Determine the efficiency of a ternary shannon-fano code

Question

Assignment:

1. Let x_n be a message with iid symbols, a quaternary alphabet and the following PMD

Px(0) = 0.7, Px(1) = Px(1) = Px(2) = Px(3)= 0.1

a) Find a binary Huffman code with single symbols xn as input and calculate its efficiency.

b) Find the efficiency of a ternary Shannon-Fano code with pairs x = [x_n, xn+₁] as input.

2. A stationary message {x_n} with iid symbols and alphabet Ax = {0, 1, 2} has PMD

P_x(0) = 1/2, P_x(1) = 1/3, P_x(2) = 1/6.

a) Construct a binary prefix code y with Ay = {0, 1} that is optimal for coding single symbols of x. Evaluate the code efficiency.

b) Consider the vector x_m= [x_2m, x_2m+1] made of two consecutive symbols taken from {x_n} and find a binary code that is optimal for x_m. Compare the efficiencies η_y and η_y.