2019•arXiv (Cornell University)Open access

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

Xiang‐Gen Xia

Abstract

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

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Key concepts: Free probability, Random matrix, Simple (philosophy), Notation, Eigenvalues and eigenvectors, Computer science, Probability theory, Information theory