Constantin Paleologu

A Robust Variable Forgetting Factor Recursive Least-Squares Algorithm for System Identification

The performance of the recursive least-squares (RLS) algorithm is governed by the forgetting factor. This parameter leads to a compromise between (1) the tracking capabilities and (2) the misadjustment and stability. In this letter, a variable forgetting factor RLS (VFF-RLS) algorithm is proposed for system identification. In general, the output of the unknown system is corrupted by a noise-like signal. This signal should be recovered in the error signal of the adaptive filter after this one converges to the true solution. This condition is used to control the value of the forgetting factor. The simulation results indicate the good performance and the robustness of the proposed algorithm.

A Variable Step-Size Affine Projection Algorithm Designed for Acoustic Echo Cancellation

The adaptive algorithms used for acoustic echo cancellation (AEC) have to provide (1) high convergence rates and good tracking capabilities, since the acoustic environments imply very long and time-variant echo paths, and (2) low misadjustment and robustness against background noise variations and double-talk. In this context, the affine projection algorithm (APA) and different versions of it are very attractive choices for AEC. However, an APA with a constant step-size parameter has to compromise between the performance criteria (1) and (2). Therefore, a variable step-size APA (VSS-APA) represents a more reliable solution. In this paper, we propose a VSS-APA derived in the context of AEC. Most of the APAs aim to cancel p (i.e., projection order) previous a posteriori errors at every step of the algorithm. The proposed VSS-APA aims to recover the near-end signal within the error signal of the adaptive filter. Consequently, it is robust against near-end signal variations (including double-talk). This algorithm does not require any a priori information about the acoustic environment, so that it is easy to control in practice. The simulation results indicate the good performance of the proposed algorithm as compared to other members of the APA family.

An Efficient Proportionate Affine Projection Algorithm for Echo Cancellation

Proportionate-type normalized least-mean-square algorithms were developed in the context of echo cancellation. In order to further increase the convergence rate and tracking, the ¿proportionate¿ idea was applied to the affine projection algorithm (APA) in a straightforward manner. The objective of this letter is twofold. First, a general framework for the derivation of proportionate-type APAs is proposed. Second, based on this approach, a new proportionate-type APA is developed, taking into account the ¿history¿ of the proportionate factors. The benefit is also twofold. Simulation results indicate that the proposed algorithm outperforms the classical one (achieving faster tracking and lower misadjustment). Besides, it also has a lower computational complexity due to a recursive implementation of the ¿proportionate history¿.

Variable Step-Size NLMS Algorithm for Under-Modeling Acoustic Echo Cancellation

In acoustic echo cancellation (AEC) applications, where the acoustic echo paths are extremely long, the adaptive filter works most likely in an under-modeling situation. Most of the adaptive algorithms for AEC were derived assuming an exact modeling scenario, so that they do not take into account the under-modeling noise. In this letter, a variable step-size normalized least-mean-square (VSS-NLMS) algorithm suitable for the under-modeling case is proposed. This algorithm does not require any a priori information about the acoustic environment; as a result, it is very robust and easy to control in practice. The simulation results indicate the good performance of the proposed algorithm.

On Regularization in Adaptive Filtering

Regularization plays a fundamental role in adaptive filtering. An adaptive filter that is not properly regularized will perform very poorly. In spite of this, regularization in our opinion is underestimated and rarely discussed in the literature of adaptive filtering. There are, very likely, many different ways to regularize an adaptive filter. In this paper, we propose one possible way to do it based on a condition that intuitively makes sense. From this condition, we show how to regularize four important algorithms: the normalized least-mean-square (NLMS), the signed-regressor NLMS (SR-NLMS), the improved proportionate NLMS (IPNLMS), and the SR-IPNLMS.

Study of the General Kalman Filter for Echo Cancellation

The Kalman filter is a very interesting signal processing tool, which is widely used in many practical applications. In this paper, we study the Kalman filter in the context of echo cancellation. The contribution of this work is threefold. First, we derive a different form of the 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. Second, we show how this general Kalman filter (GKF) is connected with some of the most popular adaptive filters for echo cancellation, i.e., the normalized least-mean-square (NLMS) algorithm, the affine projection algorithm (APA) and its proportionate version (PAPA). Third, a simplified Kalman filter is developed in order to reduce the computational load of the GKF; this algorithm behaves like a variable step-size adaptive filter. Simulation results indicate the good performance of the proposed algorithms, which can be attractive choices for echo cancellation.

Double-talk robust VSS-NLMS algorithm for under-modeling acoustic echo cancellation

Most of the adaptive algorithms used for acoustic echo cancellation (AEC) are designed assuming an exact modeling scenario (i.e., the acoustic echo path and the adaptive filter have the same length) and a single-talk context (i.e., the near-end speech is absent). In real-world AEC applications, the adaptive filter works most likely in an under-modeling situation, i.e., its length is smaller than the length of the acoustic impulse response, so that the under-modeling noise is present. Also, the double-talk case is almost inherent, so that a double-talk detector (DTD) is usually involved. Both aspects influence and limit the algorithms performance. Taking into account these two practical issues, a double-talk robust variable step size normalized least-mean- square (VSS-NLMS) algorithm is proposed in this paper. This algorithm is nonparametric in the sense that it does not require any information about the acoustic environment, so that it is robust and easy to control in practice.

An optimized NLMS algorithm for system identification

The normalized least-mean-square (NLMS) adaptive filter is widely used in system identification. In this paper, we develop an optimized NLMS algorithm, in the context of a state variable model. The proposed algorithm follows a joint-optimization problem on both the normalized step-size and regularization parameters, in order to minimize the system misalignment. Consequently, it achieves a proper compromise between the performance criteria, i.e., fast convergence/tracking and low misadjustment. Simulations performed in the context of acoustic echo cancellation indicate the good features of the proposed algorithm. HighlightsAn optimized normalized least-mean-square (NLMS) algorithm is developed for system identification, in the context of a state variable model.The proposed algorithm is based on a joint-optimization on both the normalized step-size and regularization parameters, in order to minimize the system misalignment.The performance of the proposed joint-optimized NLMS (JO-NLMS) algorithm is evaluated in the framework of acoustic echo cancellation.The JO-NLMS algorithm achieves both fast convergence and tracking, but also low misadjustment.

Regularization of the Affine Projection Algorithm

The affine projection algorithm (APA) is an attractive choice for echo cancellation, mainly for its convergence features. A matrix inversion is required within the APA. For practical reasons, this matrix needs to be regularized, i.e., a positive constant is added to the elements of its main diagonal. This regularization parameter is of great importance in practice since, if it is not chosen properly, the APA may never converge, especially under low-signal-to-noise-ratio conditions. In this brief, we propose a formula for choosing the value of the regularization parameter, aiming at attenuating the effects of the noise in the adaptive filter estimate. Simulations performed in an acoustic echo cancellation scenario prove the validity of the approach in different noisy environments.

Proportionate Adaptive Filters From a Basis Pursuit Perspective

In this letter, we show that the normalized least-mean-square (NLMS) algorithm and the affine projection algorithm (APA) can be decomposed as the sum of two orthogonal vectors. One of these vectors is derived from an ℓ2-norm optimization problem while the other one is simply a good initialization vector. By replacing this optimization with the basis pursuit, which is based on the ℓ1-norm optimization, we derive the proportionate NLMS (PNLMS) algorithm and the proportionate APA (PAPA). Many other adaptive filters can be derived following this approach, including new ones.