A New Approach to Adaptive Signal Processing

TL;DR

This paper introduces a unified linear algebraic framework for adaptive signal processing, deriving key algorithms from a single fundamental equation, simplifying understanding and enabling access to advanced methods in the field.

Contribution

It presents a systematic, background-free derivation of major ASP algorithms from the equation Ax=b, unifying diverse methods under one algebraic approach.

Findings

All algorithms derive from the single equation Ax=b.

The approach links basic ASP algorithms to advanced computational methods.

Provides a pathway to understanding modern research techniques in signal processing.

Abstract

A unified linear algebraic approach to adaptive signal processing (ASP) is presented. Starting from just Ax=b, key ASP algorithms are derived in a simple, systematic, and integrated manner without requiring any background knowledge to the field. Algorithms covered are Steepest Descent, LMS, Normalized LMS, Kaczmarz, Affine Projection, RLS, Kalman filter, and MMSE/Least Square Wiener filters. By following this approach, readers will discover a synthesis; they will learn that one and only one equation is involved in all these algorithms. They will also learn that this one equation forms the basis of more advanced algorithms like reduced rank adaptive filters, extended Kalman filter, particle filters, multigrid methods, preconditioning methods, Krylov subspace methods and conjugate gradients. This will enable them to enter many sophisticated realms of modern research and development. Eventually, this one equation will not only become their passport to ASP but also to many highly specialized areas of computational science and engineering.