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Communications Theory

Homework#5

1. The frequency response of a channel is given as

(1 + 0.9ej2"f   ,     - 0.5 < f  < 0.5

l0                ,      otherwise.

Binary PAM signals with basis function  0(t) =   1  sin c(| t )| are transmitted through this

T        \T )

channel with Ex  = 1 , T = 1 , SNR MFB  = 10.

a) Find the unbiased SNR for MMSE-DFE on this channel.

Now, suppose this channel, H(f ), is used for multi-carrier transmission with 4 sub-       channels, i.e., N = 4 . Note that, sub-channels zero and four are one-dimensional and the

fourth sub-channel is silenced ( i.e., X4  = 0 ). The remaining three sub-channels are all

two-dimensional. The transmitted energy per dimension is kept unchanged, .e., Ex   = 1 ,  but the energy contributed to the last one dimensional sub-channel is equally distributed over the remaining 7 dimensions..

A suitable coding scheme which results in a gap 装 = 4.4 dB is also used.

b) Find the new per dimensional transmitted energy . Compute and tabulate the sub- channel quantities En , Hn , SNR n  , bn , the aggregate b and b  for this multi-carrier  transmission.

c) Compute the aggregate SNR mu with 装 = 0 dB for the multi-carrier transmission system in part (b) and compare it with the unbiased SNR, you computed in part (a), and discuss.

d) Why do the multi-carrier pass-band basis functions remain orthogonal at the channel output irrespective of what is the channel impulse response? (Explain).