A somewhat more general model for echo than the system in Problem 13.22 is the system de...

A somewhat more general model for echo than the system in Problem 13.22 is the system depicted in Figure P13.31. The impulse response of this system is

h[n] = δ[n] + αg[n − n₀],

where αg[n] is the impulse response of the echo path.

(a) Assuming that

show that the complex cepstrum has the form

and determine an expression for g_k [n] in terms of g[n].

(b) For the conditions of part (a), determine and sketch the complex cepstrum when g[n] = δ[n].

(c) For the conditions of part (a), determine and sketch the complex cepstrum when g[n] = aⁿu[n]. What condition must be satisfied by α and a so that the result of part (a) applies?

(d) For the conditions of part (a), determine and sketch the complex cepstrum when g[n] = a₀δ[n] + a₁δ[n − n₁]. What condition must be satisfied by α, a₀, a₁, and n₁ so that the result of part (a) applies?