Fast communicationWidely linear general Kalman filter for stereophonic acoustic echo cancellation
Introduction
Stereophonic acoustic echo cancellation (SAEC) is a very challenging system identification problem [1]. Usually, an SAEC system consists of four adaptive filters aiming at identifying four echo paths from two loudspeakers to two microphones. The main difficulty comes from the fact that the loudspeaker (input) signals are linearly related, which results in the so-called nonuniqueness problem [2]. This issue can be addressed by manipulating the signals transmitted to the near-end room, e.g., using a preprocessor on the loudspeaker signals to make them less coherent [3], but without affecting much the stereo perception and the sound quality.
The adaptive filters used in SAEC should exploit the cross-correlation between the channels [4]. In this context, the most interesting solutions belong to the recursive least-squares (RLS) family. Due to their convergence features, these algorithms were preferred in many real-world applications [5], [6].
Recently, we proposed a different approach for the SAEC problem [1], [7], by using the widely linear (WL) model [8]. Basically, the classical two-input/two-output system with real random variables was recast as a single-input/single-output system with complex random variables. As a consequence, the four real-valued acoustic impulse responses are converted to one complex-valued impulse response. One advantage of this approach is that instead of handling two (real) output signals separately, we only handle one (complex) output signal, which is convenient for the main challenges of SAEC.
In this paper, we derive a general Kalman filter (GKF) with the WL model for SAEC, namely the WL-GKF. The term “general” refers to a different approach we propose, i.e., a block of time samples is considered at each iteration, instead of one time sample (as in the conventional approach). The main motivation behind this work is the appealing performance of the Kalman filter for echo cancellation [9], [10], [11]. Also, the WL complex Kalman filters [12], [13] were found to be attractive for many applications. The proposed algorithm has inherited some similarities with the WL augmented complex Kalman filter presented in [13]. However, the WL-GKF is derived based on a state variable model suitable for the SAEC problem. The proposed WL-GKF joins the advantages of the WL model (as described before) and the good features of the GKF [11]. Simulation results indicate that the developed algorithm outperforms the RLS counterpart. Consequently, it could represent an attractive alternative in SAEC.
Section snippets
The WL model for SAEC
In this section, we briefly review the WL model for SAEC (Fig. 1) [1], [7]. Let us denote the two input (or loudspeaker) signals by and (i.e., “left” and “right”), and the two output (or microphone signals) by and , where n is the time index. Therefore, the microphone signals are obtained aswhere and denote the stereo echo signals, and and are the near-end signals (i.e., noise or a combination of noise and
State variable model for WL SAEC
Let us express (11) by considering the P most recent time samples of the microphone signal, i.e.,whereis the input signal matrix of size and the noise signal vector, , is defined similar to . We also consider that the system to be identified is time dependent, i.e., .
In our context, is the measurement matrix and x(n) is considered as deterministic. Expression (13) is called the observation
WL general Kalman filter
It is known that, in the context of the linear sequential Bayesian approach, the optimum estimate of the state vector, , has the form [16]where is the Kalman gain matrix andis the a priori error signal vector, which is obtained using the adaptive filter coefficients at time . The a posteriori error signal vector is defined based on the adaptive filter coefficients at time n, i.e.,where
Simulation results
Simulations are performed in the context of SAEC, as described in Fig. 1. The acoustic impulse responses used for the far-end and near-end locations are shown in Fig. 2 [1]. Impulse responses in the far-end [i.e., and ] have 2048 coefficients, while the length of the impulse responses in the near-end [i.e., , , , and ] is L=512. The length of the WL adaptive filters used in the experiments is ; sample rate is 8 kHz. Two source signals are used: a
Conclusions
Due to their convergence features, RLS-based algorithms are frequently involved in SAEC. In this paper, we have motivated the use of the Kalman filter in this application. Indeed, we have developed a general Kalman filter (by considering, at each iteration, a block of time samples instead of one time sample as it is the case in the conventional approach) with the WL model for SAEC. As compared to the WL-RLS algorithm, the proposed WL-GKF compromises better between the tracking capability and
Acknowledgment
This work was supported by the UEFISCDI Romania under Grants PN-II-RU-TE no. 7/5.08.2010 and PN-II-ID-PCE-2011-3-0097.
The authors would like to thank the Handling Editor and the reviewers for the valuable comments and suggestions.
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