A Novel Family of Adaptive Filtering Algorithms Based on the Logarithmic Cost

Abstract

Abstract-We introduce a novel family of adaptive filtering algorithms based on a relative logarithmic cost. The new family intrinsically combines the higher and lower order measures of the error into a single continuous update based on the error amount. We introduce important members of this family of algorithms such as the least mean logarithmic square (LMLS) and least logarithmic absolute difference (LLAD) algorithms that improve the convergence performance of the conventional algorithms. However, our approach and analysis are generic such that they cover other well-known cost functions as described in the paper. The LMLS algorithm achieves comparable convergence performance with the least mean fourth (

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

“…similar to the stable-NLMF algorithm [24,25]. In practice, in order to avoid a division by zero, we also propose the regularized stable-PNLMF algorithm modifying (6) such that…”

“…Hence, we propose the Krylovproportionate normalized least mean mixed norm (KPNLMMN) algorithm having a convex combination of the mean-square and the mean-fourth error objectives. In addition, we point out that the stability of the mean-fourth error based algorithms depends on the initial value of the adaptive filter weights, the input and noise power [23][24][25]. In order to enhance the stability of the introduced algorithms, we further introduce the stable-PNLMF and the stable-KPNLMF algorithms [24,25].…”

“…In addition, we point out that the stability of the mean-fourth error based algorithms depends on the initial value of the adaptive filter weights, the input and noise power [23][24][25]. In order to enhance the stability of the introduced algorithms, we further introduce the stable-PNLMF and the stable-KPNLMF algorithms [24,25]. Finally we provide a complete performance analysis for the introduced algorithms, i.e., the transient, the steady-state and the tracking performance analyses.…”

Sparse Least Logarithmic Absolute Difference Algorithm with Correntropy-Induced Metric Penalty

“…, w M (n)] T stands for the estimate of the parameter vector, σ is the kernel width for CIM, and τ is a design parameter [22]. In (4), the LLAD term J LLAD (n) = |e(n)| − 1 τ ln (1 + τ |e(n)|) is robust to impulsive noise, of which the detailed analysis can be found in [22], and the CIM term (with smaller width) J CIM (n) is a sparsity-inducing term, and the two terms are balanced by a weight factor λ ≥ 0. Based on the adaptation cost (4), a gradient-based adaptive algorithm can be easily derived as follows:…”

“…However, the SA usually exhibits slower convergence performance especially for highly correlated input signals. To address this problem, the logarithmic least absolute deviation (LLAD) algorithm was recently developed in [22], which is robust to impulsive interferences and converges much faster than the original sign algorithm.…”