Solve the following problem:
A Gaussian memory less source is distributed according to N (0, 1). This source is to be transmitted over a binary symmetric channel with a crossover probability of ε = 0.
1. For each source output one use of channel is possible. The fidelity measure is squared-error distortion, i.e., d(x,xˆ ) = (x -xˆ )2.
1. In the first approach we use the optimum one-dimensional (scalar) quantizer. This results in the following quantization rule
Q(x)={xˆ x>0
{-xˆ x≤0
Where xˆ = 0.798 and the resulting distortion is 0.3634. Then xˆ and -xˆ are represented by 0 and 1 and directly transmitted over the channel (no channel coding). Determine the resulting overall distortion using this approach.
2. In the second approach we use the same quantizer used in part 1, but we allow the use of arbitrarily complex channel coding. How would you determine the resulting distortion in this case, and why?
3. Now assume that after quantization, an arbitrarily complex lossless compression scheme is employed and the output is transmitted over the channel (again using channel coding, as explained in part 2). How would the resulting distortion compare with part 2?
4. If you were allowed to use an arbitrarily complex source and channel coding scheme, what would be the minimum achievable distortion?
5. If the source is Gaussian with the same per-letter statistics (i.e., each letter is N (0, 1)) but the source has memory (for instance, a Gauss-Markov source), do you think the distortion you derived in part 4 would increase, decrease, or not change? Why?