BLIND SEPARATION OF LINEAR CONVOLUTIVE MIXTURES USING ORTHOGONAL FILTER BANKS. Milutin Stanacevic, Marc Cohen and Gert Cauwenberghs

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1 , BLIND SEPARATION OF LINEAR CONVOLUTIVE MIXTURES USING ORTHOGONAL FILTER BANKS Milutin Stanacevic, Marc Cohen and Gert Cauwenberghs Department of Electrical and Computer Engineering and Center for Language and Speech Processing Johns Hopins University, Baltimore MD 8 ABSTRACT We propose an algorithm and architecture for real-time blind source separation of linear convolutive mixtures using orthogonal filter bans. The adaptive algorithm derives from stochastic gradient descent optimization of a performance metric that quantifies independence not only across the reconstructed sources, but also across time within each source. The special case of a Laguerre section offers a compact representation with a small number of filter taps even under severe reverberant conditions, facilitating real-time implementation in a modular and scalable parallel architecture. Simulations of the proposed architecture and update rule validate the approach.. INTRODUCTION The signal processing problem of separating and deconvolving observed mixtures of unnown independent sources without nowledge of the mixing medium, is nown as blind source separation (BSS) or independent component analysis (ICA). The problem is addressed extensively in the literature and different algorithms for a wide range of applications in speech processing, wireless communications and biomedical signal processing exist. BSS algorithms have been studied in the informationtheoretic and statistical signal processing framewor. Maximization of entropy of transformed output signals and minimization of mutual information of output signals are main approaches in deriving learning algorithms from informationtheoretic perspective [,, 3]. Maximum lielihood estimation (MLE) approach leads to same algorithms as infomax principle. In statistical signal processing, the contrast functions are chosen with respect to statistical measures of independances, i.e. cumulants and nonlinear moments [4, 5]. For linear convolutive mixtures, algorithms have been formulated in time and frequency domain based on the above This wor was supported by ONR YIP (N46) and NSF Career (MIP-7346). # $% &' $( ) $( + + # $% ) $% + + Fig.. Problem Statement *!!" principles. Amari et. all [6] derived a time-domain algorithm based on a modified maximum entropy formulation. The same algorithm was obtained by Cohen and Cauwenberhgs [7] using nonlinear moments. If it can be assumed that the sources are non-stationary, variety of methods, based on the second-order statistics, can be used for separation [8, ]. The formulation in the frequency-domain is computationaly more appealing, but the ICA indeterminacy in each frequency been has to be solved [, ]. There are also algorithms that combine two domains, with the separation criterion expressed in time-domain, while the rest is done in frequency-domain [, 3]. Our objective is to reduce the complexity of algorithms by choosing an appropriate representation of the mixing medium. It has been shown that a Laguerre filterban offers a versatile and compact filter basis for use in adaptive filtering

2 Q 8 applications [4, 5], and the wor presented here extends the use of Laguerre and other orthogonal filter bans to the domain of Independent Component Analysis.. PROBLEM STATEMENT Figure schematizes the problem unnown independent sources propagate through an unnown medium and are observed by an array of sensors. The tas is to recover sources from observed signals using only the assumption that the source signals are mutually independent. The sensors inputs -/.3 are convolutive mixtures of channel impulse response and input signals 4. -/.3/5768.BA@ 4.DC'AE () <;>=@ where FHGBI.BA@ denotes channel impulse response between source J and sensor K at time A. Matrix.BA@ is an L xm di- is the number of sources and L mensional matrix, where M is number of sensors. The assumption is that LONPM, since in the case of more sensors than sources prior information about sources is necessary for separation [6]. To recover the sources, the observed signals are processed by a transformation matrix Q RS.S5T68 U; =.VAEW-/.XDCYA@/5T68 Z[;>= Q.^] 4.XDC_] where ` GaI denotes the filter that is the inverted channel impulse response FbGBI and `cgbi is a total impulse response from source deg to output signal fgi. We can rewrite the equations (), in the operator form [] with -/.3S5.hH<i 4.3Wj (3) RS.X3S5 Q.^hbUi -/.3Wj>5 Q.^hbUi 4.X3"j (4) Q.h*3l5 Q.BA@[hm.hH/5.BA@[hm Q.^hn5 Q.^h.hH (5) representing the h -transform of the channel, unmixing transformation and total impulse response, respectively. The following notation h m i 4.3WjY5 4.XoCPAE is used. We can formulated our aim as optimizing Q prqts u<v.hh such that Q.^h/5xwzy{.hH (6) where w is M xm permutation matrix and yo.^hb is a diagonal matrix, where diagonal entries represent delayed delta impulses.dc'ae. Fig.. Room impulse response 3. REPRESENTATION The mixing matrix.^hb represents the physics of propagation between sources and sensors. As an example, consider a sound source recorded in a room using a microphone. The recorded signal will consist of a direct (delayed) copy of the sound source and multi-path copies of signal, modified by the environment. The channel impulse response in this case is the room impulse response, which is dependent on reverberation and absorption characteristics of the room. An FIR filter representing typical room impulse response, as shown in Figure, requires a large number of delay elements (8 in this case) [7]. This damped response can be compactly represented using a Laguerre filter ban, a cascade of a lowpass filter followed by identical all-pass filters. The transfer function of a Laguerre filter is given by } u.hh/5~ =.hh<i ~.hhwj u 7ƒ5x < * ˆ t r (7) with ~ =.hh/5š m@ Œ m@ Ž H and ~.hh/5 Ž H m> m> Ž H (8) Parameter F represents the pole location, and for F 5 the Laguerre filter is reduced to a simple delay line. Table compares the mean square error between the 8-tap FIR filter and Laguerre filter responses for different length of Laguerre filters and different pole location F. For F 5T and filter length 5 H, the approximated response is shown in Figure 3. To implement the inverting or unmixing matrix of filters, we can employ different structures using FIR filters to approximate the inverse solution requires a large number of taps, and IIR adaptive filters can result in instability,

3 ž ž 8 8 Similarly, the total impulse response is `%GBI.^ /5 68 u ; = GBI. u.^ () Fig. 3. Approximation of the room impulse response of Figure using a Laguerre filter ban with a =.5 and filter length = 4. filter length a =. a =.5 a = Table. Mean square error between the room impulse response and its approximation with Laguerre sections of finite length. especially if the impulse response has non-minimum phase. Laguerre filters provide local stability due to fixed poles, but still have advantages of IIR filter. As shown, the room impulse response can be represented by using fewer Laguerre filters, and therefore a lower number of filters for inverting the response. 4. ADAPTIVE SOURCE SEPARATION The mixing coefficients `cgai. can be expanded through a set of orthogonal functions u with coefficients ` GBI. /5 68 GBI.3 u. () u ;>=E GBI.3 equal to GaI.X3S5 68 ;>= `cgbi. u. U () GBI.3/5 68 ;>= ` GaI.^ u.^ () Substituting () in equation, we obtain Rš. š5 68 <;>=.BAEœ.^hbUi 4.^ "j" (3) The cost function used as an optimization criterion is a scalar measure of output signal independence [5] 5 G ; I ; Ÿ ; m.^ i f G.3 f I. C WjC GBI Ÿ = (4) where ihj is the expectation operator and is a normalization constant. This cost function not only attempts to separate, but also to deconvolve (whiten) the outputs. For the simplicity of the derivation, we will also assume that the sources are white to start with, that is i d.xdc deª. C d "j>5 ª <«* (5) which transforms (4) into 5 G I Ÿ Z G.BA@ I.^]. Z.X C_ XC <GBIe Ÿ = Gradient descent of (6) produces an update rule (6).^]±5 Cc²³7 µ 5 ² g.^] (7) Ç ¹ œ.hh>i RS.X3"jœ Z.hHcºtR»(..BA@"¼n½U¾ where ² is the learning rate constant. A stochastic on-line weight adaptation rule is obtained by removing the expectation operator in (8) [5]. Independence of output signals beyond second-order statistics (removal of higher order cummulants) is obtained by applying component-wise antisymmetric nonlinear functions.à and Á. [4, 5]. The selection of the functions. and Á. depends on the statistics of source signals and have been studied extensively in.hhl5.^hb.^hb literature. Finally, substituting.^]o5 ² g.] (8) C œ.hh>i..rs.3ââwj œ Z.^hb ¹ Á.^R»š.X3.BA@ ½¾

4 C Z Ä Ë G Z Ê Î Ï È Å Ð Ã H + + H + H + H + H + H H cì () cì () cì c () c () c H cì () cì () cì c () c () c H cì 3 () c Ì 3 () cì 3 c 3 () c 3 () c 3 g Óg HÍ H Ò f Ò f Åy à ( ) Åy à ( ) Ésum È j ( γ + Ã, q) à c ij ( q) È Ã z (, q) j Ñ g (y ()) i H (z)[f(y ()] i H H H H H H H H H Ä Äx ( à ) x à ( ) Ç ( a) xæ à ( ) 3 Ç ( b) à c ij ( q) + L É i s um ( y q(z)[x j()] ) Fig. 4. Parallel architecture (a) example system bloc diagram for M 5T ˆ L 5ÕÔ, and } 5Ô, (b) unit cell diagram. When applying delay line for the filters œ.hhö5 h m we retrieve the convolutive ICA algorithm derived in [6, 7]. 4.. Laguerre Filter Ban Laguerre filter ban is a special case of orthogonal filter ban that offers compact representation. The update rule (8) for Laguerre filter becomes.]o5 ².^] () C }.hh>i. Rš.3Â"j } Z.hH ¹ Á. R»S..VAE ½¾ which can be rewritten by rearranging contributions to the update over time as.]o5 ².^] C ~ =.^hbui. RS.X3 Wj } Z m.^hbui Á. R»(.3Â.VAE"jؾl The problem with this form is that it is non-causal. Whenever ] Ù A, the update of weight.] depends on the future outputs RS. Ú{ÛÝÜÞCÝß. The problem can be solved by delaying the update on the RHS side of () [6]. We propose a modification of the rule, by omitting the noncausal terms for which ] Ù A [7]. Therefore () simplifies.]o5 ².^] () ~ =.^hbui. RS.X3 Wj U; = } Z m.^hbui Á. R»(.3Â.VAE"j ¾ For this learning rule, we propose the architecture shown in Figure 4F. An enlarged view of the unit cell is shown in Figure 4à. The sensor inputs -/.X3 are presented at the bottom of the system and fed to Laguerre filter bans. The signals from the filter bans on bottom are projected across the columns of the array. The outputs RS.X3 are obtained by summing across the rows, from left to right. The output signals are passed through nonlinearities. and Á., and they propagate along the rows from the rigth. The inner product of output signals and weights á I.@]/5 GaI.^] Á. f G. () is accumulated along the columns of array and fed into the filter ban on the top of system. The signals h I.@]S5 U; = } Z m á I.@AE (3) generated from the filter ban on top are projected along the columns and multiplied with the low-pass version of signal. Rš.3 to generating the weight update. The advantages of this architecture are local instantaneous computations, reduced complexity and scalability. The architecture lends itself to efficient implementation using either DSPs or custom parallel VLSI. 5. SIMULATIONS We simulated our proposed architecture and learning rule in a small system two input source } and two outputs (M 5â ) with Laguerre filter length 5 ã. Inputs d.x3 and d.x3 are uniform white noise signals ä ivcz À< <j. The weights.] ]5T c < } CÝ, are initialized with uniform random weights ä ircz Uj. For simple implementation, the functions and Á are the identity map.^rs. å  æçrs.^å and the signum function Á.^R».^å SæÕèÂéê. R».^å. All simulation results are referenced to the sources, in terms of, because of the equivalence of the equations

5 h() h() h() h() Fig. 5. Trajectory of the coefficients the triangularized update rule GaI.^] over time for of the learning updates under any transformtion.^hb_5.^hb.^hb. Figure 5 show the trajectories of all ìë Yëoã weights in over time. Figure 6 shows the impulse responses of the ëý filters. It is clear that f. corresponds to C d.x3 and f.x3 to C d.x3, which is one of many valid solutions to this unmixing/deconvolution tas. The rate of convergence for the proposed architecture using Laguerre filters is approximately ten times faster than the same architecture using simple delay line. One interesting side effect of breaing time symmetry by ommitting the non-causal terms in the update rule is giving rise to a minimum phase response with minumum delay in the reconstruction of the sources. 6. CONCLUSION In this paper, we have addressed the problem of blind source separation of linear convolutive mixtures using general orthogonal filter bans. The implementation using Laguerre filter bans offers a compact representation with reduced number of taps, and a faster convergence, compared with tapped delay line. Laguerre filters have a free parameter, the pole location, which can be optimized for a particular application. The proposed algorithm can be efficiently implemented in a scalable parallel architecture, with local updates. 7. REFERENCES [] S. Amari and A. Cichoci, Adaptive blind signal processing - neural networ approaches, Proc. of the IEEE, vol. 86, no., pp. 6 48, Oct Fig. 6. Impulse response of each filter after convergence [] A. Bell and T. Sejnowsi, An information maximization approach to blind separation and blind deconvolution, Neural Computation, vol. 7, pp. 5, 5. [3] S. Amari, A. Cichoci, and H. Yang, A new learning algorithm for blind signal separation, in Adv. Neural Information Processing Systems, Cambridge MA, 6, vol. 8, pp , MIT Press. [4] J. Herault and J. Jutten, Space or time adaptive signal processing by neural networ models, in Neural Networs for Computing AIP conference proceedings 5, J.S. Dener, Ed., New Yor, 86, American Institute of Physics. [5] A. Cichoci and R. Unbehauen, Robust neural networs with on-line learning for blind identification and blind separation of sources, IEEE Trans. Circuits and Systems-I Fundamental Theory and Applications, vol. 43, no., pp. 84 6, 6. [6] S. Amari, S. Douglas, A. Cichoci, and H.H. Yang, Novel on-line adaptive learning algorithms for blind deconvolution using the natural gradient algorithm, in Proc. of th IFAC Symposium on System Identification, Kitayushu City, Japan, July 7, pp [7] M. Cohen and G. Cauwenberghs, Blind separation of linear convolutive mixtures through parallel stochastic optimization, in Proc. IEEE Int. Symp. Circuits and Systems (ISCAS 8), 8, vol. 3, pp. 7. [8] E. Weinstein, M. Feder, and A. Oppenheim, Multichannel signal separation by decorrelation, IEEE

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