Prerequisites & Notation

Prerequisites for Chapter 21

This chapter introduces random matrix theory from the perspective of wireless communications. The reader needs linear algebra (eigenvalues, SVD), probability at the level of FSP Chapters 5--11, and some familiarity with complex Gaussian vectors (Ch. 8). No prior exposure to random matrix theory is assumed.

Eigenvalue decomposition and singular value decomposition
Self-check: Can you compute the eigenvalues of a Hermitian matrix and state the relationship $\sigma_i(\mathbf{A}) = \sqrt{\lambda_i(\mathbf{A}^H \mathbf{A})}$ ?
Complex Gaussian vectors and covariance matrices(Review ch08)
Self-check: Can you write the density of $\mathbf{x} \sim \mathcal{CN}(\mathbf{0}, \boldsymbol{\Sigma})$ and compute $\mathbb{E}[\mathbf{x}\mathbf{x}^H]$ ?
Convergence in distribution, weak law of large numbers(Review ch11)
Self-check: Do you know the difference between convergence in distribution and convergence in probability?
Moment generating functions and characteristic functions(Review ch09)
Self-check: Can you compute the MGF of a Gaussian random variable?
Logarithm and determinant of positive definite matrices
Self-check: Do you know that $\log\det(\mathbf{A}) = \sum_i \log \lambda_i(\mathbf{A})$ for Hermitian PD $\mathbf{A}$ ?

Notation for This Chapter

The following notation is used throughout Chapter 21. We write $\mathbf{H}$ for a generic random matrix (not necessarily a channel matrix in this chapter). The notation $F^{\mathbf{A}}$ denotes the empirical spectral distribution of a Hermitian matrix $\mathbf{A}$ .

Symbol	Meaning	Introduced
$\mathbf{H}$	Random matrix (generic), often $n \times m$ with i.i.d. entries	s01
$\mathbf{W} = \frac{1}{m}\mathbf{H}^H\mathbf{H}$	Sample covariance (Wishart-type) matrix	s01
$\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n$	Ordered eigenvalues of a Hermitian matrix	s01
$F^{\mathbf{A}}(\lambda) = \frac{1}{n}\sum_{i=1}^n \mathbf{1}_{\{\lambda_i \leq \lambda\}}$	Empirical spectral distribution (ESD) of $n \times n$ Hermitian $\mathbf{A}$	s01
$\beta = n/m$	Aspect ratio of the matrix $\mathbf{H} \in \mathbb{C}^{n \times m}$	s02
$f_{\mathrm{MP}}(\lambda; \beta)$	Marchenko-Pastur density with parameter $\beta$	s02
$\lambda_{\pm} = (1 \pm \sqrt{\beta})^2$	Endpoints of the Marchenko-Pastur support	s02
$m_F(z) = \int \frac{1}{\lambda - z}\, dF(\lambda)$	Stieltjes transform of a distribution $F$	s03
$\sigma^2$	Noise variance / noise power
$\mathcal{N}(\mu, \sigma^2)$	Gaussian distribution with mean $\mu$ and variance $\sigma^2$
$\mathcal{CN}(\mathbf{0}, \mathbf{I})$	Standard circularly symmetric complex Gaussian

← Ch 20 Why Random Matrices Matter for Communications