Prerequisites & Notation

Before You Begin

This chapter generalizes the binary detection theory of Chapter 1 and the Gaussian-noise detection theory of Chapter 2 to $M$ hypotheses. The reader should be comfortable with the likelihood ratio test, the Neyman-Pearson viewpoint, and the matched-filter sufficient statistic before proceeding.

MAP and ML decision rules for binary hypothesis testing(Review ch01)
Self-check: Can you write the MAP rule as a log-likelihood-ratio test with threshold depending on the priors?
The Q-function and Gaussian tail integrals(Review ch01)
Self-check: Can you express $P(X > a)$ for $X \sim \mathcal{N}(\mu,\sigma^2)$ in terms of $Q(\cdot)$ ?
Matched filter and correlator receivers for known signals in AWGN(Review ch02)
Self-check: Why is $\int y(t) s(t)\,dt$ a sufficient statistic when $Z(t)$ is white Gaussian?
KL divergence and its role as the expected log-likelihood ratio(Review ch01)
Self-check: State the two non-negativity properties: $D(f\|g)\geq 0$ and $D=0 \iff f=g$ a.e.
Linear algebra: inner products, orthonormal bases, Gram-Schmidt procedure
Self-check: Can you orthonormalize three linearly independent vectors in $\mathbb{R}^4$ by hand?
Moment generating function (MGF) of a random variable
Self-check: What is the MGF of an exponential random variable with mean $\bar\gamma$ ?
Complex baseband representation and symbol energy
Self-check: Given a constellation $\mathcal{X}$ , what is the average symbol energy $E_s$ ?

Notation for This Chapter

Symbols introduced or used in a specialized sense in this chapter. Symbols with $\ntn{}$ tokens follow the book-wide defaults; see the front-matter notation table for the canonical definitions.

Symbol	Meaning	Introduced
$M$	Number of hypotheses / constellation points	s01
$\mathcal{H}_m$	The $m$ -th hypothesis, $m \in \{0,1,\ldots,M-1\}$	s01
$\pi_m$	Prior probability of hypothesis $\mathcal{H}_m$	s01
$\mathcal{X} = \{\mathbf{x}_0,\ldots,\mathbf{x}_{M-1}\}$	Signal constellation (set of signal-space points)	s02
$\mathcal{R}_m$	Decision region for hypothesis $\mathcal{H}_m$	s01
$g$	Decision rule, $g(\mathbf{y}) \in \{0,\ldots,M-1\}$	s01
$N$	Dimension of signal space after Gram-Schmidt ( $N \leq M$ )	s02
$d_{m,j}$	Euclidean distance $\\|\mathbf{x}_m - \mathbf{x}_j\\|$	s02
$d_{\min}$	Minimum pairwise distance of the constellation	s03
$Q(\cdot)$	Gaussian tail probability	s03
$P(m \to j)$	Pairwise error probability: prob. of deciding $j$ given $m$ sent	s03
$M_\gamma(s)$	MGF of the instantaneous SNR $\gamma$	s04
$D(f\\|g)$	KL divergence between densities $f$ and $g$	s05
$D_{\min}$	Minimum pairwise KL divergence between hypothesis distributions	s05
$\text{SNR} = E_s/N_0$	Symbol SNR	s03

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