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# lect7 - Kevin Buckley 2007 1 ECE 8770 Topics in Digital...

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Kevin Buckley - 2007 1 ECE 8770 Topics in Digital Communications - Sp. 2007 Lecture 7 3 MLSE with Intersymbol Interference (ISI) 3.2 Viterbi Algorithm for MLSE with ISI Channels In Section 3.1 of the course notes we established the equivalent discrete-time lowpass channel representation of an ISI communication channel which is applicable for mod- ulations schemes for which the equivalent lowpass transmitted signal is of the form v ( t ) = summationdisplay n I n g ( t - nT ) . (1) This representation is reproduced below in Figure 1. The output is v n = f H I n,L + η n (2) where f = [ f 0 , f 1 , · · · , f L ] T , I n,L = [ I n , I n - 1 , · · · , I n - L ] T and η n is discrete-time, complex- valued AWGN. f 0 f 1 f L z -1 z -1 z -1 I n v n η n present input symbol ..... I I n-1 n-L past input symbols Figure 1: DT ISI channel model including noise whitening.

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Kevin Buckley - 2007 2 Since we know that the sequence { v n } forms a sufficient statistic for MLSE of the sequence { I n } , we can formulate this MLSE problem, at time n , in terms of the joint PDF of v n = [ v 1 , v 2 , · · · , v n ] T conditioned on I n = [ I 1 , I 2 , · · · , I n ] T : p ( v n /I n ) = 1 (2 πσ 2 n ) n e - n k =1 | v k - f H I k,L | 2 / 2 σ 2 n . (3) Concerning notation, here we represent current time (i.e. the most recent symbol time that we want to optimuze up to) as n , and k represents all symbol time up to n . We then consider incrementing to to the next current time n + 1. The MLSE problem, at symbol time n , is max I n p ( v n /I n ) . (4) Taking the negative natural log and eliminating constant terms that do not effect the relative costs for different I n ’s, we have the equivalent problem 1 min I n Λ n ( I n ) = CM ( I n ) = - PM ( I n ) = n summationdisplay k =1 | v k - f H I k,L | 2 , (5) where Λ n ( I n ) is the cost of sequence I n . 1 In Chapters 5 & 10 of the Course Text, where MLSE and the Viterbi algorithm are discussed, several notations are used to represent the measure or metric to be optimized to. When PM is used, it is maximized, and typically refers to a probability metric. When CM is used, it is minimized, and is sometimes referred to as a corelation of cross-correlation metric. CM is often equivalent to the Euclidean distance D . Here, I start using the notation Λ and refer to it as a cost to be mimimized. As used, it is close to if not the same as CM or D of the Course Text. I choose Λ so as to get away from the variety of Course Text notations, and because I’ve used this notation in other places to represent cost.
Kevin Buckley - 2007 3 The first n - 1 elements of I n are equal to I n - 1 , i.e. I n [1 : n - 1] = I n - 1 . This suggests that at time n we may be able to time-recursively extend time n - 1 results. The Viterbi algorithm efficiently solves this MLSE problem time-recursively (i.e. as n increases). Let Λ n ( I n ) = n summationdisplay k =1 | v k - f H I k,L | 2 (6) = n summationdisplay k =1 λ ( I k,L ) = Λ n - 1 ( I n - 1 ) + λ ( I n,L ) .

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lect7 - Kevin Buckley 2007 1 ECE 8770 Topics in Digital...

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