We have observed the first six samples of a causal signal s[n] given by s[0] = 4, s[1] =...

We have observed the first six samples of a causal signal s[n] given by s[0] = 4, s[1] = 8, s[2] = 4, s[3] = 2, s[4] = 1, and s[5] = 0.5. For the first parts of this problem, we will model the signal using a stable, causal, minimum-phase, two-pole system having impulse response and system function

The approach is to minimize the modeling error given by

where g[n] is the response of the inverse system to s[n], and the inverse system has system function

(a) Write g[n] − Gδ[n] for 0 ≤ n ≤ 5.

(b) Based on your work in part (a), write the linear equations for the desired parameters a₁, a₂, and G.

(c) What is G?

(d) For this s[n], without solving the linear equations in part (b), discuss whether you expect that the modeling error will be zero. For the rest of this problem, we will model the signal using a different stable, causal, minimum-phase system having impulse response and system function

The modeling error to be minimized in this case is ₂ given by

where g[n] is the response of the inverse system to s[n], and the inverse system now has system function

A(z) = 1 − a^z−1.

Furthermore, r[n] is the impulse response of a system with system function

B(z) = b₀ + b₁^z−1.

(e) For this model, write g[n] − r[n] for 0 ≤ n ≤ 5.

(f) Calculate the parameter values a, b₀, and b₁ that minimize the modeling error.

(g) Calculate the modeling error in part (f).