Prerequisites & Notation

Before You Begin

This chapter builds directly on Chapter 1's channel hardening and favorable propagation analysis, but now asks: what does the actual channel matrix look like? Before proceeding, make sure you are comfortable with the following.

MIMO channel model: $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{w}$ , dimensions, role of channel matrix(Review mimo/ch01)
Self-check: Can you write the received signal model for $N_r \times N_t$ MIMO and identify each term?
Complex Gaussian random vectors: $\mathcal{CN}(\boldsymbol{\mu}, \mathbf{C})$ distribution, density, whitening(Review fsp/ch08)
Self-check: What is the covariance matrix of $\mathbf{Ax}$ when $\mathbf{x} \sim \mathcal{CN}(\mathbf{0}, \mathbf{C})$ ?
Array steering vector for a ULA: $\mathbf{a}(\theta) = [1, e^{j\psi}, \ldots, e^{j(N-1)\psi}]^T$ , $\psi = \pi \sin\theta$ (Review telecom/ch07)
Self-check: Why does the steering vector depend on $\sin\theta$ and not $\theta$ directly?
Propagation basics: path loss, delay spread $\sigma_\tau$ , coherence bandwidth $B_c$ , coherence time $T_c$ (Review telecom/ch05)
Self-check: How is $B_c$ related to $\sigma_\tau$ ?
Eigenvalue decomposition and positive semidefinite matrices(Review telecom/ch01)
Self-check: If $\mathbf{A} \succeq 0$ , what can you say about its eigenvalues?
Basic understanding of channel hardening and favorable propagation (Chapter 1 of this book)(Review mimo/ch01)
Self-check: Under what channel model do channel hardening and favorable propagation hold exactly?

Notation for This Chapter

Symbols introduced or specialized in this chapter. Global conventions are in NGlobal Notation Table. The $\\ntn{}$ token enables per-user symbol customization.

Symbol	Meaning	Introduced
$\mathbf{H} \in \mathbb{C}^{N_r \times N_t}$	MIMO channel matrix ( $N_r$ receive × $N_t$ transmit)	s01
$\mathbf{R}_t \in \mathbb{C}^{N_t \times N_t}$	Transmit-side spatial covariance matrix	s02
$\mathbf{R}_r \in \mathbb{C}^{N_r \times N_r}$	Receive-side spatial covariance matrix	s02
$\Delta\theta$	Angular spread (half-angle) at base station	s02
$\theta_0$	Mean angle of departure / arrival	s02
$\mathbf{G} \in \mathbb{C}^{N_r \times N_t}$	i.i.d. $\mathcal{CN}(0,1)$ random matrix (fast-fading component)	s01
$\mathbf{U} \in \mathbb{C}^{N \times N}$	Unitary DFT matrix for virtual channel transformation	s03
$\tilde{\mathbf{H}} = \mathbf{U}_r^H \mathbf{H} \mathbf{U}_t$	Virtual (angular-domain) channel matrix	s03
$\kappa$	Ricean $K$ -factor (ratio of LOS to diffuse power)	s01
$d$	Inter-element spacing (typically $\lambda/2$ )	s01
$L$	Number of propagation paths (clusters)	s02
$\alpha_\ell$	Complex gain of path $\ell$	s02
$\phi_\ell, \psi_\ell$	Angle of departure and angle of arrival of path $\ell$	s02
$\mathbf{W}_W \in \mathbb{C}^{N_t \times N_t}$	Weichselberger coupling matrix (elementwise power in angle-angle domain)	s02

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