A problem that often arises in practice is one in which a distorted signal y[n] is the output that results when a desired signal x[n] has been filtered by an LTI system. We wish to recover the original signal x[n] by processing y[n]. In theory, x[n] can be recovered from y[n] by passing y[n] through an inverse filter having a system function equal to the reciprocal of the system function of the distorting filter. 

Suppose that the distortion is caused by an FIR filter with impulse response 

h[n] = δ[n] – ½ δ[n – n₀],

 where n₀ is a positive integer, i.e., the distortion of x[n] takes the form of an echo at delay n₀.

(a) Determine the z-transform H(z) and the N-point DFT H[k] of the impulse response h[n]. Assume that N = 4n₀.

(b) Let H_i(z) denote the system function of the inverse filter, and let h_i[n] be the corresponding impulse response. Determine h_i[n]. Is this an FIR or an IIR filter? What is the duration of h_i[n]?

(c) Suppose that we use an FIR filter of length N in an attempt to implement the inverse filter, and let the N-point DFT of the FIR filter be 

G[k] = 1/ H[k],             k = 0, 1….N – 1,

What is the impulse response g[n] of the FIR filter?

(d) It might appear that the FIR filters with DFT G[k] = 1/H [k] implements the inverse filter perfectly. After all, one might argue that the FIR distorting filter has an N-point DFT H[k] and the FIR filter in cascade has an N-point DFT G[k] = 1/H[k], and since G[k]H[k] = 1 for all k, we have implemented an all-pass, nonedistorting filter. Briefly explain the fallacy in this argument.

(e) Perform the convolution of g[n] with h[n], and thus determine how well the FIR filters with N-point DFT G[k] = 1 / H[k] implements the inverse filter.