Consider a frequency-selective fading channel with total bandwidth 12 MHz and coherence bandwidth Bc = 4 MHz. Divide the total bandwidth into 3 subchannels of bandwidth Bc , and assume that each subchannel is a Rayleigh flat-fading channel with independent fading on each subchannel. Assume the subchannels have average gains E[|H1(t)|2] = 1, E[|H2(t)|2] = .5, and E[|H3(t)|2] = .125. Assume a total transmit power of 30 mW, and a receiver noise spectral density of .001µW per Hertz.
(a) Find the optimal two-dimensional water-filling power adaptation for this channel and the corresponding Shannon capacity, assuming both transmitter and receiver know the instantaneous value of Hj(t), j = 1, 2, 3.
(b) Compare the capacity of part (a) with that obtained by allocating an equal average power of 10 mW to each subchannel and then water-filling on each subchannel relative to this power allocation.