EE_564_F09_HW7 - EE-564 Fall 2009 Homework#7 Due Wednesday 1 Digital Communications 5th Edition Proakis Salehi Problem 10.2 2(Course Reader Problem 51

EE-564 Fall 2009 Homework #7 Due Wednesday, November 18, 2009 1. “Digital Communications”, 5 th Edition, Proakis & Salehi, Problem 10.2 2. (Course Reader Problem 51) Consider an ISI channel, where the sequence of sampled matched filter outputs is given by ࢟ = ࡳ࢞ + ࢠ Where ࢟ = (ݕ ଵ , ݕ ଶ , … , ݕ ௡ିଵ ) ࢀ , ࡳ is a Hermitian symmetric Toeplitz matrix with (݅, ݆) element [ࡳ] ௜,௝ = ݃ ௜ି௝ , and where ࢠ denotes the noise contribution to the output of the matched filter. Assume there exists a lower triangular matrix ࡸ such that ࡳ = ࡸ ࡴ ࡸ . This decomposition is called Cholesky factorization and ࡸ is referred to as the Cholesky factor of G . L = ۉ ۈ ۈ ۈ ۈ ۇ L ଴ 0 L ଵ L ଴ 0 0 0 0 L ଶ L ଵ 0 L ଶ L ଴ 0 L ଵ L ଴ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 L ଶ L ଵ 0 L ଶ 0 0 0 0 0 0 0 0 L ଴ 0 L ଵ L ଴ 0 0 0 0 L ଶ L ଵ 0 L ଶ L ଴ 0 L ଵ L ଴ ی ۋ ۋ ۋ ۋ ۊ In which L ଴ = 1 , L ଵ = −1/2 , and L ଶ = 1/4 . Input signal set is {-1, 1} and the output sequence is y=(1.5, −1, 2.5, −2, −2.5 , 0, 1, 0) ୘ . Assume that the initial state is x ିଵ = x ିଶ = 0 . Find the optimum MAP decoded sequence.