Notation Preferences — RF Imaging

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Key	Meaning	Default
gauss	Real Gaussian distribution	$\mathcal{N}$
cgauss	Circularly symmetric complex Gaussian	$\mathcal{CN}$
covmat	Covariance matrix	$\boldsymbol{\Sigma}$
fim	Fisher information matrix	$\mathbf{J}$
snr	Signal-to-noise ratio	$\text{SNR}$
n0	One-sided noise power spectral density	$N_0$
bw	Signal bandwidth (Hz)	$W$
fc	Carrier frequency	$f_0$
wl	Wavelength ( $\lambda_0 = c/f_0$ at carrier)	$\lambda$
pl	Path loss coefficient (power)	$\beta$
noise	AWGN noise vector (system model)	$\mathbf{w}$
steer_tx	Transmit array steering vector	$\mathbf{a}$
steer_rx	Receive array steering vector	$\hat{\mathbf{a}}$
bf	Beamforming / precoding vector	$\mathbf{v}$
bf_gain	Complex beamforming gain: $g_{i,q,k} = \mathbf{a}^H \mathbf{v}$	$g_{i,q,k}$
fk	$k$ -th OFDM subcarrier frequency	$f_k$
ntx	Number of transmit antennas	$N_t$
nrx	Number of receive antennas	$N_r$
noisevar	Noise variance / noise power	$\sigma^2$
refl_fn	Complex reflectivity function	$c$
refl	Discretized reflectivity vector	$\mathbf{c}$
scat	Scattering coefficient (Tx $i$ , Rx $j$ , voxel $q$ )	$c_{i,j,q}$
scat_pow	Scattering power (variance of $c_{i,j,q}$ )	$\gamma_{i,j,q}$
sens	Measurement / sensing matrix	$\mathbf{A}$
img_model	Linear imaging observation model: $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$	$\mathbf{y}$
wavenum	Wavenumber: $\kappa = 2\pi/\lambda$	$\kappa$
wavenum_c	Combined Tx-Rx wavenumber vector: $\kappa_{\mathbf{s},\mathbf{r}} = \kappa_\mathbf{s} + \kappa_\mathbf{r}$	$\kappa_{\mathbf{s},\mathbf{r}}$
txpos	Transmitter position (subscripted $\mathbf{s}_i$ for Tx $i$ )	$\mathbf{s}$
rxpos	Receiver position (subscripted $\mathbf{r}_j$ for Rx $j$ )	$\mathbf{r}$
tgtpos	Target / voxel position (subscripted $\mathbf{p}_0$ for a reference point)	$\mathbf{p}$
tgt_rgn	Target region in space	$\Omega$
voxel	Voxel (pixel) index: $q = 1, \ldots, Q$	$q$
delay	Round-trip delay for Tx $i$ , Rx $j$	$\tau_{i,j}$
pilot	Pilot signal matrix (Tx $i$ , subcarrier $k$ )	$\mathbf{S}_{i,k}$
tgt_snr	Target SNR threshold in link budget	$\rho_{\text{dB}}$
gtx	Transmit antenna gain	$G^{\text{tx}}$
grx	Receive antenna gain	$G^{\text{rx}}$
bp	Backpropagation (matched filter) image: $\hat{\mathbf{c}}^{\text{BP}} = \mathbf{A}^H \mathbf{D}^{-1} \mathbf{y}$	$\hat{\mathbf{c}}^{\text{BP}}$
lasso	LASSO solution: $\hat{\mathbf{z}} = \arg\min \\|\mathbf{y} - \mathbf{A}\boldsymbol{\Phi}\mathbf{z}\\|^2 + \lambda\\|\mathbf{z}\\|_1$	$\hat{\mathbf{z}}_{\text{LASSO}}$
reg	Regularization parameter	$\lambda$

Global Notation — RF Imaging

Notation conventions used throughout the RF Imaging textbook, consistent with Book 1 (Fundamentals of Wireless Communication).

Inverse Problems

Symbol	Meaning
$\mathcal{A}$	Forward operator (continuous)
$\mathbf{A}$	Sensing / measurement matrix (discrete)
$\mathbf{A}^H$	Adjoint (conjugate transpose) of the sensing matrix
$\mathbf{g}$	Scene vector (ground truth reflectivity)
$\hat{\mathbf{g}}$	Reconstructed scene estimate
$\mathbf{y}$	Measurement vector
$\mathbf{n}$	Noise vector
$\lambda$	Regularization parameter
$R(\mathbf{g})$	Regularization functional
$R_\alpha$	Regularized inverse operator with parameter $lpha$

Scattering and Wave Physics

Symbol	Meaning
$\chi(\mathbf{r})$	Contrast function (object susceptibility)
$G(\mathbf{r}, \mathbf{r}')$	Green's function
$k = 2\pi/\lambda$	Wavenumber
$\lambda$	Wavelength (context distinguishes from regularization parameter)
$\mathbf{E}^{\mathrm{inc}}$	Incident electric field
$\mathbf{E}^{\mathrm{sc}}$	Scattered electric field
$\mathbf{S}$	Scattering matrix (S-matrix)

Compressed Sensing

Symbol	Meaning
$\\|\mathbf{x}\\|_0$	$\ell_0$ quasi-norm (number of nonzero entries)
$\\|\mathbf{x}\\|_1$	$\ell_1$ norm (sparsity-promoting)
$\delta_s$	Restricted isometry constant of order $s$
$\mu(\mathbf{A})$	Mutual coherence of matrix $\mathbf{A}$
$\mathrm{spark}(\mathbf{A})$	Spark of matrix $\mathbf{A}$

Proximal Operators and Optimization

Symbol	Meaning
$\mathrm{prox}_f(\mathbf{x})$	Proximal operator of $f$
$\partial f(\mathbf{x})$	Subdifferential of $f$ at $\mathbf{x}$
$f^*$	Fenchel conjugate of $f$
$\iota_C(\mathbf{x})$	Indicator function of convex set $C$
$\mathcal{S}_\tau$	Soft-thresholding operator with threshold $\tau$

Message Passing

Symbol	Meaning
$\hat{\gamma}^t$	AMP/OAMP estimate at iteration $t$
$\mathbf{r}^t$	AMP/OAMP residual at iteration $t$
$\eta_t(\cdot)$	Denoiser at iteration $t$
$\mathrm{div}(\eta)$	Divergence of the denoiser (Onsager correction)
$\tau_t$	State evolution noise variance at iteration $t$

Deep Learning for Imaging

Symbol	Meaning
$f_\theta$	Neural network with parameters $\theta$
$D_\sigma$	Denoiser network for noise level $\sigma$
$\nabla_{\mathbf{x}} \log p(\mathbf{x})$	Score function
$\mathcal{L}(\theta)$	Loss function
$T$	Number of unrolling layers / diffusion steps

3D Representations

Symbol	Meaning
$\phi(\mathbf{r})$	Signed distance function (SDF)
$\sigma(\mathbf{r})$	Volume density (NeRF) or reflectivity
$\mathbf{c}(\mathbf{r}, \mathbf{d})$	View-dependent color/reflectance
$\alpha_i$	Opacity of Gaussian splat $i$

General Conventions

Symbol	Meaning
$\mathbf{x}$	Lowercase bold: vector
$\mathbf{X}$	Uppercase bold: matrix
$(\cdot)^T, (\cdot)^H$	Transpose, conjugate transpose
$\otimes$	Kronecker product
$\odot$	Hadamard (element-wise) product
$\mathbb{E}[\cdot]$	Expectation
$\mathcal{CN}(\boldsymbol{\mu}, \mathbf{C})$	Complex Gaussian distribution