Prerequisites & Notation

Before You Begin

This chapter develops the signal model that the rest of the book will optimize. If any prerequisite below is unfamiliar, a quick refresher will save time later — every optimization chapter assumes fluency with the cascaded-channel formulas derived here.

ULA / UPA steering vectors: $\mathbf{a}(\theta) \in \mathbb{C}^{N}$ (Review ch07)
Self-check: Can you write the steering vector of an $N_1 \times N_2$ UPA at AoA $(\theta, \phi)$ as the Kronecker product of two ULA vectors?
Rayleigh and Ricean fading: LoS component + i.i.d. Gaussian(Review ch06)
Self-check: Given a K-factor $K_{\text{Rice}}$ , can you decompose the channel into deterministic and random parts?
Kronecker correlation model: $\mathbf{H} = \mathbf{R}_r^{1/2} \tilde{\mathbf{H}} \mathbf{R}_t^{1/2}$ (Review ch02)
Self-check: What does the rank of $\mathbf{R}_t$ tell you about the number of significant transmit eigendirections?
Fraunhofer distance: $d_F = 2D^2/\lambda$ where $D$ is the aperture dimension(Review ch05)
Self-check: Compute $d_F$ for a $1\text{ m}^2$ aperture at 28 GHz.
Rank and condition number of a matrix; keyhole channels(Review ch15)
Self-check: Why does a rank-1 channel limit MIMO multiplexing to a single stream?

Notation for This Chapter

Channel-model symbols. We reuse $\boldsymbol{\Phi}, \mathbf{H}_1, \mathbf{H}_2, N$ from Chapters 1–2 and introduce steering vectors, correlation matrices, and near-field coordinates.

Symbol	Meaning	Introduced
$\mathbf{a}_{\text{BS}}(\theta)$	BS array steering vector at AoD $\theta$ , $\in \mathbb{C}^{N_t}$	s02
$\mathbf{a}_{\text{RIS}}(\theta, \phi)$	RIS array steering vector at AoA/AoD $(\theta, \phi)$ , $\in \mathbb{C}^{N}$	s02
$\mathbf{a}_{\text{UE}}(\phi)$	UE array steering vector, $\in \mathbb{C}^{N_r}$ (scalar 1 for single-antenna UE)	s02
$K_{\text{Rice}}$	Ricean K-factor of the cascaded path	s03
$\mathbf{R}_{\text{RIS}} \in \mathbb{C}^{N \times N}$	Spatial correlation matrix of the RIS elements in a rich-scattering field	s03
$d_F = 2D^2/\lambda$	Fraunhofer (far-field) distance for an aperture of dimension $D$	s04
$D$	Largest dimension of the RIS aperture	s04
$P_t, \sigma^2$	Transmit power and noise variance (carried over)	s01

← Ch 2 The Cascaded Channel, Dimensions, and Linearity