Unformatted text preview: EE 806, Detection and Estimation Theory OSU, Spring 2008 Problem Set 4 Problem 1 May 5, 2008 Due: May 12, 2008 Consider the simple binary hypothesis testing problem for which the receiver operating characteristics (ROC) and the derivative ( dPD ) are given in the below figure. Based on dPF these plots, provide (approximate) answers to the following questions.
1 0.9 0.8 0.7
5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 P 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 dPD/dPF 0.6 D PF 0.1 0.2 0.3 0.4 0.5 P 0.6 0.7 0.8 0.9 1 F (a) With uniform costs (C00 = C11 = 0 and C01 = C10 = 1), find the probability of misdetection for the minimax decision rule. (b) Find the minimum Bayes risk for 0 = Problem 2 Consider the binary hypothesis testing problem based on the observation of the pair of random variables (Y1 , Y2 ). The pair (Y1 , Y2 ) is jointly uniform in the shaded regions as shown in the following graph under H0 and H1 .
y2 1 H0 1 y2 H1
2 3 and uniform costs. 1 y1 (a) For equal priors and uniform costs, find and plot the decision regions 0 and 1 that minimize the Bayes risk for hypotheses H0 and H1 respectively. 1 1 y1 (b) Argue that if two points on ROC, (PF,1 , PD,1 ) and (PF,2 , PD,2 ) are achievable, then any convex combination, (PF,1 + (1 - )PF,2, PD,1 + (1 - )PD,2) is also achievable for any [0, 1]. (c) Find the possible operating points for the (PF , PD ) pair and based on the observation given in part (b), plot the ROC. Problem 3 A radar detects aircrafts based on the received signal Yk , 1 k n. If there is no aircraft (null hypothesis, H0 ), the radar receives noise, Nk for each k. If an aircraft is present (hypothesis H1 ), a scaled version, Nk , of the noise is received. More precisely, H1 : Yk = Nk H0 : Y k = N k , for k = 1, 2, . . . , n; the constant is unknown and Nk iid N (0, 1). (a) Assuming that > 1, determine the form of the locally most powerful (LMP) test. Do not try to solve for threshold . (b) Does the UMP test exist when > 1? Explain. If your answer is yes, give the test. (c) Does the UMP test exist if it is not known whether > 1 or < 1? Explain. Problem 4 - (Poor, Ch. 3, Pr. 20) Problem 5 - (Poor, Ch. 3, Pr. 21) Use equation (III.C.18) for Pe and recall the binary symmetric channel covered in class and in Example II.B.1 in Poor. 2 ...
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This note was uploaded on 07/17/2008 for the course EE 806 taught by Professor Unknown during the Spring '08 term at Ohio State.
- Spring '08