2007 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics October 21-24, 2007, New Paltz, NY SUBBAND METHOD FOR MULTICHANNEL LEAST SQUARES EQUALIZATION OF ROOM TRANSFER FUNCTIONS Nikolay D. Gaubitch, Mark R. P. Thomas and Patrick A. Naylor Imperial College London Exhibition Road, SW7 2AZ, London, UK {ndg,mrt102,p.naylor}@imperial.ac.uk ABSTRACT Equalization of room transfer functions (RTFs) is important in many speech and audio processing applications. It is a challenging problem because RTFs are several thousand taps long and non- minimum phase and in practice only approximate measurements of the RTFs are available. In this paper, we present a subband multichannel least squares method for equalization of RTFs which is computationally efcient and less sensitive to inaccuracies in the measured RTFs compared to its fullband counterpart. Experi- mental results using simulated impulse responses demonstrate the performance of the algorithm. 1. INTRODUCTION Equalization of room transfer functions (RTFs) is essential in sev- eral applications in acoustic signal processing, including speech dereverberation [1] and sound reproduction [2]. Although, in the- ory, exact equalization is possible when multiple observations are available [2], there are many obstacles for RTF equalization in practice [3]. Consider the L-tap room impulse response of the acous- tic path between a source and the mth microphone in an M- element microphone array, hm =[hm,0 hm,1 ... hm,L-1], with a z-transform Hm(z) constituting the RTF. Equalization can be achieved, in principle, by an inverse system with transfer function Gm(z) satisfying Gm(z)Hm(z)= κz -τ , m =1, 2,...,M (1) where τ and κ are arbitrary delay and scale factors respectively. Equivalently, considering the Li tap impulse response of Gm(z), gm =[gm,0 gm,1 ... gm,L i -1] T , (1) can be written in the time domain as Hmgm = d, (2) where Hm is a (L + Li - 1) × Li convolution matrix, and d = [0 ... 0 | {z } τ κ 0 ... 0] T is the (L + Li - 1) × 1 output vector of the equalized RTFs. The problem of equalization is to nd Gm(z). In practice, RTF equalization is not straightforward since: (i) RTFs are non-minimum phase [4] and do not give a stable causal solution for Gm(z); (ii) the average difference between maxima and minima in RTFs are in excess of 10 dB [3, 5] and therefore RTFs typically contain spectral nulls that, after equalization, give strong peaks in the spectrum causing narrow band noise ampli- cation; (iii) equalization lters designed from inaccurate estimates of Hm(z) will cause distortion in the equalized signal [3]; (iv) the length L of hm can be several thousand taps in length [5]. Several methods for RTF equalization have been proposed. Single channel methods [4, 6, 7] typically result in large process- ing delay, which is problematic for many communications appli- cations, extremely long and non-causal inverse lters, and provide only approximate equalization [2]; inherently these only partially equalize deep spectral nulls, which makes them less sensitive to noise and inexact RTF estimates [1]. In the multichannel case, the non-minimum phase problem is eliminated and exact inver- sion can be achieved [2, 8]. However, it has been observed that exact equalization is of limited value in practice, when the RTF estimates contains even moderate errors [1, 3, 9]. Various alterna- tives have been proposed for improving robustness to RTF inaccu- racies [10, 11, 12]. In this paper, we introduce a new method for equalization lter design. Given a set of multichannel RTFs, we decompose the RTFs into their subband equivalent lters. These are then used to design the subband inverse lters and the equalization is performed in each subband before a fullband equalized signal is reconstructed. It is shown that this approach not only reduces the computational load, but also reduces the sensitivity to estimation errors and the effect of measurement noise in the RTFs. An important result is that this method accommodates multichannel equalization of large order systems, taking advantage of the shorter length of multichan- nel equalization lters and the low sensitivity to RTF inaccuracies of single channel methods. The remainder of the paper is organized as follows. Fullband multichannel least squares (LS) equalization is reviewed in Sec- tion 2. The subband multichannel LS method is described in Sec- tion 3. In Section 4, experimental results are given and conclusions are drawn in Section 5. 2. MULTICHANNEL LS EQUALIZATION In the multichannel case exact inversion can be achieved using Be- zout’s theorem [2, 8]: given a set of M RTFs, Hm(z), and as- suming that these do not have any common zeros, a set of lters, Gm(z), can be found such that [2, 8] M X m=1 Hm(z)Gm(z)=1. (3) The relation in (3) can be written in the time domain using (2) with τ =0 and κ =1 as M X m=1 Hmgm = Hg = d, (4) where H =[H1 H2 ... HM], and g =[g T 1 g T 2 ... g T M ] T . 978-1-4244-1619-6/07/$25.00 ©2007 IEEE 14 Authorized licensed use limited to: Imperial College London. Downloaded on January 4, 2010 at 11:03 from IEEE Xplore. Restrictions apply.