Plotting the lower and upper bounds

Question

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.