µ 2 = 0 Convergence performance is evaluated in terms of the EMSE, which can be defined (in db) for the multichannel case as: ( )2 EMSE n] = 0 log E e q n] v n] (6) q= where the operator E { } is the mathematical expectation; indeed the EMSE is evaluated over 000 independent runs Results for the combined MIMO filter are compared with those of individual MIMO filters using respectivel µ and µ 2 The combined MIMO filter uses a step size value µ a = 05 and the following initial setting for the adaptation of the auxiliar parameter: a q 0] = 0, r q 0]; the smoothing factor in (5) is chosen as η = 09 All the sstems use the same projection order K = 4 and the same regularization factor δ = 000 Convergence results are depicted in Fig 3, where it is worth noting that the MIMO filter with µ shows a slow convergence rate but a good precision at stead state, while the MIMO filter with µ 2 provides faster convergence rate but lower precision The combined MIMO filter is capable of exploiting the advantages of both the MIMO filters, thus showing fast convergence rate and good precision at stead state Moreover, it can be notice that, in proximit of the intersection between the two conventional sstems, the proposed combined MIMO filter shows even a performance gain with respect to the conventional filters B Evaluation of the Combined MAEC In the second set of experiments we use the prosed combined architecture for an MAEC application The experimental scenario is that of a reverberant room with size 6 5 3, 3 m where multiple loudspeakers and microphones are placed In particular, a uniform linear arra (ULA) of P = 8 loudspeakers is adopted with a spacing of 20 cm In front of the loudspeaker arra, at a distance of 4, 5 m, is located a ULA of = 8 microphones, in which the spacing is 5 cm The P = 64 acoustic impulse responses (AIRs) are measured b an image source method, using a reflection factor of the walls of ρ = 078 with a sampling rate of 8 khz Each AIR is truncated after M = 280 samples As far-end input is used a coloured signal generated b using the same autoregressive model described in the previous set of experiments The length of the input signal is L = 6000 samples An additive iid white Gaussian noise signal v n] is added to each microphone signal in order to provide 20 db of signal to noise ratio (SNR) In order to introduce an abrupt change in the acoustic environment we shift the AIRs circularl to the right b 30 samples, at the time instant n = L/2 Similarl to the previous experiment, we compare the performance of two conventional MIMO filters with the combination of them, ie the CMAEC Again we differentiate the two MIMO filters according to their step size value In particular, we choose a small value for the first MIMO filter, µ = 0, and a large value for the second one, µ 2 = The rest of the parameter setting for all the sstems is the same of the previous setup In this case, the performance is evaluated in terms of the echo return loss enhancement (ERLE), which can be defined (in db) for the multichannel case as: Multichannel Audio Algorithms: New Trends and Innovative Applications 78

5 ERLE n] = ERLE q n] q= ( E { d 2 q n] } ) (7) = 0 log E { e 2 q n] } q= Results are depicted in Fig 4, in which it is possible to notice the behaviour of the proposed CMAEC which alwas follows the best performing MIMO filter and outperforms the conventional sstems in correspondence of their performance intersection This proves the effectiveness of the proposed combined MIMO architecture for MAEC applications V CONCLUSION This paper introduces a new framework for MIMO filtering in MAEC applications The proposed approach is based on the adaptive combination of MIMO filters, each one having different settings and, hence, different capabilities The combined MIMO is capable of exploiting the best properties of each MIMO filter, thus providing superior performance The combination can be achieved according to a sstem-b-sstem scheme, which guarantees robustness to the MAEC against abrupt changes in the environment Experimental results have shown the effectiveness of the sstem in multichannel scenarios The proposed framework paves the wa for future works, since results could be further improved b involving other kinds of adaptive algorithms for each filter, eg in the frequenc-domain, or other combination schemes, eg the filter-b-filter one, or also other constrained algorithms for the adaptation of the mixing parameters REFERENCES ] Y Huang, J Chen, and J Benest, Immersive audio schemes, IEEE Signal Process Mag, vol 28, no, pp 20 32, Jan 20 2] C Kriakakis, Fundamental and technological limitations of immersive audio sstems, Proc of the IEEE, vol 86, no 5, pp 94 95, 998 3] D Comminiello, M Scarpiniti, R Parisi, and A Uncini, Intelligent acoustic interfaces for immersive audio, in 34th AES Convention, Rome, Ma 203 4] D Comminiello, Adaptive algorithms for intelligent acoustic interfaces, PhD dissertation, Sapienza Universit of Rome, Dec 20 5] Y Huang, J Benest, and J Chen, Acoustic MIMO Signal Processing Berlin: Springer-Verlag, ] J Benest, D Morgan, and M Sondhi, A better understanding and an improved solution to the specific problems of stereophonic acoustic echo cancellation, IEEE Trans Audio, Speech, Lang Process, vol 6, no 2, pp 56 65, Mar 998 7] H Buchner, J Benest, and W Kellermann, Generalized multichannel frequenc-domain adaptive filtering: Efficient realization and application to hands-free speech communication, Signal Process, vol 85, no 3, pp , Mar ] S Cecchi, L Romoli, P Peretti, and F Piazza, A combined pschoacoustic approach for stereo acoustic echo cancelllation, IEEE Trans Audio, Speech, Lang Process, vol 9, no 6, pp , Aug 20 9] D- Nguen, W-S Gan, and A W H Khong, Time-reversal approach to the stereophonic acoustic echo cancellation, IEEE Trans Audio, Speech, Lang Process, vol 9, no 2, pp , Feb 20 0] E Eweda, Comparison of RLS, LMS and sign algorithms for tracking randoml time-varing channels, IEEE Trans Signal Process, vol 42, no, pp , Nov 994 ] J Benest, P Duhamel, and Y Grenier, A multichannel affine projection algorithm with applications to multichannel acoustic echo cancellation, IEEE Signal Process Lett, vol 3, no 2, pp 35 37, Feb 996 2] J Arenas-García, A R Figueiras-Vidal, and A H Saed, Mean-square performance of a convex combination of two adaptive filters, IEEE Trans Signal Process, vol 54, no 3, pp , Mar ] J Arenas-García, M Martínez-Ramón, A Navia-Vázquez, and A R Figueiras-Vidal, Plant identification via adaptive combination of transversal filters, Signal Process, vol 86, no 9, pp , Sep ] N J Bershad, J C M Bermudez, and J Tourneret, An affine combination of two LMS adaptive filters Transient mean-square analsis, IEEE Trans Signal Process, vol 56, no 5, pp , Ma ] M T M Silva and V H Nascimento, Improving the tracking capabilit of adaptive filters via convex combination, IEEE Trans Signal Process, vol 56, no 7, pp , Jul ] D Comminiello, M Scarpiniti, R Parisi, and A Uncini, Combined adaptive beamforming schemes for nonstationar interfering noise reduction, Signal Process, 203, 006/jsigpro ], Combined adaptive beamforming techniques for noise reduction in changing environments, in Proc IEEE Int Conf Telecom Signal Process (TSP 3), Rome, Jul ] D Comminiello, M Scarpiniti, R Parisi, A Cirillo, M Falcone, and A Uncini, Multi-stage collaborative microphone arra beamforming in presence of nonstationar interfering signals, in Proc Int Works Machine List Multisource Environ (CHiME ), Florence, Ital, Sep 20, pp ] L A Azpicueta-Ruiz, A R Figueiras-Vidal, and J Arenas-García, A normalized adaptation scheme for the convex combination of two adaptive filters, in Proc IEEE Int Conf Acoust, Speech, Signal Process (ICASSP 08), Las Vegas, NV, USA, Mar 2008, pp Multichannel Audio Algorithms: New Trends and Innovative Applications 782