In this problem, you are to consider the design of low pass filters using the Pade and Prony methods.
(a) Suppose you would like to design a low pass filter with a cutoff frequency ωc = π/2. Using the Pade approximation with p + q + 1 = 20, compare the designs that result when p = 0, 2, 4, ... , 20. Which design is the best? (Note: one of the parameters that you will need to experiment with in your design is the delay, n0, of the ideal unit sample response.)
(b) Repeat part (a) for a narrowband low pass filter that has a cutoff frequency ωc = π/16. Which design is the best and how do these filters compare to those designed in part (a)?
(c) Repeat part (a) using Prony's method and compute the minimum error ?p, q for each filter. Which filter is the best? Compare your designs to those obtained using Pade approximation.
(d) Using the m-file el lip.min the Signal Processing Toolbox, design a tenth-order elliptic lowpass filter with a cutoff frequency ωc = π/2, and evaluate the Prony error. Compare this filter to the best Prony filter found in part (c) and explain what you observe.