A Simple Introduction to Free Probability Theory and its Application to Random Matrices

TL;DR

This paper provides a simplified introduction to free probability theory and demonstrates its application in analyzing large random matrices, particularly for calculating eigenvalue distributions and mutual information in massive MIMO systems.

Contribution

It offers an accessible explanation of free probability theory and illustrates its practical use in large-scale random matrix analysis for signal processing.

Findings

Eigenvalue distributions can be computed using second order statistics.

Application to mutual information calculation in massive MIMO systems.

Simplified approach makes free probability theory more accessible.

Abstract

Free probability theory started in the 1980s has attracted much attention lately in signal processing and communications areas due to its applications in large size random matrices. However, it involves with massive mathematical concepts and notations, and is really hard for a general reader to comprehend. The main goal of this paper is to briefly describe this theory and its application in random matrices as simple as possible so that it is easy to follow. Applying free probability theory, one is able to calculate the distributions of the eigenvalues/singular-values of large size random matrices using only the second order statistics of the matrix entries. One of such applications is the mutual information calculation of a massive MIMO system.