Prerequisites & Notation

Before You Begin

Chapter 2 takes the binary-hypothesis-testing machinery of Chapter 1 and specialises it to the Gaussian observation model that governs nearly every physical receiver. A reader comfortable with the LRT, the Neyman--Pearson lemma, the $Q$ -function, and with linear algebra over $\mathbb{R}^n$ and $\mathbb{C}^n$ will follow this chapter without friction.

Likelihood ratio test (LRT) and Neyman--Pearson lemma(Review ch01)
Self-check: State the Neyman--Pearson lemma and sketch its proof via the variational argument.
Gaussian distributions, $Q$ -function, and error probabilities(Review ch01)
Self-check: Compute $Q(2)$ and explain why $Q(x) \leq \tfrac12 e^{-x^2/2}$ .
Inner products, norms, projections in $\mathbb{R}^n$ (Review ch01)
Self-check: Compute the orthogonal projection of $\mathbf{y}$ onto $\text{span}(\mathbf{s})$ .
Positive-definite matrices and Cholesky factorisation(Review ch01)
Self-check: Explain why a covariance matrix admits a Cholesky factor $\mathbf{L}$ with $\mathbf{L}\mathbf{L}^{\mathsf{T}} = \mathbf{C}$ .
Linear time-invariant systems and convolution
Self-check: Relate the output of an LTI filter to the convolution of its input with the impulse response.

Chapter 2 Notation

Symbols introduced or used repeatedly in this chapter. Boldface lowercase denotes column vectors; boldface uppercase denotes matrices; $\mathbf{w}$ denotes additive noise throughout (per the book-wide convention).

Symbol	Meaning	Introduced
$\mathbf{y}$	Observation vector in $\mathbb{R}^n$	s01
$\mathbf{s}$	Known (deterministic) signal vector	s01
$\mathbf{w}$	Additive white Gaussian noise vector, $\mathbf{w}\sim\mathcal{N}(\mathbf{0},\sigma^2\mathbf{I})$	s01
$E_s$	Signal energy, $E_s = \\|\mathbf{s}\\|^2 = \sum_i s_i^2$	s01
$N_0$	One-sided noise PSD; per-sample variance is $\sigma^2 = N_0/2$	s01
$T(\mathbf{y})$	Test statistic (often the sufficient statistic)	s01
$d^2$	Deflection coefficient, $d^2 = 2E_s/N_0$ in AWGN	s01
$P_F, P_D$	False-alarm and detection probabilities	s01
$h(t)$	Matched-filter impulse response, $h(t) = s(T-t)$	s04
$\mathbf{C}_w$	Noise covariance matrix (colored case)	s03
$\mathbf{L}$	Cholesky factor of $\mathbf{C}_w$ , so $\mathbf{C}_w = \mathbf{L}\mathbf{L}^{\mathsf{T}}$	s03
$\\|\mathbf{v}\\|_{\mathbf{C}}^2$	Mahalanobis squared norm, $\mathbf{v}^{\mathsf{T}}\mathbf{C}^{-1}\mathbf{v}$	s03
$\langle y, s\rangle$	Continuous-time inner product, $\int_0^T y(t)s(t)\,dt$	s04

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