RP-HW5-sol - Kyung Hee University Department of Electronics...

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1 Kyung Hee University Department of Electronics and Radio Engineering C1002900 Random Processing Homework 5 Solutions Spring 2010 Professor Hyundong Shin Issued: May 17, 2010 Due: May 31, 2010 (No acceptance of overdue submission) Reading: Course textbook Chapter 8, Lecture Note X HW 5.1 (a) By inspection, X is equally likely to be positive or negative ( , X Y f has uniform height in the shaded region, and there are equal areas of shade for X 0 and X 0 ). Then PP  01 12 . Given H H 0 , we see that Y is twice as likely to be in the interval  ,,   21 1 2 rather than in the interval , 11 . Normalizing, we obtain the conditional density:  , , ,o t h e r w i s e . YH y fy H y   0 13 1 2 16 1 0 (5.1) Similarly, we obtain   f yH 1 : , , t h e r w i s e . y H y 1 16 1 2 13 1 0 (5.2)   H 0 1 2 1 2 0 y   H 1 1 2 1 2 0 y (b) Since the hypotheses are equiprobable, the minimum probability of error (MPE) decision rule is the same as the ML rule: ˆ ˆ Hy H f f 1 0 10 . (5.3)

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2 Trivially,  ,,    0 21 1 2 and ,  1 11 . Regardless of which hypothesis is true, this de- cision rule will be incorrect one-third of the time (for either conditional density, one-third of the density lies outside the region of correct decision). Thus  error 13 (given that the hypo- theses are equiprobable). (c) We can determine the region of operating characteristic (ROC) by forming the LRT test, and then varying the threshold of the test:    ˆ ˆ LRT : HH YH fy H Ly H  1 0 1 0 . (5.4) We can substitute in the conditional densities to obtain: , , ,o t h e r w i s e . y y   12 1 2 21 0 (5.5)   L y 12 1 2 0 y 1 2 H 0 H 1 For  2 , we obtain the decision rule: , ˆ ,. Hy 1 0 1 1 (5.6) Looking at the conditional densities, it is easy to find F P and D P for this test: ˆ F PH H H H Y H H  10 0 1 1 3  . (5.7) ˆ D H H H Y H H 1 2 1 3 . (5.8) Obviously, when  2 , the LRT has performance     FD PP 00 . Similarly, when  12 , the LRT has performance     11 . By using a perverse test (i.e., by deciding the opposite of whatever result we obtain from an optimal test), we can obtain the performance       , 1 131 23 2313 . Let us interpolate between any of these ROC boundary points:
3 13 1 0 F P D P 23 1 ROC 56 ,   52 63 The point     ,, FD PP 5623 is achievable, and conveniently lies on the interpolation line be- tween two tests:  , ˆ the reversal of the LRT for , A Hy  1 0 1 12 2 1 . (5.9)      ˆ achieves , , BF D Hy H  1 11 . (5.10) Since our desired point lies exactly between the performance of these two tests, we take our in- terpolated test to be: ˆ with probability ˆ .

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RP-HW5-sol - Kyung Hee University Department of Electronics...

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