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# HW09 - University of Illinois Spring 2011 ECE 361 Problem...

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Spring 2011 ECE 361: Problem Set 9: Solutions Maximum-Likelihood Sequence Estimation; Passband Signals 1. [Maximum-likelihood sequence estimation] (a) The squared distance is proportional to (0 . 1 x [ m - 2] + 0 . 2 x [ m - 1] + x [ m ] - y [ m ]) 2 giving the trellis shown below. 2 . 25 0 . 01 0 . 00 3 . 61 3 . 61 4 . 00 0 . 01 4 . 84 0 . 09 0 . 04 2 . 89 2 . 25 ++ - + + - -- 00 ++ ++ - + - + -- -- + - 0 - + - 0+ 0 . 9 - 0 . 7 - 1 . 1 3 . 24 0 . 8 0 . 01 3 . 61 0 . 25 0 . 04 4 . 41 5 . 76 0 , 25 0 . 01 0 . 16 (b) Using the Viterbi algorithm, we progress through the trellis as shown in the ﬁgure below. The best paths are 00 -→ 0+ -→ + - -→ - + -→ ++, 00 -→ 0+ -→ + - -→ -- -→ - +, 00 -→ 0+ -→ + - -→ - + -→ + - , 00 -→ 0+ -→ + - -→ -- -→ -- , 2. [Trellis diagram structure] (a) The ﬁrst statement is true: the two edges correspond to data bits 0 and 1 (or ± 1 or ± E T ). The second statement is false: the two edges have the data bit since the data bit labels on the edges become part of the label of the node being entered. (b) The squared-distance labels on the two edges leaving a node at depth m in the trellis are of the form M X i =1 h [ i ] x [ m - i ] + 1 - y [ m ] ! 2 and M X i =1 h [ i ] x [ m - i ] - 1 - y [ m ] ! 2 and these are obviously diﬀerent except when the two quantities being squared are negatives of each other. Thus, we have M X i =1 h [ i ] x [ m - i ] + 1 - y

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HW09 - University of Illinois Spring 2011 ECE 361 Problem...

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