Prerequisites & Notation

Before You Begin

This chapter takes the hardware-aware design thread of Part IV to its most extreme form: what if almost every antenna element were passive? A Reconfigurable Intelligent Surface (RIS) replaces costly RF chains with tunable phase shifters that reflect an incoming wave toward a chosen direction. On paper this delivers enormous aperture at near-zero power, but the Tx $\to$ RIS $\to$ Rx link suffers the infamous double-fading loss $|\beta_1 \beta_2|$ that erases most of the nominal aperture gain. The CommIT array-fed RIS architecture of Caire and collaborators fixes this by placing a small active array just in front of the passive RIS: the active array does the heavy radiating, the passive RIS does the large-aperture focusing, and the end-to-end link only pays one path-loss factor. We assume familiarity with the following prior material.

Hybrid beamforming: analog/digital split, fully-connected and subarray structures, phase-shifter codebooks(Review ch20)
Self-check: Can you write the two-stage hybrid transmit signal $\mathbf{x} = \mathbf{F}_{\text{RF}} \mathbf{F}_{\text{BB}} \mathbf{s}$ and state when the analog stage is purely phase-constrained?
Near-field vs far-field propagation, spherical wavefronts, Fraunhofer distance(Review ch17)
Self-check: Can you compute $d_F = 2 D^2 / \lambda$ for an RIS with $D = 1$ m at $f_0 = 28$ GHz and decide whether a user at $5$ m sits in the near field?
Steering vectors and array manifolds for ULAs and UPAs(Review ch07)
Self-check: Can you write the steering vector $\mathbf{a}(\phi,\theta)$ of a half-wavelength ULA with $N_t$ elements?
Linear precoding (ZF, MMSE, MRT) and SVD-based eigenmode transmission(Review ch06)
Self-check: Can you derive the ZF precoder $\mathbf{W} = \mathbf{H}^{H} (\mathbf{H} \mathbf{H}^{H})^{-1}$ and explain why it decouples users at high SNR?
Shannon capacity $\log_2(1 + \text{SNR})$ and sum-rate maximization(Review ch13)
Self-check: Can you write the Gaussian MIMO capacity $C = \log_2 \det(\mathbf{I} + \text{SNR}\, \mathbf{H} \mathbf{H}^{H} / N_t)$ in closed form?
Path loss models and link budgets for mmWave/sub-THz bands(Review ch15)
Self-check: Can you write the Friis path loss in dB and explain why $\beta \propto 1/r^2$ gives the far-field power decay?

Notation for This Chapter

Symbols introduced or specialized in this chapter. Throughout the chapter we use $\mathbf{H}_{\text{Tx-RIS}}$ for the forward (active array $\to$ RIS) channel, $\mathbf{H}_{\text{RIS-Rx}}$ for the reflected (RIS $\to$ user) channel, and $\boldsymbol{\phi}$ for the RIS phase profile. The chapter-local $N_{\text{RIS}}$ and $N_a$ are not global \ntn{} symbols; they are defined for this chapter only. See NGlobal Notation Table for the master table.

Symbol	Meaning	Introduced
$N_{\text{RIS}}$	Number of passive RIS elements (tiles). Typically $N_{\text{RIS}} \in \{256, 1024, 4096\}$	s01
$N_a$	Number of active elements in the array feed (array-fed RIS). Typically $N_a \ll N_{\text{RIS}}$	s02
$\Phi_n = e^{j\phi_n}$	Reflection coefficient of the $n$ -th RIS element (unit modulus, programmable phase)	s01
$\boldsymbol{\phi} \in [0, 2\pi)^{N_{\text{RIS}}}$	Vector of RIS phase shifts	s01
$\boldsymbol{\Phi} = \text{diag}(e^{j\phi_1}, \ldots, e^{j\phi_{N_{\text{RIS}}}})$	Diagonal RIS reflection matrix	s01
$\mathbf{H}_{\text{Tx-RIS}}$	Forward channel matrix from the Tx (or array feed) to the RIS, $N_{\text{RIS}} \times N_t$ (or $\times N_a$ )	s01
$\mathbf{H}_{\text{RIS-Rx}}$	Reflected channel matrix from the RIS to the receiver, $N_r \times N_{\text{RIS}}$	s01
$\mathbf{H}_{\text{eff}}$	End-to-end effective channel $\mathbf{H}_{\text{RIS-Rx}}\, \boldsymbol{\Phi}\, \mathbf{H}_{\text{Tx-RIS}}$	s01
$\beta_1, \beta_2$	Large-scale path-loss amplitudes of the Tx $\to$ RIS and RIS $\to$ Rx links (double fading)	s01
$d_1, d_2$	Physical ranges Tx-to-RIS and RIS-to-Rx (metres)	s01
$\mathbf{W}$	Active-array precoding matrix at the array feed, $N_a \times K$	s04
$\mathbf{v}_{k}$	Per-user active-array precoder for user $k$ , $\mathbf{v}_{k} \in \mathbb{C}^{N_a}$	s04
$G_{\text{RIS}}$	Aperture (array) gain of the passive RIS, $G_{\text{RIS}} \approx N_{\text{RIS}}$	s01
$\mathbf{a}(\phi, \theta)$	Steering vector of the active array toward angle $(\phi, \theta)$	s02
$\mathbf{a}_{\text{RIS}}(\phi, \theta)$	Steering vector of the RIS toward angle $(\phi, \theta)$	s02
$P_c^{\text{RF}}$	DC power per active RF chain (used in digital vs array-fed comparison)	s05
$P_c^{\text{RIS}}$	Static DC power per RIS element (phase-shifter control, near zero)	s05

← Ch 20 RIS Fundamentals and the Double-Fading Problem