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Key	Meaning	Default
gauss	Real Gaussian distribution	$\mathcal{N}$
cgauss	Circularly symmetric complex Gaussian	$\mathcal{CN}$
covmat	Covariance matrix	$\boldsymbol{\Sigma}$
ch	Channel matrix (MIMO, OFDM, general)	$\mathbf{H}$
snr	Signal-to-noise ratio	$\text{SNR}$
n0	One-sided noise power spectral density	$N_0$
bw	Signal bandwidth (Hz)	$W$
fc	Carrier frequency	$f_0$
noise	AWGN noise vector (system model)	$\mathbf{w}$
fk	$k$ -th OFDM subcarrier frequency	$f_k$
doppler_max	Maximum Doppler frequency	$f_D$
coh_bw	Coherence bandwidth	$B_c$
coh_time	Coherence time	$T_c$
noisevar	Noise variance / noise power	$\sigma^2$
delta_f	OFDM subcarrier spacing	$\Delta f$
n_ofdm	Number of OFDM subcarriers / DFT size	$N$
delay	Round-trip delay for Tx $i$ , Rx $j$	$\tau_{i,j}$
dd_spread	Delay-Doppler spreading function: $h(\tau, \nu) = \sum_{i=1}^P h_i\,\delta(\tau-\tau_i)\,\delta(\nu-\nu_i)$	$h$
zak	Zak transform: $Z_f(t, \nu) = \sum_{k \in \mathbb{Z}} f(t - k/\nu_0)\,e^{-j2\pi k\nu/\nu_0}$	$Z$
sft	Symplectic Fourier transform (2D Fourier on the delay-Doppler plane)	$\mathcal{F}_s$
isft	Inverse symplectic Fourier transform	$\mathcal{F}_s^{-1}$
dd_delay	Delay variable (continuous) on the DD plane	$\tau$
dd_doppler	Doppler variable (continuous) on the DD plane	$\nu$
otfs_m	Number of delay bins in an OTFS frame	$M$
otfs_npaths	Number of resolvable propagation paths	$P$
dd_res_delay	Delay resolution: $\Delta\tau = 1/W$	$\Delta\tau$
dd_res_doppler	Doppler resolution: $\Delta\nu = 1/T$	$\Delta\nu$
otfs_frame	OTFS frame duration (total signaling time)	$T$

Universal Conventions for the OTFS Book

Fixed conventions used throughout this book. These are standard across the OTFS and delay-Doppler communications literature and are not customizable.

General Mathematics

Symbol	Meaning
$\mathbb{R}, \mathbb{C}, \mathbb{Z}$	Real, complex, integer number fields
$\mathbb{C}^{M \times N}$	Space of complex $M \times N$ delay-Doppler matrices
$j = \sqrt{-1}$	Imaginary unit (engineering convention)
$\delta(\cdot)$	Dirac delta (continuous) or Kronecker delta (discrete)
$\star\star$	Two-dimensional convolution (delay and Doppler)
$\otimes$	Kronecker product
$\lfloor \cdot \rfloor, \lceil \cdot \rceil$	Floor and ceiling functions
$\triangleq$	Defined as

Symbol	Meaning
$\mathbf{x}, \mathbf{y}, \mathbf{w}$	Column vectors (always boldface lowercase)
$\mathbf{H}, \mathbf{A}$	Matrices (always boldface uppercase)
$(\cdot)^T, (\cdot)^*, (\cdot)^H$	Transpose, complex conjugate, conjugate transpose (Hermitian)
$\\|\cdot\\|$	Euclidean ( $\ell_2$ ) norm
$\text{vec}(\cdot)$	Column-wise vectorization of a matrix
$\mathbf{I}_N$	$N \times N$ identity matrix
$\mathbf{F}_N$	$N \times N$ unitary DFT matrix

Symbol	Meaning
$\tau, \nu$	Continuous delay and Doppler variables
$\ell, k$	Discrete delay and Doppler indices: $\ell = 0, \ldots, M-1$ ; $k = 0, \ldots, N-1$
$M, N$	Number of delay bins, number of Doppler bins
$P$	Number of resolvable propagation paths
$\Delta\tau = 1/W$	Delay resolution (inverse of bandwidth)
$\Delta\nu = 1/T$	Doppler resolution (inverse of frame duration)
$T$	OTFS frame duration (seconds)
$W$	Signal bandwidth (Hz)
$h(\tau, \nu)$	Delay-Doppler spreading function
$h_i, \tau_i, \nu_i$	Complex gain, delay, Doppler shift of the $i$ -th path
$x_{DD}, y_{DD}$	Transmit / receive signals in the DD domain
$x_{TF}, y_{TF}$	Transmit / receive signals in the time-frequency domain

Symbol	Meaning
$Z_f(t, \nu)$	Zak transform of $f(t)$
$\mathcal{F}_s$	Symplectic Fourier transform (2D DFT on the DD plane)
$\mathcal{F}_s^{-1}$	Inverse symplectic Fourier transform (ISFFT)
$\nu_0$	Zak transform period along the time axis: $\nu_0 = 1/T_0$ for fundamental period $T_0$

Symbol	Meaning
$f_0, \lambda_0$	Carrier frequency, carrier wavelength
$f_D$	Maximum Doppler frequency: $f_D = (v_{\max}/c)\,f_0$
$T_c, B_c$	Coherence time, coherence bandwidth
$N_0, \sigma^2$	Noise PSD (one-sided), noise variance
$\text{SNR}$	Signal-to-noise ratio (linear unless in dB)
$\mathbf{w}$	AWGN noise vector (not $\mathbf{n}$ — avoids collision with antenna count)
$\mathcal{CN}(\boldsymbol{\mu}, \mathbf{R})$	Circularly symmetric complex Gaussian distribution