Free deconvolution for signal processing applications

TL;DR

This paper explores how free probability theory, particularly free deconvolution, can be applied to analyze large random matrices in signal processing, aiding in source detection and eigenvalue distribution estimation.

Contribution

It demonstrates the use of multiplicative free deconvolution to derive limit eigenvalue distributions and improve estimators for covariance matrices in noisy systems.

Findings

Free deconvolution simplifies eigenvalue distribution estimation.

It provides asymptotic descriptions for large sample covariance matrices.

The method enhances source detection in noisy environments.

Abstract

Situations in many fields of research, such as digital communications, nuclear physics and mathematical finance, can be modelled with random matrices. When the matrices get large, free probability theory is an invaluable tool for describing the asymptotic behaviour of many systems. It will be shown how free probability can be used to aid in source detection for certain systems. Sample covariance matrices for systems with noise are the starting point in our source detection problem. Multiplicative free deconvolution is shown to be a method which can aid in expressing limit eigenvalue distributions for sample covariance matrices, and to simplify estimators for eigenvalue distributions of covariance matrices.