PS09 - University of Illinois Spring 2011 ECE 361: Problem...

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University of Illinois Spring 2011 ECE 361: Problem Set 9 Maximum-Likelihood Sequence Estimation; Passband Signals Due: Tuesday April 26 at 9:30 a.m. 1. [Maximum-likelihood sequence estimation] The received signal in a communication system operating over a band-limited channel is Y [ m ] = X [ m ] + 0 . 2 X [ m - 1] + 0 . 1 X [ m - 2] + N [ m ] where { X [ m ] } is a sequence of independent random variables taking on values ± 1 with equal probability, and { N [ m ] } is a sequence of N (0 , 1) random variables independent of each other and of the sequence { X [ m ] } . Transmission begins at 0 and so X [ - 1] = X [ - 2] = 0. Assume that the received sequence is Y [0] , Y [1] , Y [2] , Y [3] ··· = 0 . 9 , - 0 . 7 , - 1 . 1 , 0 . 8 , ··· (a) Sketch the initial portion of the trellis that would be used for maximum-likelihood detection of the received sequence and mark on each branch the squared distance that would be added as the computations proceed. (b) Find the best path from the start node (at depth 0) to each of the four nodes at depth 3 in the
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