hw1 - EEL 6537 Homework #1 ML Parameter Estimation and CRB...

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Unformatted text preview: EEL 6537 Homework #1 ML Parameter Estimation and CRB Due 2/4/10 1. Consider the data set x l = a s l + e l , l = 1 , 2 , , L, where x l denote the l th M 1 data vector, a = [ 1 1 1 ] T (an M 1 vector) with () T denoting the transpose, is a complex-valued unknown scalar, { s l } is the known signal waveform, and e l denotes the l th error vector. The error vectors e l , l = 1 , 2 , , L , are independently and identically distributed zero-mean circularly symmetric complex Gaussian random vectors with an unknown and arbitrary covariance matrix Q . The problem of interest herein is to determine the maximum likelihood (ML) estimate of from { x l } L l =1 and its Cramer-Rao bound (CRB). a) Determine the log-likelihood function (logarithm of the pdf) l 1 of { x l } L l =1 . b) Derive the ML estimate of . (Consider first setting l 1 / Q ij = 0 , where Q ij denotes the ij th element of Q .) c) Calculate the CRB of ....
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