# Romain Couillet

Romain Couillet and Mérouane Debbah

Random Matrix Methods for Wireless Communications
Cambridge University Press
September 2011.

Cambridge University Press online ordering (UK)
Amazon online ordering (US/FR)

## Abstract

Blending theoretical results with practical applications, this book provides an introduction to random matrix theory and shows how it can be used to tackle a variety of problems in wireless communications. The Stieltjes transform method, free probability theory, combinatoric approaches, deterministic equivalents and spectral analysis methods for statistical inference are all covered from a unique engineering perspective. Detailed mathematical derivations are presented throughout, with thorough explanation of the key results and all fundamental lemmas required for the reader to derive similar calculus on their own. These core theoretical concepts are then applied to a wide range of real-world problems in signal processing and wireless communications, including performance analysis of CDMA, MIMO and multi-cell networks, as well as signal detection and estimation in cognitive radio networks. The rigorous yet intuitive style helps demonstrate to students and researchers alike how to choose the correct approach for obtaining mathematically accurate results.

The book chapters are being constantly updated. Here are the latest of these.
• Chapter 6 - Deterministic equivalents: This chapter will be profoundly changed in the future in order to better introduce the Gaussian tools and to confront it to the Bai-Silverstein approach. Some changes were already implemented. [chapter]
• The statement and proofs of Theorems 6.16-6.17 were incorrect and have been updated.
• Chapter 7 - Spectrum Analysis: A better consistency between the results for the sample covariance matrix and the information plus noise models has been implemented. [chapter]
• The statement of Theorem 7.8 was erroenous and has been updated.
• Chapter 9 - Extreme eigenvalues and eigenspace projections: Formerly called "Extreme Eigenvalues", this chapter was largely revisited. [chapter]
• Many new results concerning the spiked model are introduced.
• The spiked model results are now written in a more consistent manner.
• The method of orthogonal polynomials is pushed further to explain the convergence to the Tracy-Widom law.

## List of Typos

Some errors concerning statements of theorems of the 2011 version of the book are listed below.
• Theorem 3.18. The correct expression of $\Theta^2$ is $$\Theta^2=-\log\left(1-\frac{cm_F(-1/x)^{\color{red}2}}{(1+cm_F(-1/x))^2}\right)+\kappa \frac{cm_F(-1/x)^{\color{red}2}}{(1+cm_F(-1/x))^2}.$$
• Theorems 6.14. The entries of ${\bf Y}_N$ have variance $\sigma_{ij}^2/\color{red}n$.
• Theorems 6.16-6.17. The results only hold for $c_i<1$ and $\lim\sup_n c_i<1$, instead of $0\leq c_i\leq 1$ as previously stated.
• Theorem 7.8. The statement was mixed up with the statement of Theorem 7.2. See updates in the new Chapter 7.
• Theorems 9.2-9.3,9.10. Some sign and division symbols are erroneous. See updates in the new Chapter 9.