Solve the following problem:
It can be shown that if X is a zero-mean continuous random variable with variance σ2, its rate distortion function, subject to squared-error distortion measure, satisfies the lower and upper bounds given by the inequalities
H(X)-1/2 log(2πeD) ≤ R(D) ≤ 1/2 log σ 2/2
where H(X) denotes the differential entropy of the random variable X (see Cover and Thomas, 2006).
1. Show that, for a Gaussian random variable, the lower and upper bounds coincide.
2. Plot the lower and upper bounds for a Laplacian source with σ = 1.
3. Plot the lower and upper bounds for a triangular source with σ = 1.