An efficient stabilized fast Newton adaptive filtering algorithm for stereophonic acoustic echo cancellation SAEC

Abstract

This paper addresses the field of stereophonic acoustic echo cancellation (SAEC) by adaptive filtering algorithms. Recently, we have proposed a new version of the fast Newton transversal FNTF algorithm for SAEC applications. In this paper, we propose an efficient modification of this algorithm for the same applications. This new algorithm uses a new proposed and simplified numerical stabilization technique and takes into account the cross-correlation between the inputs of the channels. The basic idea is to introduce a small nonlinearity into each channel that has the effect of reducing the inter-channel coherence while not being noticeable for speech due to self masking. The complexity of the proposed algorithm does not alter the complexity of the original version and is kept less than half the complexity of the fastest two-channel FTF filter version. Simulation results and comparisons with the extended two-channel normalized least mean square NLMS and FTF algorithms are presented.

Introduction

Acoustic echo cancellers are indispensable for communication systems such as teleconferencing in order to decrease echoes which impair the quality of communications. Theoretically, stereophonic acoustic echo cancellation SAEC can be viewed as a simple generalization of the usual single-channel acoustic echo cancellation principle to the two channel case [1], [2], [3]. In SAEC, there is a desire to have far better sound quality and sound localization than what has been provided before. The improvements in quality can be achieved by increasing the signal bandwidth and also by adding more audio channels to the system. This last fact spurred the need for multi-channel acoustic echo cancellers. Two-channel SAEC is most interesting since only complexity issues differ for the more general multi-channel case. A basic scheme for SAEC is sketched in Fig. 1, where we illustrate the concept with a transmission room on the left and a receiving room on the right. The transmission room is sometimes referred to as the far-end and the receiving room as the near-end. In this figure, when we have a signal in the transmission room (that means the source send a signal, which can be a man or a woman speaker), the two microphones mic1 and mic2 receive two amounts of signal, the first received signal amount is the direct source signal modified by the path G^v1 (then captured by mic1) and G^V2 (then captured by mic2), respectively. The second signal amounts is the presents diffuse noises B^v1 and B^v2 in the transmission room and captured by mic1 and mic2, respectively. We note that in this paper we do not take into account these two quantities of diffuse noises B^v1 and B^v2 and they are beyond the scope of this paper. On the other hand, in the receiving room, and as depicted in Fig. 1, the echo is due to acoustic coupling between the loud-speakers and the incorporated microphones in this room. In this scheme of Fig. 1, the acoustic echo paths ${\mathit{Ch}}_1$ and ${\mathit{Ch}}_2$ in the local room are modeled by adaptive FIR filters ${\mathit{h}}^{\nu 1}$ and ${\mathit{h}}^{\nu 2}$, from which their added outputs produces an estimate $\hat{y}$ of the true echo y. Indeed, the physical impulse responses ${\mathit{Ch}}_1$ and ${\mathit{Ch}}_2$ are of infinite length; nevertheless it is assumed that the filters ${\mathit{h}}^{\nu 1}$ and ${\mathit{h}}^{\nu 2}$ are “sufficiently long”, in the sense that the tails of ${\mathit{Ch}}_1$ and ${\mathit{Ch}}_2$ not modeled by ${\mathit{h}}^{\nu 1}$ and ${\mathit{h}}^{\nu 2}$ have low energy and thus can be neglected. Speaking in the sequel of “true” impulse responses means that we only consider the first parts of ${\mathit{Ch}}_1$ and ${\mathit{Ch}}_2$ which contain most of the energy, and which are assumed to be of the same size L as the model filters ${\mathit{h}}^{\nu 1}$ and ${\mathit{h}}^{\nu 2}$. In SAEC for teleconferencing, we have a fundamental problem of the possibility to identify the true impulse responses of the acoustic echo paths. This problem arises from the correlation between the two signals picked up in the remote room in this request. SAEC is fundamentally different from traditional mono echo cancellation. A SAEC, straightforwardly implemented, not only would have to track changing echo paths in the receiving room but also in the transmission room. For example, the canceller has to converge adaptively if one talker stops talking and another starts talking at a different location in the transmission room. There is no adaptive algorithm that can track such a change sufficiently fast and this scheme therefore results in poor echo suppression. Thus, a generalization of the mono AEC in the stereo case does not result in satisfactory performance. The problems of SAEC were first described in an early paper [4], and later on in [3]. The fundamental problem is that the two channels may carry linearly related signals which in turn may make the normal equations, to be solved by the adaptive algorithm, singular. This implies that there is no unique solution to the equation but an infinite number of solutions and it can be shown that all solutions (but the physically true one) depend on the transmission room. As a result, intensive studies have been made of how to handle this properly.

Generalization of the solution to the normal equations in a more practical sense was addressed in Refs. [5] and [6]. It was explained that in practice, the problem is not actually singular but extremely ill-conditioned due to the fact that the length of the adaptive filter is shorter than the echo paths of the transmission room. Furthermore, in practice, the transmission room is not completely stationary, i.e. smooth continuous changes exist, which slightly improves the situation by making the problem somewhat less ill-conditioned [7], [8]. A complete theory of non-uniqueness and characterization of the SAEC solution was presented in Refs. [9] and [10]. It is shown that the only solution to the non-uniqueness problem is to reduce the correlation between the stereo signals and an efficient low complexity method for this purpose was also given in [8] and [9]. Ref. [11] presents a combination of mono and stereo echo cancellation which has the benefit of lower complexity than a pure stereo solution. Currently, attention has been focused on the investigation of other methods that decrease the cross-correlation between the channels in order to get well-behaved estimates of the echo paths [12]. The main problem is how to reduce the correlation sufficiently without affecting stereo perception and sound quality. Early examples of SAEC implementations can be found in [13], [14], [15]. These solutions were presented before the theory and limitations of SAEC were fully understood, and were mainly based on the use of a single adaptive filter for each return channel. The performance of the SAEC is strictly affected by the choice of algorithm more than in the monophonic case. This is easily recognized since the performance of most adaptive algorithms depends on the condition number of the input signal covariance matrix. We have to recall here that there are several efficient other techniques that allow to resolve these problems differently, one of these techniques is the partial update of the filter coefficients techniques as explained in [16] and [17], and the use of the two errors filtering to avoid the problem of channel coherence as described in [18].

In the SAEC application, the condition-number is very high, and algorithms such as the LMS or the NLMS that do not take the coherence between the input signals into account, converge very slowly to the theoretical solution. It is consequently very interesting to study multi-channel adaptive filtering algorithms. A framework for multi-channel adaptive filtering can be found in Refs. [2], [3], [4], [5], [16] and [19].

In [1], we have proposed a new version of the fast Newton transversal FNTF algorithm for SAEC applications. Here, we propose an efficient modification of this algorithm for the same applications. The new proposed algorithm has good performances in SAEC case. This new algorithm takes into consideration the correlation effect of the impulse responses. We also propose, in this paper, a new numerical stabilization technique that allows good properties of the prediction part of the proposed algorithm even with speech signal as input. We describe the basic FNTF algorithms and its modified version and show simulation results to demonstrate the good performance properties of the proposed algorithm in SAEC applications in which the acoustic channels are highly correlated.

This paper is organized as follows: Section 2 explains the SAEC problem and describes the fundamental differences between mono and stereo acoustic echo cancellation. In Section 3, we present two-channel adaptive filtering algorithms with a particular and detailed presentation for the two-channel 2CFNTF algorithm. In Section 4, we give in first, the existing decorrelating versions of the algorithms used, and then we describe the proposed modified two-channel fast Newton transversal M2CFNTF algorithm which takes into account the correlation effects of the channels. In Section 5, we present a new numerical technique that stabilizes the proposed M2CFNTF algorithm. In Section 6, we give a comparison between the proposed and others algorithms in terms of complexity. Finally, simulations results are presented in Section 7. The notations that we have used in this paper are fairly standard. Boldface symbols are used for vectors and matrices. We also have the following notations:

L: length of the adaptive filter;

N: length of the predictors;

t: discrete time index;

(.)^T: transpose.