The sequence h[n] = δ[n] + αδ[n − n0] is a simplified model for the impulse respon...

The sequence

h[n] = δ[n] + αδ[n − n0]

is a simplified model for the impulse response of a system that introduces an echo.

(a) Determine the complex cepstrum for this sequence. Sketch the result.

(b) Determine and sketch the cepstrum c_h[n].

(c) Suppose that an approximation to the complex cepstrum is computed using N-point DFTs as in Eqs. (13.46a) to (13.46c). Obtain a closed-form expression for the approximation for the case n₀ = N/6. Assume that phase unwrapping can be accurately done. What happens if N is not divisible by n₀?

(d) Repeat part (c) for the cepstrum approximation c_xp[n], 0 ≤ n ≤ N − 1, as computed using Eqs. (13.60a) and (13.60b).

(e) If the largest impulse in the cepstrum approximation c_xp[n] is to be used to detect the value of the echo delay n₀, how large must N be to avoid ambiguity? Assume that accurate phase unwrapping can be achieved with this value of N.