The Krylov-proportionate normalized least mean fourth approach: Formulation and performance analysis

Abstract

a b s t r a c tWe propose novel adaptive filtering algorithms based on the mean-fourth error objective while providing further improvements on the convergence performance through proportionate update. We exploit the sparsity of the system in the mean-fourth error framework through the proportionate normalized least mean fourth (PNLMF) algorithm. In order to broaden the applicability of the PNLMF algorithm to dispersive (non-sparse) systems, we introduce the Krylov-proportionate normalized least mean fourth (KPNLMF) algorithm using the Krylov subspace projection technique. We propose the Krylov-proportionate normalized least mean mixed norm (KPNLMMN) algorithm combining the mean-square and mean-fourth error objectives in order to enhance the performance of the constituent filters. Additionally, we propose the stable-PNLMF and stable-KPNLMF algorithms overcoming the stability issues induced due to the usage of the mean fourth error framework. Finally, we provide a complete performance analysis, i.e., the transient and the steadystate analyses, for the proportionate update based algorithms, e.g., the PNLMF, the KPNLMF algorithms and their variants; and analyze their tracking performance in a non-stationary environment. Through the numerical examples, we demonstrate the match of the theoretical and ensemble averaged results and show the superior performance of the introduced algorithms in different scenarios.

A Novel Generalized Group-Sparse Mixture Adaptive Filtering Algorithm

“…In the well-known SGM, we attempt to seek out the minimum of the cost functions regardless of the unknown parameter space. Herein, we use a Riemannian metric structure (RMS) presented in [24] instead of the Euclidean space for providing a rapid convergence within the Pt updating method.…”

“…In the RMS, the cost functions are given by J (d, x,ĥ) [24], where the distance betweenĥ(n + 1) and h(n) becomes D ĥ (n + 1),ĥ(n)…”

“…with Riemannian metric tensor G −1 (n) [24]. According to the proportionate-type AF updating scheme, we can write the reported proportionate-type algorithms via finding out the solution of the following generic updating [25,26]ĥ…”

Sparse-Aware Bias-Compensated Adaptive Filtering Algorithms Using the Maximum Correntropy Criterion for Sparse System Identification with Noisy Input

“…The correntropy-induced metric (CIM) [ 5 ], as an effective SPE, has been utilized to improve the performance of the algorithm in SSI, resulting in the CIM-NLMS, CIM-NLMF, CIM-LMMN, and CIM-MCC algorithms [ 22 , 24 , 25 ]. Correspondingly, the latter algorithms are proportionate-type AFAs (including proportionate NLMS [ 19 ], proportionate NLMF [ 26 ], proportionate MCC [ 27 ], and so on) which use the gain matrix to improve performance. Although those algorithms above make full use of the sparsity of the system, they lack consideration of the noisy input problem.…”

Stochastic behavior of the nonnegative least mean fourth algorithm for stationary Gaussian inputs and slow learning

“…The LMF algorithm was proposed in [20], where it was verified that it could outperform the LMS algorithm in the presence of non-Gaussian measurement noise. This desirable property has led to a series of studies about the convergence behaviors of the LMF algorithm and some of its variants [21]- [31]. Recently, a nonnegative LMF (NNLMF) algorithm was proposed in [32] to improve the performance of the NNLMS algorithm under non-Gaussian measurement noise.…”