Determine an optimal binary prefix code for efficiency

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/T_s = 8kHz, L = 6 and = V0/2.

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

c) Show that when L is even Δ = v₀ In 2, and there exists a prefix code with unit efficiency for x_q(nTs).