Notation Preferences — Reconfigurable Intelligent Surfaces

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Key	Meaning	Your Symbol	Default
gauss	Real Gaussian distribution		$\mathcal{N}$
cgauss	Circularly symmetric complex Gaussian		$\mathcal{CN}$
covmat	Covariance matrix		$\boldsymbol{\Sigma}$
snr	Signal-to-noise ratio		$\text{SNR}$
n0	One-sided noise power spectral density		$N_0$
bw	Signal bandwidth (Hz)		$W$
wl	Wavelength ( $\lambda_0 = c/f_0$ at carrier)		$\lambda$
bf	Beamforming / precoding vector		$\mathbf{v}$
prec	Precoding matrix (MU-MIMO downlink)		$\mathbf{W}$
ntx	Number of transmit antennas		$N_t$
nrx	Number of receive antennas		$N_r$
noisevar	Noise variance / noise power		$\sigma^2$
tx_power	Transmit power		$P_t$
nusers	Number of users		$K$

Universal Conventions

Fixed conventions used throughout this book. These are standard across all telecommunications literature and are not customizable.

General Mathematics

Symbol	Meaning
$\mathbb{R}, \mathbb{C}$	Real and complex number fields
$\mathbb{R}^n, \mathbb{C}^n$	Real/complex $n$ -dimensional vector spaces
$\mathbb{R}^{m \times n}, \mathbb{C}^{m \times n}$	Spaces of real/complex $m \times n$ matrices
$j = \sqrt{-1}$	Imaginary unit (engineering convention)
$\Re\{\cdot\}, \Im\{\cdot\}$	Real and imaginary parts
$\|\cdot\|$	Absolute value (scalar) or cardinality (set)
$\triangleq$	Defined as
$\mathcal{O}(\cdot)$	Big-O asymptotic notation

Vectors and Matrices

Symbol	Meaning
$\mathbf{x}, \mathbf{y}, \mathbf{z}$	Column vectors (boldface lowercase)
$\mathbf{A}, \mathbf{H}, \boldsymbol{\Phi}$	Matrices (boldface uppercase or Greek)
$(\cdot)^T$	Transpose
$(\cdot)^H$	Conjugate transpose (Hermitian)
$(\cdot)^{-1}, (\cdot)^{\dagger}$	Matrix inverse / Moore-Penrose pseudoinverse
$\\|\cdot\\|, \\|\cdot\\|_2$	Euclidean norm
$\\|\cdot\\|_F$	Frobenius norm
$\text{tr}(\cdot), \det(\cdot), \text{rank}(\cdot)$	Trace, determinant, rank
$\text{diag}(\cdot)$	Diagonal matrix from vector, or diagonal of matrix
$\odot$	Hadamard (element-wise) product
$\mathbf{I}_n$	$n \times n$ identity matrix
$\mathbf{A} \succeq 0$	Positive semidefinite ordering

RIS-Specific Symbols

Symbol	Meaning
$\boldsymbol{\Phi}$	RIS phase-shift matrix, $\boldsymbol{\Phi} = \text{diag}(e^{j\theta_1}, \ldots, e^{j\theta_N})$
$\boldsymbol{\theta}$	Vector of RIS phase shifts, $[\theta_1, \ldots, \theta_N]^T \in [0, 2\pi)^N$
$N$	Number of RIS reflecting elements
$\mathbf{H}_1$	BS-to-RIS channel matrix, $N \times N_t$
$\mathbf{H}_2$	RIS-to-UE channel matrix, $N_r \times N$ (or $\mathbf{h}_2 \in \mathbb{C}^N$ for single UE)
$\mathbf{h}_d$	Direct BS-to-UE channel (LoS or NLoS), $N_t \times 1$
$\mathbf{h}_{\text{eff}}$	Effective cascaded channel: $\mathbf{h}_{\text{eff}}^H = \mathbf{h}_d^H + \mathbf{h}_2^H \boldsymbol{\Phi} \mathbf{H}_1$
$\|\phi_n\| = 1$	Unit-modulus constraint on each RIS element

Probability Basics

Symbol	Meaning
$\mathbb{E}[\cdot]$	Expectation
$\Pr(\cdot)$	Probability of an event
$\sim$	Distributed as