Parallel Gaussian channels. Consider two parallel complex Gaussian channels with channel gains h1 = 1 + j, h2 = −3j and noise covariances N1 = 1, N2 = 2. Assume that the transmitter knows the channel characteristics.

(a) At “low” SNR, which of the two channels would you use?

(b) For what values of net input power P would you start using both channels?

(c) Plot the capacity as a function of net input power P using the waterfilling power allocation. Also plot for comparison the capacity attained if the transmitter does not know the channel characteristics, and splits power evenly across the two channels.