Solve the following problem:
A discrete memory less source U is to be transmitted over a memory less communication channel. For each source output, the channel can be used only once. Determine the minimum theoretical distortion achievable in transmission of the source over the channel in each of the following cases.
1. The source is a binary source with 0 and 1 as its outputs with p(U = 0) = 0.1; the channel is a binary symmetric channel with crossover probability ε = 0.1; and the distortion measure is the Hamming distortion (probability of error).
2. The channel is as in part 1, but the source is a zero-mean Gaussian source with variance 1. The distortion is the squared-error distortion.
3. The source is as in part 2, and the channel is a discrete-time AWGN channel with input power constraint P and noise variance σ2.