Determine an optimal binary prefix code

Question

Assignment:

A discrete time rp x(nTs) with iid symbols and Laplacian distribution

Px(a) = 1/2V₀e^-|a|/V0

is quantized with an uniform quantizer with L levels and step size  Δ. Let x_q(nT_s) be the quantizer output rp.

a) Find alphabet, entropy and information rate of x_q(nT_s) in the case 1/Ts = 8kHz, L = 6 and = V0/2.

b) Find an optimal binary prefix code for x_q(nT_s), with the above values for L and and evaluate its efficiency.

c) Show that when L is even and Δ = V₀ ln 2, there exists a prefix code with unit efficiency for x_q(nTs).