Silviu Ciochină

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.

An overview on optimized NLMS algorithms for acoustic echo cancellation

Acoustic echo cancellation represents one of the most challenging system identification problems. The most used adaptive filter in this application is the popular normalized least mean square (NLMS) algorithm, which has to address the classical compromise between fast convergence/tracking and low misadjustment. In order to meet these conflicting requirements, the step-size of this algorithm needs to be controlled. Inspired by the pioneering work of Prof. E. Hänsler and his collaborators on this fundamental topic, we present in this paper several solutions to control the adaptation of the NLMS adaptive filter. The developed algorithms are “non-parametric” in nature, i.e., they do not require any additional features to control their behavior. Simulation results indicate the good performance of the proposed solutions and support the practical applicability of these algorithms.

Recursive Least-Squares Algorithms

Thanks to their fast convergence rate, recursive least-squares (RLS) algorithms are very popular in SAEC [1]. Indeed, it is well known that the convergence rate of RLS-type algorithms are not much affected by the nature of the input signal, even when this one is ill-conditioned. Actually, the very first SAEC prototype was based on the fast RLS (FRLS) algorithm, which was implemented in subbands [2]. In this chapter, we derive the RLS and FRLS algorithms in the WL context.

Adaptive filtering for the identification of bilinear forms

Abstract Bilinear systems are involved in many interesting applications, especially related to the approximation of nonlinear systems. In this context, the bilinear term is usually defined in terms of an input–output relation (i.e., with respect to the data). Recently, a different approach has been introduced, by defining the bilinear term with respect to the impulse responses of a spatiotemporal model, which resembles a multiple-input/single-output (MISO) system. Also, in this framework, the Wiener filter has been studied to address the identification problem of these bilinear forms. Since the Wiener filter may not be always very efficient or convenient to use in practice, we propose in this paper an adaptive filtering approach. Consequently, we develop and analyze some basic algorithms tailored for the identification of bilinear forms, i.e., least-mean-square (LMS), normalized LMS (NLMS), and recursive-least-squares (RLS). Simulations performed in the context of system identification (based on the MISO system approach) support the theoretical findings and indicate the appealing performance of these algorithms.

Efficient recursive least-squares algorithms for the identification of bilinear forms

Abstract Due to its fast convergence rate, the recursive least-squares (RLS) algorithm is very popular in many applications of adaptive filtering, including system identification scenarios. However, the computational complexity of this algorithm represents a major limitation in applications that involve long filters. Moreover, when the parameter space becomes large, the system identification problem is more challenging and the adaptive filters should be able to cope with this aspect. In this paper, we focus on the identification of bilinear forms, where the bilinear term is defined with respect to the impulse responses of a spatiotemporal model. From this perspective, the solution requires a multidimensional adaptive filtering technique. Recently, the RLS algorithm tailored for bilinear forms (namely RLS-BF) was developed for this purpose. In this framework, the contribution of this paper is mainly twofold. First, in order to reduce the computational complexity of the RLS-BF algorithm, two versions based on the dichotomous coordinate descent (DCD) method are proposed; due to its arithmetic features, the DCD algorithm represents one of the most attractive alternatives to solve the normal equations. However, in the bilinear context, we need to consider the particular structure of the input data and the additional related challenges. Second, in order to improve the robustness of the RLS-BF algorithm in noisy environments, a regularized version is developed, together with a method to find the regularization parameters, which are related to the signal-to-noise ratio (SNR). Furthermore, using a proper estimation of the SNR, a variable-regularized RLS-BF algorithm is designed and two DCD-based low-complexity versions are proposed. Due to their nature, these variable-regularized algorithms have good robustness features against additive noise, which make them behave well in different noisy condition scenarios. Simulation results indicate the good performance of the proposed low-complexity RLS-BF algorithms, with appealing features for practical implementations.

Echo and Noise Suppression as a Binaural Noise Reduction Problem

This chapter deals with the important problem of residual-echo-plus-noise suppression. When reformulated with the WL model, this problem becomes similar to binaural noise reduction. The most useful filters for suppression are then derived.

System Identification with the Wiener Filter

The main objective of SAEC is to identify the four acoustic impulse responses, h t, ll, h t, rl, h t, lr, h t, rr, or, equivalently, the complex acoustic impulse response, h t, of the stereophonic system. In this chapter, we show how to estimate h t with the Wiener approach [1], which has been extremely useful in many problems, in general, and in echo cancellation, in particular. The Wiener filter is derived from the mean-square error (MSE) criterion. We will discuss the well-known nonuniqueness problem that occurs in SAEC but reformulated in the WL model context. Because of this problem, some pre-processing of the complex input signal may be necessary. We also study, in the context of SAEC, the deterministic algorithm, which is an iterative approach to find the Wiener filter. Finally, we end this chapter by discussing the regularized MSE criterion, which can be very useful for the derivation of filters that promote sparsity. This approach has the potential to solve the SAEC problem without distorting the input signals.

A Class of Stochastic Adaptive Filters

In this chapter, we derive, study, and analyze a class of stochastic adaptive filters for SAEC with the WL model. All developed algorithms try to converge to the optimal Wiener filter. We start with the classical stochastic gradient algorithm, which is a good approximation of the deterministic algorithm studied in the previous chapter.

A Class of Affine Projection Algorithms

Affine projection algorithms (APAs) are very good candidates for echo cancellation. The two main reasons for that are: they may converge and track much faster than the NLMS algorithm and they can be efficient from an arithmetic complexity viewpoint. In this chapter, we derive some useful APAs for SAEC with the WL model.

Variable Step-Size Adaptive Filters for Echo Cancellation

The principal issue in acoustic echo cancellation (AEC) is to estimate the impulse response between the loudspeaker and the microphone of a hands-free communication device. Basically, we deal with a system identification problem, which can be solved by using an adaptive filter. The most common adaptive filters for AEC are the normalized least-mean-square (NLMS) algorithm and the affine projection algorithm (APA). These two algorithms have to compromise between fast convergence rate and tracking on the one hand and low misadjustment and robustness (against the near-end signal) on the other hand. In order to meet these conflicting requirements, the step-size parameter of the algorithms needs to be controlled. This is the motivation behind the development of variable step-size (VSS) algorithms. Unfortunately, most of these algorithms depend on some parameters that are not always easy to tune in practice. The goal of this chapter is to present a family of non-parametric VSS algorithms (including both VSS-NLMS and VSS-APA), which are very suitable for realistic AEC scenarios. They are developed based on another objective of AEC application, i.e., to recover the near-end signal from the error signal of the adaptive filter. As a consequence, these VSS algorithms are equipped with good robustness features against near-end signal variations, like double-talk. This idea can be extended even in the case of the recursive least-squares (RLS) algorithm, where the overall performance is controlled by a forgetting factor. Therefore, a variable forgetting factor RLS (VFF-RLS) algorithm is also presented. The simulation results indicate that these algorithms are reliable candidates for real-world applications.