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电子科技大学信号检测与估计课后习题答案（英文版）.ppt

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    Problem 4.3 ( ) p y 1 1 , 0    0 ,    y 1 otherwise ( ) p y 0     2 3 (y +1) , 0 4 0 , otherwise   y 1 a. Find the Bays decision and minimum Bays risk for testing H1 versus H0 with equal a priori probabilities, =1 and C C 11 =0 ， C C 10 = 01 = 00 b. Find the minimax decision and minimax risk for the same costs as in part (a). c. Find the Neyman-Pearson decision and the for a false-alarm Pd probability of UESTC-何子述等 0 f  1 1
    Problem 4.3 Solution: The likelihood ratio for this problem is , 0 ( ) L y =  ( ) p y 1 ( ) p y 2 1 2 3 （y +1） 4   y 1 (a) For this case, the threshold is one. We choose H1 if ( )= L y 1 2  3 （y +1） 4 1 or 2 y  1 3 Thus, the decision regions are :  R : 0 The Bayes’ risk is:   R : 1/ 3 1/ 3 and 0.4519       y 1) dy dy 1 2 ( y r  1/ 3 1  1/ 3    y 1 0 2 1 2 0 3 4  1 2 UESTC-何子述等
    Problem 4.3 (b) For this part, we want to find a threshold such that the two conditional risks are equal, i.e., y 0  0 3( 4 2 y  1) dy =  1 y 0 dy Solving this, we arrive at a cubic equation which must solve, namely 3 y 0 07 y   4 0 Two of the solutions are imaginary, and thus unacceptable. If we model the cubic equation as    ax b 0 UESTC-何子述等 3 x 3
    Problem 4.3 the real root can be written as =x A B  b   2  where and b 2     B A  - 2 b 4  3 a 27 1/3        2 b 4 1/3  3 a   27  0=0.54793 y resulting in The average risk is r =0.45207 (c) In this case, we must find a threshold such that y 0  3 y  4 UESTC-何子述等 0 2   1 dy  f  , or 3 y 0  03 y  f 4   0 4
    Problem 4.3 The solution is where and A  f  2 0=y A B   2   B  f  2 4  f  2 4  f  1/3 1 1/3 1 The probability of detection is dP  y 0  0 dy  y 0 UESTC-何子述等 5
    Problem 4.6 Repeat Problem 4.3for the hypothesis H 1 : y = s + n, where s is a fixed real positive number and n has a pdf : H 0 : y = s + n  ( ) p n  1  n (1 2 ) Solution :  p y H  1  1 y  2  s    1      p y H 0   1 y  2  s    1     UESTC-何子述等 ( ) L y  1 (  1 (  y y   2 2 s s ) ) 6
    Problem 4.6 (a) With equal a priori probabilities and uniform costs, we make D1 decision if As s is positive The Bayes’ risk is then  L y  1   ys  0 y D 1   D 0 0 r  1 2   0 1 y dy 2 ) ]  s  1 2  [1 (  0   1 [1 ( y   dy 2 ) ]  s   1 2 s 1  tan  UESTC-何子述等 7
    Problem 4.6 (b) According to the given cost, we have P 01 P 10   y 0  [1 (  1 y dy 2 ) ]  s  y 0   1 [1 ( y  dy 2 ) ]  s  0 y  0 (c) The probability of false alarm is determined by picking a threshold such that    f dy 0y  1 [1 ( y   s 2 ) ] y 0  UESTC-何子述等 8
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