This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Detection and Estimation (Spring, 2009) Statistical Detection Theory I NCTU EE P.1 Statistical Detection Theory (I) ~ NeymanPearson Theorem Observe a realization of a random variable whose pdf is either N (0,1) or N (1,1). Binary Hypotheses: H : µ = 0 null hypothesis H 1 : µ = 1 alternative hypothesis or H : x[0] = w[0] null hypothesis H 1 : x[0] = 1 + w[0] alternative hypothesis A reasonable approach is to decide H 1 if x[0] > 1/2. Threshold = 1/2 Type I error : We decide H 1 but H is true. ( False Alarm ) Type II error : We decide H but H 1 is true. ( Miss ) Type I error probability: P(H 1 ;H ) Type II error probability: P(H ;H 1 ) P(H i ;H j ): the probability of deciding H i when H j is true. It is not possible to reduce both error probabilities simultaneously. Detection & Estimation (Spring, 2009) Statistical Detection Theory I NCTU EE P.2 As the threshold changes, one error increases while the other decreases. P(H 1 ;H ) Probability of false alarm ( P FA ) P(H 1 ;H 1 ) = 1 – P(H ;H 1 ) Probability of detection (P D ) Example, if P FA = 103 , we have γ = 3. With this choice, we have Detection & Estimation (Spring, 2009) Statistical Detection Theory I NCTU EE P.3 NeymanPearson (NP) Approach: We wish to maximize P D subject to the constraint P FA = α . General operation of a detector: decide either H or H 1 based on an observed set of data {x[0], x[1], .., x[N1]} R 1 = { x : decide H 1 or reject H } critical region R = { x : decide H or reject H 1 } α : significance level or size of the test. P D : the power of the test. There exist many critical regions that has the same significance level. ⇒ choose the one that maximizes P D . Remarks: 9 The critical region that attains the maximum power is the best critical region . 9 The NP theorem tells us how to choose R 1 if we are given p( x ;H ), p( x ;H 1 ), and α . Detection & Estimation (Spring, 2009) Statistical Detection Theory I NCTU EE P.4 (Proof) We use Lagrangian multipliers to maximize P D for a given P...
View Full
Document
 Spring '09
 ShengJyhWang

Click to edit the document details