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

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.