Prerequisites & Notation

Before You Begin

The array-fed RIS is the CommIT group's signature architecture for high-frequency wireless. It combines a small active BS array with a large passive RIS in a carefully engineered near-field configuration. Chapter 11 develops the theory, the eigenmode analysis, and the multi-user performance story. The prerequisites below are essential: near-field geometry (Ch. 3.4), SVD, and the AO framework are all used throughout.

Near-field vs. far-field channel models (Chapter 3.4)(Review ch03)
Self-check: Recall the Fraunhofer distance and what changes about the channel when $d < d_F$ .
Joint active-passive beamforming framework (Chapter 5)(Review ch05)
Self-check: State the outer AO loop and inner WMMSE for joint RIS optimization.
Hybrid analog-digital beamforming (MIMO Ch. 20–21)(Review ch20)
Self-check: For a hybrid BS with $N_t$ antennas and $N_{\text{RF}}$ RF chains, what is the effective precoder structure?
Singular value decomposition and eigenmode parallel channels(Review ch01)
Self-check: For a channel $\mathbf{H} = \mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^H$ , how many parallel sub-channels does it support and what are their capacities?
Multi-user RIS for multi-stream (Chapter 7)(Review ch07)
Self-check: What is the multiplexing gain of a $K$ -user MU-RIS system? What limits it?

Notation for This Chapter

Array-fed RIS notation. The key new object is the eigenmode decomposition of $\mathbf{H}_1$ — we use SVD throughout.

Symbol	Meaning	Introduced
$N_t$	Active array size at BS (small: $\sim 8$ - $32$ elements)	s01
$N$	Passive RIS elements (large: $\sim 256$ - $4096$ )	s01
$d_{\text{AR}}$	Array-to-RIS distance (short: few cm to few m, near-field regime)	s01
$r = \text{rank}(\mathbf{H}_1)$	Rank of the BS-RIS channel; governs the number of independent eigenmodes	s02
$\boldsymbol{\sigma}_k(\mathbf{H}_1)$	$k$ -th singular value of $\mathbf{H}_1$ , $k = 1, \ldots, r$	s02
$\mathbf{u}_k, \mathbf{v}_k$	$k$ -th left/right singular vectors of $\mathbf{H}_1$ (natural beams)	s02
$K$	Number of users served simultaneously	s03
$\mathbf{D} = \mathbf{H}_1^H \boldsymbol{\Phi}^H \mathbf{h}_2$	Multi-user effective channel matrix (cascaded BS-to-UE)	s03

← Ch 10 The Array-Fed RIS Architecture