Prerequisites & Notation

Before You Begin

STAR-RIS extends the passive RIS framework to a full-space coverage architecture: each element simultaneously reflects to one half-space and transmits to the other. This chapter builds on the optimization machinery of Chapters 5–7 and adds the amplitude-splitting constraint unique to STAR-RIS.

Joint active-passive beamforming framework (Ch. 5-7)(Review ch05)
Self-check: State the AO framework for joint $\mathbf{W}$ and $\boldsymbol{\Phi}$ optimization.
Electromagnetic wave transmission and reflection at interfaces(Review ch05)
Self-check: For a dielectric interface with permittivities $\epsilon_1, \epsilon_2$ , write the Fresnel coefficients.
Power conservation: $|r|^2 + |t|^2 \leq 1$ for passive elements
Self-check: Why must a lossless bi-directional passive element satisfy $|r|^2 + |t|^2 = 1$ ?
Discrete-phase RIS (Chapter 8)(Review ch08)
Self-check: Recall the $\text{sinc}^2(\pi/2^B)$ quantization loss.
Time-sharing and resource allocation fundamentals
Self-check: Given a time-sharing split $\alpha, 1-\alpha$ , what is the average rate of users served in each slot?

Notation for This Chapter

STAR-RIS adds two coefficients per element (one for reflection, one for transmission) instead of passive RIS's one. We use $r_n, t_n$ for the per-element reflection/transmission complex coefficients.

Symbol	Meaning	Introduced
$r_n = a_n^r e^{j\theta_n^r}$	Reflection coefficient of element $n$ (amplitude + phase)	s01
$t_n = a_n^t e^{j\theta_n^t}$	Transmission coefficient of element $n$	s01
$\boldsymbol{\Phi}^r = \text{diag}(r_n)$	Reflection-side phase-shift matrix	s01
$\boldsymbol{\Phi}^t = \text{diag}(t_n)$	Transmission-side phase-shift matrix	s01
$\|a_n^r\|^2 + \|a_n^t\|^2 = 1$	Energy-conservation constraint (energy-splitting protocol)	s02
$\beta_n \in \{0, 1\}$	Mode-switching binary indicator: 1 = reflect, 0 = transmit	s02
$\tau$	Time fraction allocated to reflection in time-switching protocol	s02
$\mathcal{U}^r, \mathcal{U}^t$	Reflection-side and transmission-side user sets	s03

← Ch 9 The STAR-RIS Concept