Assignment:
1. The signal a(t) has probability density function
Pa(u) = {oKe-2|u|, -3 < u < 3 otherwise
a) Find the value of the signal-to-quantization noise ratio using an uniform quantizer with three bits.
b) Find the value of the signal-to-quantization noise ratio using a non-uniform quantizer with three bits and standard µ-law .
2. Consider an audio signal with bandwidth B = 4 kHz and uniformly distributed amplitude in (-A, +A), with A = 5 V. The signal is PCM encoded with a uniform quantizer having L = 2b levels.
a) Determine the relation between the signal to quantization noise ratio (in dB) and the number of bits b.
b) Discuss whether, with the same number of levels L, the system is improved if a non-uniform quantizer is used.
3.18 Prove that
1. a) All functions of the type (3.58) satisfy properties P1-P4 in section 3.3.1.
b) Any continuous and differentiable function that satisfies properties P1-P4 must be of the type (3.58). Observe that this does not constitute a proof of the "only" part as continuity and differentiability were not among the axioms.
2. a) Find the entropy of a geometric rv with
Ax = {0,1,2,...}, Px(k) = (1-p)pk
b) Prove that the entropy of a binomial rv with parameters n ∈ N and p ∈ (0, 1), having
Ax ={0,1,....,n}, Px(k)= (nk) (1-p)n-k Pk is bounded by H(x) ≤ n [p log1/2 p+(1-p) log1/2(1-p)]
c) Calculate the values of the entropy and the upper bound above for a binomial rv with parameters n = 8 and p = 1/2. For which of the above rvs is it possible to determine the efficiency?