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Communications Theory

Homework#6

1. The frequency response of a channel is given as

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

l0                ,      otherwise.

This channel is used for multi-carrier transmission with 8 complex sub-channels, i.e.,     N = 8 . Note that, sub-channels zero and the last one are one-dimensional. The last one-  dimensional sub-channel is silenced ( i.e., X8  = 0 ). The remaining 7 sub-channels are all two-dimensional (so, with the first and the last one dimensional sub-channels counted as one two-dimensional sub-channel, there are 8 two dimensional sub-channels, altogether). The transmitted energy per dimension is Ex   = 1 .  The energy contributed to the last one   dimensional sub-channel is equally distributed over the remaining 15 dimensions..           A suitable coding scheme which results in a gap r = 4.4 dB is also used and

SNR       = 10.

2         N0

2

b) What is the symbol period T over each of the sub-channels of this transmission system? And why? (Note that the base-band equivalent basis function used in this system

is  0(t) =   1  sin c(| t )| ).

c) Compute and tabulate the sub-channel quantities En , Hn , SNR n  , bn , the aggregate b and b  for this multi-carrier transmission.

d) Compute the aggregate SNR mu with r = 0 dB .