Prerequisites & Notation

Before You Begin

This chapter blends the communication-theoretic model of Chapter 1 with the electromagnetic and circuit hardware that realizes it. Basic familiarity with the tools below will let you understand why a unit cell looks the way it does, not just what the resulting $\phi_n$ is.

Reflection coefficient from impedance mismatch: $\Gamma = (Z_L - Z_0)/(Z_L + Z_0)$
Self-check: For a matched load ( $Z_L = Z_0$ ), what is $\Gamma$ ? For an open circuit?
Varactor diodes: capacitance vs. reverse bias voltage
Self-check: Why does $C(V)$ determine the resonant frequency of an LC tank?
PIN diodes as RF switches
Self-check: What are the typical ON and OFF resistances of a PIN diode?
Array factor of an $N$ -element ULA (Telecom Ch. 7)(Review ch07)
Self-check: Can you write the array factor as $\sum_n e^{j(kn d \cos\theta + \theta_n)}$ and explain each term?
Uniform-rounding quantization noise and SNR loss (Telecom Ch. 23)(Review ch23)
Self-check: What is the mean-squared error of a $B$ -bit uniform quantizer on $[-\pi, \pi)$ ?

Notation for This Chapter

Hardware-specific symbols. The core RIS notation ( $\boldsymbol{\Phi}, \boldsymbol{\theta}, N$ ) carries over from Chapter 1.

Symbol	Meaning	Introduced
$B$	Number of control bits per RIS element (phase resolution)	s02
$\theta^{(B)}_n$	$B$ -bit quantized phase at element $n$	s02
$\Delta\theta$	Quantization step size, $\Delta\theta = 2\pi / 2^B$	s02
$a_n(\theta_n)$	Element-level amplitude response as a function of the commanded phase	s03
$\mathbf{Z}$	Inter-element mutual impedance matrix, $N \times N$ , used to model coupling	s04
$\boldsymbol{\Phi}_{\text{full}}$	Full (non-diagonal) RIS response matrix accounting for mutual coupling	s04
$Z_0$	Free-space impedance, $\approx 377\,\Omega$	s01
$\lambda$	Carrier wavelength	s01

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