HW6Solutions

HW6Solutions - ECEN 661 - Modulation Theory Homework #6...

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ECEN 661 - Modulation Theory Homework #6 Solutions 1. (a) Given the sequence of matched filter outputs, , the ML detector will find the sequence which optimizes . The trellis diagram and the appropriate branch metrics for the MLSE is shown below. (b) The energy transmitted per bit is . The minimum distance will occur when two sequences differ in two bits. For example, and . Then the resulting precoded sequences (assuming ) would be and . This will result in transmitted waveforms that differ (by an amplitude of 2A) during two different symbol inter- vals. The total energy in the difference waveforms will be . The resulting asymptotic efficiency is . (c) For this format, each path has . nearest neighbors. X () t d n 1 T s nT s rt r n 2 ω c t cos MLSE (Viterbi Algorithm) a ˆ n r r 1 r 2 r 3 , ,, = λ a r n 1 b n n r n 1 a n a n 1 n == a n 1 0 = a n 1 1 = r 1 / a 1 0 = r 1 / a 1 1 = r 1 / a 1 0 = r 1 / a 1 1 = r 2 / a 2 0 = r 2 / a 2 1 = r 2 / a 2 0 = r 2 / a 2 1 = Each branch is labelled with “branch metric / input bit.” E b s 2 t t d 0 T s A 2 T s 2 ----------- a 00000 ,,,,, = a ' 01000 = a 0 0 = b = b ' 01100 = d min 2 4 A 2 T s 8 E b η d 2 4 E b ---------- 2 N d N 1 N 2 1 ++ + NN 1 2 --------------------- N 2   =
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(d) For each path there are nearest neighbors which cause one bit error, neighbors which cause 2 errors, neighbors which cause 3 bit errors, . .., and 1 neighbor which causes bit errors. Hence the average number of bit errors is
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This note was uploaded on 02/21/2012 for the course ECEN 661 taught by Professor Miller during the Spring '11 term at Texas A&M.

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HW6Solutions - ECEN 661 - Modulation Theory Homework #6...

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