Prerequisites & Notation

Before You Begin

Reconfigurable intelligent surfaces (RIS) sit at the intersection of three disciplines: electromagnetic propagation, array processing, and non-convex optimization. Before starting, make sure the tools below are familiar — we will use them without re-derivation.

Free-space path loss: the $1/d^2$ law and the Friis equation(Review ch05)
Self-check: Given a transmit power $P_t$ , antenna gains $G_t, G_r$ , and distance $d$ , can you write the received power $P_r$ without looking it up?
Phased-array beamforming: steering vectors and array factor(Review ch07)
Self-check: Can you write the steering vector of an $N$ -element ULA at angle $\theta$ in terms of the element spacing and wavelength $\lambda$ ?
MIMO channel matrix, beamforming vectors, and matched filtering(Review ch15)
Self-check: For a single-user MISO channel $\mathbf{h} \in \mathbb{C}^{N_t}$ , what beamforming vector maximizes $|\mathbf{h}^H \mathbf{v}|^2$ under $\|\mathbf{v}\|=1$ ?
Complex Gaussian random variables and circular symmetry(Review ch02)
Self-check: If $X \sim \mathcal{CN}(0, \sigma^2)$ , what is the distribution of $|X|^2$ ?
Inner products, Cauchy–Schwarz, and unit-norm constraints(Review ch01)
Self-check: Can you state the equality condition of the Cauchy–Schwarz inequality $|\mathbf{a}^H \mathbf{b}|^2 \leq \|\mathbf{a}\|^2 \|\mathbf{b}\|^2$ ?

Notation for This Chapter

Symbols introduced in this chapter. See also the NGlobal Notation Table master table. Customizable symbols use $\ntn{}$ tokens; the values shown are the defaults from the registry. RIS-specific symbols ( $\boldsymbol{\Phi}$ , $\boldsymbol{\theta}$ , $\mathbf{H}_1$ , $\mathbf{H}_2$ ) are not yet in the token registry — we use raw LaTeX for these consistently across all RIS chapters.

Symbol	Meaning	Introduced
$N$	Number of RIS reflecting elements	s01
$\boldsymbol{\Phi}$	RIS phase-shift matrix, $\boldsymbol{\Phi} = \text{diag}(e^{j\theta_1}, \ldots, e^{j\theta_N})$ — the programmable object	s01
$\boldsymbol{\theta} = [\theta_1, \ldots, \theta_N]^T$	Vector of RIS phase shifts, $\theta_n \in [0, 2\pi)$	s01
$\mathbf{H}_1 \in \mathbb{C}^{N \times N_t}$	BS-to-RIS channel matrix (rows = RIS elements, columns = BS antennas)	s02
$\mathbf{H}_2 \in \mathbb{C}^{N_r \times N}$	RIS-to-UE channel matrix; written $\mathbf{h}_2 \in \mathbb{C}^N$ for single-antenna UE	s02
$\mathbf{h}_d \in \mathbb{C}^{N_t}$	Direct BS-to-UE channel (possibly zero under blockage)	s02
$\mathbf{h}_{\text{eff}}^H$	Effective end-to-end channel: $\mathbf{h}_{\text{eff}}^H = \mathbf{h}_d^H + \mathbf{h}_2^H \boldsymbol{\Phi} \mathbf{H}_1$	s02
$\mathbf{v}$	BS beamforming vector, $\mathbf{v} \in \mathbb{C}^{N_t}$ , $\\|\mathbf{v}\\| = 1$	s02
$d_1, d_2$	BS-to-RIS and RIS-to-UE distances	s04
$d_0$	Direct BS-to-UE distance	s04
$\lambda$	Carrier wavelength	s04
$P_t$	Transmit power	s03
$\sigma^2$	Receiver noise variance	s03
$\text{SNR}$	Received signal-to-noise ratio	s03

Back to chapter overview What is a Reconfigurable Intelligent Surface?