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Unformatted text preview: ECE 562 Fall 2011 Linear Equalization • Recall that the effective ISI model is given by: Z k = L- 1 X ` =0 h ` s m k- ` + W k , k = 1 , 2 ,... (1) To make the channel symmetric across N symbols s m 1 ,...,s m N , we append the frame with L- 1 null symbols, i.e., in those symbol periods we do not transmit anything. We collect the channel outputs in the extended frame into the column vector: Z = [ Z 1 Z 2 ··· Z N + L- 1 ] > We can rewrite (1) in matrix vector form as: Z = Hs ( m ) + W (2) where s ( m ) = [ s m 1 ··· s m N ] > , W = [ W 1 ··· W N + L- 1 ] > , and H = h ... h 1 h ... . . . . . . . . . . . . h L- 1 ... ... h ... . . . . . . . . . . . . ... h L- 1 ... h . . . . . . . . . . . . . . . ... . . . ... h L- 1 Note that H is an ( N + L- 1) × N matrix with full column rank. We may also rewrite (2) in terms of the columns of H as Z = N X j =1 s m j h j + W (3) where h j is the j-th column of H . • Even though computing the MLSE via the Viterbi algorithm has a complexity that is linear in the frame size N , it is still exponential in the length of the channel L . Furthermore, in order to compute the branch metric, it is necessary to know precisely the channel gain from the transmitter to the receiver (even for PSK constellations.) This motivates the study of suboptimal solutions that are based on linear operations on the received vector Z ....
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