Prerequisites & Notation

Before You Begin

Real RIS hardware supports only a finite set of phase states — typically $2^B$ levels for $B$ -bit control. The continuous-phase algorithms of Chapters 5–7 need a quantization layer before deployment. This chapter formalizes discrete-phase optimization and quantifies its loss.

Continuous-phase AO and passive algorithms (Chapters 5–6)(Review ch05)
Self-check: Write the continuous-phase passive subproblem and state how element-wise BCD solves it.
Hardware phase quantization (Chapter 2)(Review ch02)
Self-check: Recall the $\text{sinc}^2(\pi/2^B)$ coherent SNR loss for $B$ -bit quantization.
Projection onto discrete sets, rounding
Self-check: For a set $\Theta_B = \{0, \Delta\theta, 2\Delta\theta, \ldots\}$ , write the projection $\arg\min_{\vartheta \in \Theta_B} |\vartheta - \theta|_{\text{mod } 2\pi}$ .
Combinatorial optimization: greedy algorithms, exhaustive search complexity
Self-check: For $N$ RIS elements and $L$ phase levels per element, what is the brute-force search space?
SDR relaxation (Chapter 6)(Review ch06)
Self-check: Recall the lifted PSD variable and the relaxation of the rank-1 constraint.

Notation for This Chapter

Discrete-phase notation. We introduce the level count $L$ , bit depth $B$ , and projection operators; the basic RIS symbols carry over from Chapters 1–6.

Symbol	Meaning	Introduced
$B$	Bits of phase resolution per RIS element	s01
$L = 2^B$	Number of phase levels per element	s01
$\Delta\theta = 2\pi/L$	Phase step size	s01
$\Theta_B = \{0, \Delta\theta, \ldots, (L-1)\Delta\theta\}$	Set of admissible phase values	s01
$\mathcal{M}_B = \{e^{j\theta} : \theta \in \Theta_B\}^N$	Discrete feasible set for $\boldsymbol{\phi}$ (a finite discrete torus of $L^N$ points)	s01
$\mathcal{P}_{\Theta_B}(\theta)$	Nearest-level projection of $\theta \in [0, 2\pi)$ onto $\Theta_B$	s02
$\eta_B$	Discrete-phase efficiency factor: $\text{sinc}^2(\pi/2^B)$ for coherent case	s04
$f^\star_B, f^\star_\infty$	Best achievable objectives with $B$ -bit and continuous phases	s04

← Ch 7 The Discrete-Phase RIS Problem