Prerequisites

Before You Begin

This chapter brings together linear algebra (Chapter 1), probability theory (Chapter 2), fading channel models (Chapter 6), antenna arrays (Chapter 7), and information-theoretic foundations (Chapter 11). MIMO theory is the synthesis of all these threads: linear algebra provides the SVD and matrix decompositions; probability gives us the random channel model; fading channels define the propagation environment; array theory establishes the spatial dimension; and information theory supplies the capacity formulas.

Singular value decomposition (SVD) and eigenvalue decomposition(Review ch01)
Self-check: Can you compute the SVD $\mathbf{A} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H$ of a $3 \times 2$ matrix and identify its rank from the singular values?
Matrix rank, null space, and condition number(Review ch01)
Self-check: Can you determine the rank of a matrix from its singular values and explain what a large condition number implies about numerical stability?
Complex Gaussian random vectors and covariance matrices(Review ch02)
Self-check: Can you write the PDF of a circularly symmetric complex Gaussian vector $\mathbf{z} \sim \mathcal{CN}(\mathbf{0}, \mathbf{R})$ and compute expectations like $\mathbb{E}[\mathbf{z}\mathbf{z}^H]$ ?
Rayleigh and Ricean fading models(Review ch06)
Self-check: Can you describe how Rayleigh fading arises from the sum of many scattered paths and state the distribution of the fading envelope and instantaneous SNR?
Antenna array response vectors and beamforming(Review ch07)
Self-check: Can you write the array steering vector $\mathbf{a}(\theta) = [1, e^{j2\pi d\sin\theta/\lambda}, \ldots]^T$ for a uniform linear array and explain spatial filtering?
Channel capacity, mutual information, and water-filling(Review ch11)
Self-check: Can you state the capacity of a scalar AWGN channel $C = \log_2(1 + \text{SNR})$ and explain the water-filling principle for parallel Gaussian channels?

Chapter 15 Notation

Key symbols introduced or heavily used in this chapter. Bold uppercase denotes matrices; bold lowercase denotes column vectors.

Symbol	Meaning	Introduced
$\mathbf{H}$	MIMO channel matrix ( $n_r \times n_t$ )	s01
$n_t$	Number of transmit antennas	s01
$n_r$	Number of receive antennas	s01
$\mathbf{x}$	Transmitted signal vector ( $n_t \times 1$ )	s01
$\mathbf{y}$	Received signal vector ( $n_r \times 1$ )	s01
$\mathbf{n}$	Additive noise vector, $\mathbf{n} \sim \mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I})$	s01
$\sigma_i$	$i$ -th singular value of $\mathbf{H}$	s01
$\kappa(\mathbf{H})$	Condition number $\sigma_{\max}/\sigma_{\min}$ of the channel matrix	s01
$\mathbf{R}_{t}, \mathbf{R}_{r}$	Transmit and receive spatial correlation matrices	s02
$\mathbf{Q}$	Input covariance matrix $\mathbb{E}[\mathbf{x}\mathbf{x}^H]$	s03
$P$	Total transmit power constraint	s03
$C_{\mathrm{MIMO}}$	MIMO channel capacity (bits/s/Hz)	s03
$r$	Multiplexing rate (in DMT context)	s06
$d(r)$	Diversity gain as a function of multiplexing rate	s06
$\text{SNR}$	Signal-to-noise ratio $P / \sigma^2$	s01

Notation Note: Noise Vector

This chapter uses $\mathbf{n}$ for the additive noise vector, following the convention of Tse & Viswanath (2005) and most MIMO literature. The Ferkans library standard (see notation.yaml) designates $\mathbf{w}$ for noise to avoid collision with the number of antennas $n_t$ , $n_r$ . In this chapter, the subscripts on the antenna counts ( $n_t$ , $n_r$ ) provide sufficient disambiguation.

← Ch 14 The MIMO Channel Matrix