A unified framework for adaptive filter algorithms with variable step-size

Abstract

Employing a recently introduced framework within which a large number of classical and modern adaptive filter algorithms can be viewed as special cases, we extend this framework to cover block normalized LMS (BNLMS) and normalized data reusing LMS (NDRLMS) adaptive filter algorithms. Accordingly, we develop a generic variable step-size adaptive filter. Variable step-size normalized LMS (VSSNLMS) and VSS affine projection algorithms (VSSAPA) are particular examples of adaptive algorithms covered by this generic variable step-size adaptive filter. In this paper we introduce two new VSS adaptive filter algorithms named the variable step-size BNLMS (VSSBNLMS) and the variable step-size NDRLMS (VSSNDRLMS) based on the generic VSS adaptive filter. The proposed algorithms show the higher convergence rate and lower steady-state mean square error compared to the ordinary BNLMS and NDRLMS algorithms.

Introduction

Adaptive filtering has been, and still is, an area of active research that plays an important role in an ever increasing number of applications such as the noise cancellation, channel estimation, the channel equalization and the acoustic echo cancellation [1], [2]. The least mean square (LMS) and its normalized version (NLMS) are the workhorses of adaptive filtering. In the presence of colored input signals, the LMS and NLMS algorithms have extremely slow convergence rates. To solve this problem a number of adaptive filtering structures based on affine subspace projections [3], [4], [5], data reusing adaptive algorithms [6], [7], [8], Block adaptive filters [2] and multirate techniques have been proposed in the literature [9], [10], [11]. In all these algorithms the selected fixed step-size can change the convergence rate and the steady-state mean square error. It is well known that the steady-state mean square error (MSE) decreases, when the step-size decreases while the convergence speed increases when the step-size increases. By optimally selecting the step-size, during the adaptation, we can obtain the both fast convergence rate and low steady-state mean square error. Important examples of the two new variable step-size (VSS) versions of the NLMS and the APA algorithm (APA) can be found in [12].

In [13], the generic adaptive filter based on the weighted, estimated Wiener–Hopf equation is proposed and then we used this framework to study the transient analysis of different adaptive filter algorithms covered by our generic adaptive filter [14]. The affine projection algorithms (APA), the transform domain adaptive filters (TDAF) [15] and the Pradhan Reddy Subband Adaptive Filters (PRSAF) [16] are the particular examples that can be covered with this framework. This framework obscures the relations, commonalities and differences, between the numerous adaptive algorithms available today [17].

Our objectives in this paper are

• Extend the generic adaptive filter of [14] to cover the block and data reusing adaptive filter algorithms.

• Develop the generic variable step-size version of the generic adaptive filter. The VSSNLMS and VSSAPA of [12] can be easily derived from this generic variable step-size adaptive filter.

• Based on the generic VSS adaptive filter, we introduce the two new VSS adaptive filter algorithms named the variable step-size block normalized least mean square (VSSBNLMS) and the variable step-size NDRLMS (VSSNDRLMS), which are characterized by the fast convergence speed, and reduced steady-state mean square error when compared to the ordinary BNLMS and NDRLMS adaptive filter algorithms.

We have organized our paper as follows: In the following section the generic framework with extension of more compatibility, will be briefly reviewed. In this section we will show that the Block and data reusing adaptive filter algorithms can be derived with the extension of this generic adaptive filter. In the next section, we present the generic variable step-size update equation forming the basis of our development of the VSSBNLMS and VSSNDRLMS adaptive filter algorithms. Subsequently, we derive the VSSBNLMS and VSSNDRLMS adaptive algorithms. Finally, before concluding the paper, we demonstrate the advantages of the proposed algorithms by presenting several experimental results.

Throughout the paper, the following notations are adopted:

∥ · ∥

Euclidean norm of a vector.

∥ · ∥²

squared Euclidean norm of a vector.

$\parallel \underline{t}{\parallel }_{\mathbf{\Sigma }}^2$

Σ-weighted Euclidean norm of a column vector t defined as t^TΣt.

Tr(·)

trace of a matrix.

(·)^T

transpose of a vector or a matrix.

diag{ … }

diagonal matrix of its entries { … }.

A ⊗ B

Kronecher product of matrices A and B.

vec(T)

creating an M² × 1 column vector t through stacking the columns of the M × M matrix T.

vec(t)

creating an M × M matrix T form the M² × 1 column vector t.

λ_max

the largest eigenvalue of a matrix.

${\mathfrak{R}}^{+}$

the set of positive real numbers.

E{ · }

expectation operator.