A Unified Bayesian Perspective for Conventional and Robust Adaptive Filters

TL;DR

This paper introduces a unified Bayesian framework for deriving and interpreting both conventional and robust adaptive filters, encompassing well-known algorithms and proposing new robust variants under different noise models.

Contribution

It provides a novel Bayesian perspective that unifies existing adaptive filters and derives new robust algorithms for non-Gaussian noise scenarios.

Findings

Unified Bayesian derivation of adaptive filters including LMS, NLMS, and Kalman filter.

Introduction of new robust adaptive algorithms under Laplacian noise.

Numerical results demonstrate the effectiveness and properties of the proposed filters.

Abstract

In this work, we present a new perspective on the origin and interpretation of adaptive filters. By applying Bayesian principles of recursive inference from the state-space model and using a series of simplifications regarding the structure of the solution, we can present, in a unified framework, derivations of many adaptive filters that depend on the probabilistic model of the measurement noise. In particular, under a Gaussian model, we obtain solutions well-known in the literature (such as LMS, NLMS, or Kalman filter), while using non-Gaussian noise, we derive new adaptive algorithms. Notably, under the assumption of Laplacian noise, we obtain a family of robust filters of which the sign-error algorithm is a well-known member, while other algorithms, derived effortlessly in the proposed framework, are entirely new. Numerical examples are shown to illustrate the properties and provide a better insight into the performance of the derived adaptive filters.