OPTIMAL BEAMFORMING AS A TIME DOMAIN EQUALIZATION PROBLEM WITH APPLICATION TO ROOM ACOUSTICS

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1 OPTIMAL BEAMFORMING AS A TIME DOMAIN EQUALIZATION PROBLEM WITH APPLICATION TO ROOM ACOUSTICS Mark R P Thomas, Ivan J Tashev Microsoft Research Redmond, WA 985, USA {markth, ivantash}@microsoftcom Felicia Lim, Patrick A Naylor Dept of Electrical and Electronic Enineerin Imperial Collee London, UK {felicialim6, pnaylor}@imperialacuk ABSTRACT Sinals captured by microphone arrays provide spatial diversity that can be exploited by multichannel processin alorithms to suppress noise and reverberation Beamformin is a class of approaches that treats the problem with respect to the spatial location of wanted and competin sources, leverain properties of propaation of waves in free space A related class of alorithms is channel equalization that exploits knowlede of the acoustic impulse response between a source and microphones with a view to near-perfect dereverberation Beamformin has been shown to be a very powerful and practical tool in a number of domains, whereas channel equalizers are notoriously sensitive to noise and channel mismatch leadin to limited practical applicability This paper investiates some of the common properties of these alorithms and presents a solution incorporatin approaches from both disciplines Index Terms Beamformin, channel equalization, dereverberation INTRODUCTION Speech sinals captured by hands-free devices are typically corrupted by noise and reverberation that impair the perceived quality and intelliibility of the received speech Microphone arrays are attractive because the sinals they receive exhibit spatial diversity that can be exploited to suppress components that are not spatially collocated with the source Processin of multichannel sinals can be divided into two broad classes: beamformin [], which is larely motivated by the propaation of waves in free space and/or the spatial correlation of noise sinals, and channel equalization [], motivated by the acoustic impulse response between the source and receivers in a reverberant environment Beamformer desin often operates under certain common assumptions: that all sources lie in the farfield, all sensors are omnidirectional, and propaation from a source to the array is characterized by a pure delay This is particularly true in the desin of non-adaptive superdirective beamformers [] Some assumptions are circumvented with adaptive (data-dependent solutions, however both adaptive and non-adaptive approaches often incorporate distortionless constraints to ensure that waves propaatin in the wanted direction are left undistorted under the assumption that the source-receiver transfer function is a pure delay Conversely, channel equalizers such as the Multichannel Input-Output Inverse Theorem ( [] use prior knowlede of the reverberant impulse responses from source to receivers with a view to near-perfect equalization, but without explicit constraints for spatial selectivity Channel inversion techniques are notoriously sensitive to errors in channel estimates and can often increase the level of reverberation under channel mismatch, for example due to a different source location, the relocation of furniture, or a chane in the ambient temperature [, 5] To this end, channel shortenin/reshapin techniques [6, 7] have been proposed to improve robustness Both beamformin and channel inversion can be viewed as filterand-sum operations with different optimization criteria The concept of Formin [8] was introduced to combine both concepts into a sinle alorithm that controls the tradeoff between the channel inversion provided by the alorithm and the spatial and noise performance of an optimal filter-and-sum beamformer The approach was to formulate both and beamformin as frequency domain desin problems, to combine their respective cost functions, and to evaluate the performance as a channel equalizer In order to apply frequency domain desins to real world sinals it is necessary to obtain a finite impulse response (FIR approximation [9] to the frequency domain filter This approximation inherently introduces error into the desin, producin suboptimal behavior Several works have proposed time-domain beamformer desins to circumvent this issue [9,, ] Conversely, channel equalization alorithms such as are also usually posed as time-domain problems [,, 7] so no approximation is required This paper considers and non-adaptive superdirective beamformin purely as both time domain desin problems As a reference, a third hybrid case is considered whereby a beamformer is desined usin knowlede of the reverberant impulse response for sources in several locations The performance is evaluated both in terms of channel equalization in the wanted direction and in terms of spatial selectivity The remainder of this paper is oranized as follows The equalization and beamformin problems are formulated in Sec In Section,, an optimal filter-and-sum beamformer and a hybrid reference alorithm are formulated in the time domain The alorithms are evaluated in Section and conclusions are drawn in Section 5 PROBLEM FORMULATION Consider an array of M microphones placed in a reverberant environment Let ȟ p,m = [ȟp,m( ȟp,m(l ]T R L ( hp,m = [ h p,m( h p,m(l ] T R L ( be the impulse responses of lenth L samples between a source with position index p {,,, P } and receiver m {,,, M}

2 in the reverberant and anechoic cases respectively The source at index p is considered to be a wanted source and all other values of p are considered to be unwanted noise sources It is assumed that ȟ p,m and h p,m contain all propaation delays and have not been truncated Additionally, let ȟ p = [(ȟp,t (ȟp,m T ] T R ML ( hp = [( h p, T ( h p,m T ] T R ML ( contain stacked impulse responses between source at p and all M microphones The aim for both filter-and-sum beamformer and channel equalizer desin is to synthesize filters m = [ m( m(l i ] T R L i that produce a desired response at the system output, which will be referred to in this paper as equalization filters irrespective of whether they were desined for beamformin or equalization Their stacked representation over all microphones is defined in a similar way to ( (5 = [ T T M ] T R ML i (6 Now let Ȟp,m R(L+L i L i be a convolution matrix derived from ȟp,m so that Ȟp,mm and hp,m(n m(n, where denotes linear convolution, are equivalent: h p,m( h p,m( h p,m( Ȟ p,m = h p,m(l h p,m(l h p,m(l (7 A similar formulation is used for H p,m Convolution matrices can be stacked for all M channels, Ȟ p = [Ȟp, Ȟ p,m ] R (L+L i ML i (8 H p = [ H p, H p,m ] R (L+L i ML i, (9 then further stacked over all P source positions to form universal convolution matrices Ȟ = [ȞT ȞT P ] T R P (L+L i ML i ( H = [ H T H T P ] T R P (L+L i ML i ( The equalization filters are then synthesized in such a way that the response of the equalized system in direction p is found by a filterand-sum operation: y p = M Ȟ p,m m = Ȟp R(L+L i, ( m= where y p = [y p( y p(l + L i ] T For notational convenience, the equalized output can be found for all P directions in a sinle operation y = Ȟ RP (L+L i R P (L+L i, ( where y = [y T y T y T P ] T Accents ˇ ( and ( depict a reflection and the direct path ALGORITHMS The Multichannel Input-Output Inverse Theorem ( alorithm [] was proposed as a means of providin exact inverse filterin of room acoustics from multichannel observations A stable, causal inverse of a sinle-channel system does not enerally exist due to the nonminimum phase characteristic of room transfer functions In the multichannel case, it was shown that under certain conditions a stable, finite, causal and exact inverse always exists Considerin only the wanted direction p, equalization filters are defined usin to produce the equalized output { if l = τ; d p (l = ( otherwise, where τ is an arbitrary inteer delay with vector representation d p = [d p ( d p (L + L i ] T R (L+L i (5 The equalizer desin can then be stated as a least-squares convex optimization problem ĝ = ar min Ȟp d p, (6 yieldin theoretically exact equalization providin the filters ȟm,p are known, that there are no common zeros between ȟm,p (n and ȟ m+,p (n, and L L i (7 M is satisfied [] Optimal Filter-and-Sum Beamformer ( In a typical superdirective beamformin application, it is desirable to have a response y yieldin unit ain in the look direction p and minimized ain elsewhere Capture vectors describin the array behavior for a source in look direction p are usually defined in terms of pure delays of the direct-path sinal In the time domain the filter desin can be stated as the convex optimization problem with distortionless constraint ĝ = ar min H d subject to H p = d p, (8 where the desired spatial desired response d is d = [ d T p ] T R P (L+L i (9 Other desired responses such as a taperin towards the look direction can help to reduce the amplitude of sidelobes Time domain formulations for filter-and-sum beamformers are not common; it is derived in this way to make this and mutually compatible Case Usin a similar approach to the optimal beamformer, the problem can be restated assumin knowlede of reverberant impulse responses ȟp in all P directions, thereby usin all available information: ĝ = ar min subject to Ȟp = d p (

3 5 z WNG (db y x Frequency (Hz Fi White noise ains Fi Microphone array (reen, evaluation points (blue and look direction (red Notice the constraint in ( is identical to the requirement in (6 In the case that the filter lenth requirement (7 is just satisfied, it is clear that the constraint in ( consumes every available deree of freedom so that ( and (6 will yield the same filter In order to introduce spatial selectivity, either the equalizer lenth L i should be increased to increase the available derees of freedom, or an inequality introduced to the constraint as ĝ = ar min Ȟ d subject to Ȟp d p < ɛ, ( where ɛ is an arbitrary constant EVALUATION The aim of this experiment is to evaluate the alorithms under test in ideal conditions, ie when the impulse responses are known exactly and no robustness constraints are applied This was chosen to provide insiht into shortcomins that limit their ultimate performance Definin all three cases as time domain problems also helps to makes informed comparisons Metrics White noise ain measures the sensitivity of the approaches to sensor noise Assumin the distortionless constraint is met, WNG = lo ((k H (k, ( where (k = [ (k M (k] T C M is a vector of discrete Fourier transforms of m(n and ( H is a Hermitian (conjuate transpose The alorithms under test should be evaluated both as channel equalizers and a beamformers since they draw upon ideas from both fields To this end, the equalized impulse response (EIR is first calculated for all look directions under reverberant conditions: y = Ȟĝ ( Ideally y p d p = and y p p There are several measures derived from ( for a sinle source [] Here we shall use the Direct to Reverberant Ratio (DRR ( yp(τ DRR p = lo τ n= y p(n + L+L i db, n=τ+ y p(n ( in which the direct component is defined as a sinle sample at index τ As a measure of beamformer performance, spatial selectivity is often evaluated with the directivity index (DI that measures the ratio of the sensitivity in the look direction to the mean of sensitivity over the entire space Lettin y p(k be the discrete Fourier transform of y p(n, the Reverberant Directivity Index (RDI is ( y p (k RDI(k = lo P db (5 p= yp(k Experimental Setup P A -channel ULA with inter-mic spacin cm centered at [,, ] m was placed in a m room with reverberation time T 6 = ms as shown in Fi Impulse responses were simulated usin the the source-imae method [] for P = 6 anles on the horizontal plane at radius m from the centre of the array The look direction was chosen as the endfire steerin anle (p = 8 The followin parameters were used: samplin frequency f s = 8 khz, L = 56 samples, L i = L samples The taret response d p was a perfect impulse with delay τ = L/ samples The desin was solved for the three alorithms under test, then evaluated with ( The inequality constraint ɛ was set to db Results and Discussion The results in Fi show that introduces the least white noise ain and is therefore least sensitive to sensor noise; this is an intuitive result as the optimization problem is unconstrained and therefore better conditioned than the constrained cases The result introduces the reatest WNG at all frequencies, with the oracle solution lyin in between; this is most likely due to the constant ɛ relaxin the distortionless constraint Fis and 5 show equalized impulse responses stacked across anles and the correspondin DRRs respectively They reveal that robustness to sensor noise is not correlated with spatial robustness as

4 Direct to Reverberant Ratio EIR, Stacked Anles EIR, Stacked Anles 7 8 DRR (db EIR, Stacked Anles Anle (de 5 5 Case Case RDI (db Anle (de Anle (de Case 5 Fi 5 Direct-to-reverberant ratios Fi Equalized impulse responses 8 5 Fi Reverberant directivity patterns the solution provides perfect equalization in the look direction and a very poor DRR elsewhere The oracle solution provides a similar spatial DRR to the but with a db improvement in the look direction Fi shows the reverberant directivity patterns: the manitude responses of the equalized systems as a function of frequency and anle under reverberant conditions The results for the and oracle cases are similar to a classic (anechoic directivity pattern, showin a main lobe in the look direction but with added distortions due to reverberation The reverberant directivity pattern is hihly chaotic in the case but close inspection reveals unit ain in a thin horizontal stop correspondin to the look direction only Fi 6 shows the correspondin reverberant directivity index, hihlihtin the poor performance of and revealin that the oracle case provides a 5 db improvement over These preliminary results suest that the and paradims can be combined into a sinle oracle case that provides the performance in the look direction of and the spatial robustness of a without increasin WNG The close spacin of the array microphones was chosen to exaerate WNG; a practical array would most likely have much hih inter-microphone spacin with Frequency (Hz 5 Fi 6 Reverberant directivity indices accordinly reduced WNG These results motivate a deeper investiation into WNG constraints, a wider rane of eometries and conditions, the introduction of channel mismatch, variable numbers of control points and comparisons with frequency domain approaches 5 CONCLUSIONS Multichannel equalization and optimal beamformin can be formulated as filter and sum operations with differin optimization criteria Derivin both cases in the time domain, an additional oracle case was proposed that exploits known impulse responses in multiple directions Under simulated reverberant conditions, it was shown that performs well in the desined look direction but lacks the spatial robustness of the optimal beamformer Conversely the optimal beamformer lacks channel equalization performance in the look direction The oracle cases exhibits ood properties from both cases without increased sensitivity to sensor noise This work motivates a deeper investiation under a wider rane of conditions

5 6 REFERENCES [] M S Brandstein and D B Ward, Eds, Microphone Arrays: Sinal Processin Techniques and Applications, Spriner- Verla, Berlin, Germany, [] M Miyoshi and Y Kaneda, Inverse filterin of room acoustics, IEEE Trans Acoust, Speech, Sinal Process, vol 6, no, pp 5 5, Feb 988 [] H Cox, R M Zeskind, and T Kooij, Practical superain, IEEE Trans Acoust, Speech, Sinal Process, vol, pp 9 98, June 986 [] W Zhan, E A P Habets, and P A Naylor, A systemidentification-error-robust method for equalization of multichannel acoustic systems, in Proc IEEE Intl Conf on Acoustics, Speech and Sinal Processin (ICASSP, Dallas, TX, USA, Mar [5] M R P Thomas, N D Gaubitch, and P A Naylor, Application of channel shortenin to acoustic channel equalization in the presence of noise and estimation error, in Proc IEEE Workshop on Applications of Sinal Processin to Audio and Acoustics, New Paltz, New York, USA, Oct [6] W Zhan, E A P Habets, and P A Naylor, On the use of channel shortenin in multichannel acoustic system equalization, in Proc Intl Workshop Acoust Echo Noise Control (IWAENC, Tel Aviv, Israel, Au [7] J O Junmann, T Mei, S Goetze, and A Mertins, Room impulse response reshapin by joint optimization of multiple p-norm based criteria, in Proc European Sinal Processin Conf (EUSIPCO, Barcelona, Spain, Au, pp [8] F Lim, M R P Thomas, and P A Naylor, Former: A spatially aware channel equalizer, in Proc IEEE Workshop on Applications of Sinal Processin to Audio and Acoustics, New Paltz, New York, Oct [9] E Mabande, A Schad, and W Kellermann, A time-domain implementation of data-independent robust broadband beamformers with low filter order, Edinburh, Scotland, May [] C Lai, S Nordholm, and Y Leun, Desin of robust steerable broadband beamformers with spiral arrays and farrow filter structure, in Proc Intl Workshop Acoust Echo Noise Control (IWAENC, Tel-Aviv, Israel, Au [] S Yan, Y Ma, and C Hou, Optimal array pattern synthesis for broadband arrays, J Acoust Soc Am, vol, no 5, pp , Au 7 [] P A Naylor and N D Gaubitch, Eds, Speech Dereverberation, Spriner, [] J B Allen and D A Berkley, Imae method for efficiently simulatin small-room acoustics, J Acoust Soc Am, vol 65, no, pp 9 95, Apr 979

JOINT DEREVERBERATION AND NOISE REDUCTION BASED ON ACOUSTIC MULTICHANNEL EQUALIZATION. Ina Kodrasi, Simon Doclo

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