Prerequisites & Notation

Before You Begin

This chapter builds on the fixed-sample-size detection framework developed in Chapters 1-3. We now relax the assumption that the sample size is fixed in advance and consider detectors that observe the data sequentially, stopping as soon as a decision can be made with sufficient confidence. If any item below feels unfamiliar, revisit the linked chapter before proceeding.

Binary hypothesis testing, likelihood ratio test, Neyman-Pearson lemma(Review ch01)
Self-check: Can you state the Neyman-Pearson lemma and derive the threshold for a target false alarm probability?
Log-likelihood ratio and its distribution under each hypothesis(Review ch01)
Self-check: Can you compute $\mathbb{E}[\ell(Y) | \mathcal{H}_0]$ and $\mathbb{E}[\ell(Y) | \mathcal{H}_1]$ for a Gaussian shift-in-mean problem?
KL divergence and its role as the expected LLR(Review ch01)
Self-check: Why is $\mathbb{E}[\ell(Y) | \mathcal{H}_1] = D(f_1 \| f_0)$ ?
Chernoff bound and error exponents(Review ch02)
Self-check: Can you sketch the Chernoff exponent $e^*$ as a function of $\lambda$ ?
Stopping times, martingales (light familiarity)
Self-check: Can you state the optional stopping theorem for bounded stopping times?
Q-function and Gaussian tail probabilities(Review ch01)
Self-check: Can you compute $Q(x)$ for $x = 1, 2, 3$ to two decimal places?
Order statistics of i.i.d. samples
Self-check: Can you write the distribution of the $k$ -th order statistic of $n$ i.i.d. uniforms?

Notation for This Chapter

Symbols introduced or used in a specialized sense in this chapter. We adopt the conventions of Chapter 1 for likelihood ratios, error probabilities, and decision rules.

Symbol	Meaning	Introduced
$N$	Stopping time (random number of samples before deciding)	s01
$\ell^{(n)}$	Cumulative log-likelihood ratio after $n$ samples	s01
$A, B$	SPRT thresholds ( $B < 0 < A$ , accept $\mathcal{H}_1$ above $A$ , $\mathcal{H}_0$ below $B$ )	s01
$\alpha, \beta$	Target Type-I and Type-II error probabilities	s01
$\mathbb{E}_i[N]$	Average sample number (ASN) under $\mathcal{H}_i$	s01
$S_n$	CUSUM statistic after $n$ samples	s02
$\nu$	True change-point instant	s02
$\tau$	Detection alarm time (stopping rule for change detection)	s02
$\mathrm{ARL}_0, \mathrm{ARL}_1$	Average run length under $\mathcal{H}_0$ (false alarm) and $\mathcal{H}_1$ (detection delay)	s02
$T$	CFAR test cell under test (CUT)	s03
$Z$	CFAR reference-window noise estimate	s03
$\alpha_{\mathrm{cfar}}$	CFAR threshold multiplier ( $\gamma = \alpha_{\mathrm{cfar}} \cdot Z$ )	s03
$N_{\mathrm{ref}}$	Number of reference cells in CFAR window	s03
$P_f, P_d$	False-alarm and detection probabilities	s01

← Ch 3 Wald's Sequential Probability Ratio Test (SPRT)