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Communications Theory

Homework#7

1. Assume P(D) = 1+ aD-1 , which corresponds to the following channel:

h(t)一——( F.T)) H(f ) =〈(1+ aej2"f   ,     - 0.5 < f  < 0.5

l0                 ,      otherwise.

In this problem, we would like to design optimal digital multi-channel transceiver (transmitter and receiver) structures for the above given channel.

a) Assume N = 4 , a = 0.9 and compute F, Λ and M . Plot the magnitude spectrum of the column vectors ofM , i.e., the singular vectors using MATLAB command “abs(fft)” . Use at least 32 point FFT and discuss the shapes.

b) Repeat part (a) for N = 4 , a = 0. 1and compare the resulting spectrums.

c) Assume N = 8 , a = 0.9 and repeat part (a) with 64 point FFT . Using the diagonal

elements of Λ , calculate 入1 ,   2            3  ,   4            5  ,   6            7    and compare these values with

2            2            2

values you got in homework#5 for Hn  , n = 0,1,2,3 and explain the similarities.

d) Do you see any special structure in F matrices, you have computed so far? Explain.