Determine efficiency of ternary shannon-fano code with pairs

Question

Assignment:

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

P_x(0) = 0.7, P_x(1) = P_x(2) = P_x(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, xn₊₁] as input.

2. A stationary message {xn} 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 A_y = {0, 1} that is optimal for coding single symbols of x. Evaluate the code efficiency.

b) Consider the vector 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.