Notation Preferences — Statistical Inference, Detection & Estimation

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Key	Meaning	Default
pmf	Probability mass function of discrete RV $X$	$P$
pdf	Probability density function	$f$
cdf	Cumulative distribution function	$F$
cpdf	Conditional PDF	$f$
gauss	Real Gaussian distribution	$\mathcal{N}$
cgauss	Circularly symmetric complex Gaussian	$\mathcal{CN}$
qfn	Gaussian tail probability: $Q(x) = 1 - \Phi(x) = \frac{1}{\sqrt{2\pi}}\int_x^{\infty} e^{-t^2/2}\,dt$	$Q$
var	Variance: $\text{Var}(X) = \sigma_X^2$	$\text{Var}$
cov	Covariance of two RVs	$\text{Cov}$
covmat	Covariance matrix	$\boldsymbol{\Sigma}$
corrmat	Correlation matrix $\mathbb{E}[\mathbf{X}\mathbf{X}^H]$	$\mathbf{R}$
xcov	Cross-covariance matrix	$\boldsymbol{\Sigma}_{xy}$
mi	Mutual information	$I$
kldiv	Kullback-Leibler divergence	$D$
markov	Markov chain relation	$X \multimap Y \multimap Z$
lr	Likelihood ratio $f_1(y)/f_0(y)$	$L$
llr	Log-likelihood ratio	$\ell$
dec	Decision rule / detector	$g$
pfa	False alarm probability	$P_f$
pd	Detection probability	$P_d$
fisher	Fisher information (scalar)	$J$
fim	Fisher information matrix	$\mathbf{J}$
mle	Maximum likelihood estimator	$g_{\text{ml}}$
mmse_est	MMSE (posterior mean) estimator: $g_{\text{mmse}}(y) = \mathbb{E}[X\|Y=y]$	$g_{\text{mmse}}$
lmmse	LMMSE estimation matrix: $\mathbf{A} = \boldsymbol{\Sigma}_{xy}\boldsymbol{\Sigma}_y^{-1}$	$\mathbf{A}$
cost	Loss / cost function	$c$
acorr	Autocorrelation (discrete-time, WSS)	$r_{xx}$
psd	Power spectral density	$P_x$
xpsd	Cross-power spectral density	$P_{xy}$
tfn	Frequency response of LTI system	$\check{h}$
snr	Signal-to-noise ratio	$\text{SNR}$
es	Energy per transmitted symbol	$E_s$
n0	One-sided noise power spectral density	$N_0$
bw	Signal bandwidth (Hz)	$W$
noise_rv	Noise random variable (theoretical)	$Z$
noisevar	Noise variance / noise power	$\sigma^2$
sens	Measurement / sensing matrix	$\mathbf{A}$
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	$\lambda$
kstate	Kalman filter state vector	$\mathbf{X}_n$
ktrans	State transition matrix (Kalman)	$\mathbf{A}_n$
kg	Kalman gain	$\mathbf{K}_n$
wiener_nc	Non-causal Wiener filter	$\check{h}_{\text{nc}}$
wiener_c	Causal Wiener filter	$\check{h}_c$

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)
$\lceil \cdot \rceil, \lfloor \cdot \rfloor$	Ceiling and floor functions
$\triangleq$	Defined as
$\propto$	Proportional to
$\mathcal{O}(\cdot)$	Big-O asymptotic notation

Vectors and Matrices

Symbol	Meaning
$\mathbf{x}, \mathbf{y}, \mathbf{z}$	Column vectors (always boldface lowercase)
$\mathbf{A}, \mathbf{B}, \mathbf{H}$	Matrices (always boldface uppercase)
$(\cdot)^T$	Transpose
$(\cdot)^*$	Complex conjugate
$(\cdot)^H$	Conjugate transpose (Hermitian)
$(\cdot)^{-1}$	Matrix inverse
$(\cdot)^{\dagger}$	Moore-Penrose pseudoinverse
$\\|\cdot\\|$ or $\\|\cdot\\|_2$	Euclidean (l_2) norm
$\\|\cdot\\|_p$	$\ell_p$ norm
$\\|\cdot\\|_F$	Frobenius norm
$\langle \mathbf{x}, \mathbf{y} \rangle$	Inner product $\mathbf{y}^H \mathbf{x}$
$\text{tr}(\cdot)$	Matrix trace
$\det(\cdot)$	Matrix determinant
$\text{rank}(\cdot)$	Matrix rank
$\text{diag}(\cdot)$	Diagonal matrix from vector, or diagonal of matrix
$\text{vec}(\cdot)$	Column-wise vectorization
$\otimes$	Kronecker product
$\odot$	Hadamard (element-wise) product
$\mathbf{I}_n$	$n \times n$ identity matrix
$\mathbf{0}_{m \times n}$	$m \times n$ zero matrix
$\mathbf{A} \succ 0$ , $\mathbf{A} \succeq 0$	Positive definite, positive semidefinite
$\lambda_i(\mathbf{A})$	$i$ -th eigenvalue of $\mathbf{A}$
$\sigma_i(\mathbf{A})$	$i$ -th singular value of $\mathbf{A}$

Probability Basics

Symbol	Meaning
$\mathbb{E}[\cdot]$	Expectation
$\Pr(\cdot)$	Probability of an event
$\sim$	Distributed as
$\stackrel{d}{=}$	Equal in distribution