Prerequisites

Before You Begin

This chapter builds on linear algebra (Chapter 1), probability and random processes (Chapter 2), and digital modulation (Chapter 8). Detection theory relies on conditional probability and Gaussian distributions from Chapter 2, signal-space geometry from Chapter 8, and linear algebra for the LMMSE estimator derivation from Chapter 1.

Gaussian random variables and the Q-function(Review ch02)
Self-check: Can you compute $P(X > a)$ for $X \sim \mathcal{N}(\mu, \sigma^2)$ in terms of the Q-function $Q\!\left(\frac{a - \mu}{\sigma}\right)$ ?
Conditional probability and Bayes theorem(Review ch02)
Self-check: Can you apply Bayes theorem to compute $P(H_i \mid y) = \frac{p(y \mid H_i) P(H_i)}{\sum_j p(y \mid H_j) P(H_j)}$ ?
Signal-space representation and ML detection(Review ch08)
Self-check: Can you express the ML detector as $\hat{m} = \arg\min_m \|\mathbf{r} - \mathbf{s}_m\|^2$ and explain why it is the minimum-distance rule?
Constellation diagrams and decision regions(Review ch08)
Self-check: Can you sketch the decision regions for QPSK and 16-QAM and identify which regions correspond to which symbols?
Matrix inverses and positive-definite matrices(Review ch01)
Self-check: Can you solve $\mathbf{x} = (\mathbf{A}^H \mathbf{A} + \sigma^2 \mathbf{I})^{-1} \mathbf{A}^H \mathbf{y}$ and explain why the matrix is invertible when $\sigma^2 > 0$ ?

Chapter 9 Notation

Key symbols introduced or heavily used in this chapter.

Symbol	Meaning	Introduced
$P_e$	Probability of (symbol or bit) error	s01
$Q(x)$	Gaussian Q-function: $Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^\infty e^{-t^2/2}\, dt$	s02
$H_0, H_1$	Null and alternative hypotheses in binary detection	s01
$\Lambda(y)$	Likelihood ratio $p(y \mid H_1) / p(y \mid H_0)$	s01
$P(H_i \mid y)$	A-posteriori probability of hypothesis $H_i$ given observation $y$	s01
$\hat{\theta}$	Estimate of the unknown parameter $\theta$	s03
$I(\theta)$	Fisher information about $\theta$	s03
$\bar{\gamma}$	Average received SNR (averaged over fading)	s04
$E_b / N_0$	Energy per bit to noise spectral density ratio	s02
$\mathbf{R}_{hh}$	Channel autocorrelation matrix	s05
$\operatorname{erfc}(x)$	Complementary error function: $\operatorname{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2}\, dt$	s02
$d$	Diversity order (high-SNR BER slope on log-log scale)	s04
$M_\gamma(s)$	Moment generating function of the SNR random variable $\gamma$	s04
$\hat{\mathbf{h}}_{\text{LS}}$	Least-squares channel estimate	s05
$\hat{\mathbf{h}}_{\text{MMSE}}$	Minimum mean-square error channel estimate	s05

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