RP-HW3-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 3 Solutions: Sinusoid Estimation Spring 2010 Professor Hyundong Shin Issued: May 3, 2010 Due: May 17, 2010 (No acceptance of overdue submission) Reading: Course textbook Chapter 8 In this homework, we explore a basic problem involving sinusoid estimation. Specifically, given noisy observation of the form   cos , , , , nn YA n W n N  0 01 1 (3.1) we wish to estimate one or more of the nonrandom parameters A , 0 , or , where A 0 and   0 0 . In (3.1), we assume that the noises n W are i.i.d.   , 2 0 random variables. This model arises in a number of applications: analog communications, Doppler radar, noise cancel- lation, interference suppression, radio astronomy, and sonar direction-finding, for example. HW 3.1 (Cramér-Rao Bound) (a) The elements of the Fisher information matrix then take the form  ; ij ij f xx         YY Ix y x 2 (3.2) where  ;l n ; l n c o s N n N ff y n A n  yx 1 2 0 2 0 1 2 2 2 . (3.3) We compute the Fisher matrix entries one at a time. First,  ;c o s c o s cos Re , NN N n fn n A n N           y x 2 11 2 00 22 2 11 1 0 122 2 1 (3.4) where we introduce the function    defined via N jn n e N    0 1 0 0 1 (3.5)
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2 As we will see, this function and its first and second derivatives, respectively  N jn n ne N    0 1 22 0 0 2 (3.6) N n ne N   0 1 2 0 0 4 (3.7) play a central role in the Fisher information for the problem. Using (3.6),  ' ; cos sin sin Im N n N n f A An n n AN nn N AN              YY Ix y x 2 12 0 1 00 2 0 1 0 2 0 0 2 1 1 2 1 2 2 (3.8) and using (3.5), ; cos sin sin Im . N n N n f A An n AN n N AN  y x 2 13 1 2 0 1 0 2 0 0 2 1 1 2 2 (3.9) Using (3.7), '' ; sin cos cos Re , N n N n NN N f n A AA N n N N T      y x 2 2 22 0 1 2 0 2 0 2 1 2 0 2 0 11 0 0 1 122 2 1 1 4 (3.10) and using (3.6),
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3   ' ; sin cos cos Re . N n N n NN nn N f An n A AA N n N S                YY Ix y x 2 23 0 1 22 0 2 0 2 1 0 2 0 11 0 00 0 1 122 2 1 2 (3.11) Finally, using (3.5), ; sin cos cos Re N n N n N n f A n N n N N  y x 2 2 33 1 0 2 0 2 1 0 2 0 1 0 0 0 1 2 1 (3.12) (b) To develop the asymptotic (large N ) behavior of the Fisher information, we explore the corres-
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RP-HW3-sol - Kyung Hee University Department of Electronics...

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