The Pade approximant provides  an exact matching of M + N – 1 values of h[n], where M and N are the orders of the numerator and denominator of the rational approximation. But there is no method for choosing the numerator and denominator orders, M and N. Also, there is no guarantee on how well the  rest of the signal is matched. Prony’s rational approximation considers  how well the rest of the signal is approximated. Let h[n] = 0.9ⁿ u[n], be  the impulse response we wish to find a rational approximation. Take  the first 100 values of this signal as the impulse response.

(a) Assume the order of the numerator and the denominators are  equal M = N = 1, use the MATLAB function prony to obtain  the rational approximation, and then use filter to verify that the  impulse response of the rational approximation is close to the  given 100 values. Plot the error between h[n] and the impulse  response of the rational approximation for the first 100 samples.  Plot the poles and zeros of the rational approximation and compare them to the poles and zeros of H(z) = Z(h[n]).

(b) Suppose that h[n] = (h₁ ∗ h₂)[n], i.e., the convolution of  h₁ [n] = 0.9ⁿ u[n] and h² [n] = 0.8ⁿ u[n]. Use again Prony to find  the rational approximation when the first 100 values of h[n]are  available. Use convfrom MATLAB to compute h[n]. Compare  the impulse response of the rational approximation to h[n].  Plot the poles and zeros of H(z) = Z(h[n]) and of the rational  approximation.

(c) Consider the h[n]given above, and perform the Prony approximation  using orders M = N = 3, explain your results. Plot poles and zeros.