Consider the design of a type I bandpass linear-phase FIR filter using the Parks–McClell...

Consider the design of a type I bandpass linear-phase FIR filter using the Parks–McClellan algorithm. The impulse response length is M+1 = 2L+1. Recall that for type I systems, the frequency response is of the form H(e^jω) = Ae(e^jω)e ^–jωm/2, and the Parks–McClellan algorithm finds the function A_e(e^jω) that minimizes the maximum value of the error function

where F is a closed subset of the interval 0 ≤ ω ≤ π, W(ω) is a weighting function, and H_d (e^jω) defines the desired frequency response in the approximation intervals F. The tolerance scheme for a bandpass filter is shown in Figure P7.63.

(a) Give the equation for the desired response H_d (e^jω) for the tolerance scheme in Figure P7.63.

(b) Give the equation for the weighting function W(ω) for the tolerance scheme in FigureP7.63.

(c) What is the minimum number of alternations of the error function for the optimum filter?

(d) What is the maximum number of alternations of the error function for the optimum filter?

(e) Sketch a “typical” weighted error function E(ω) that could be the error function for an optimum bandpass filter if M = 14. Assume the maximum number of alternations.

(f) Now suppose that M, ω₁, ω₂, ω₃, the weighting function, and the desired function are kept the same, but ω₄ is increased, so that the transition band (ω₄ − ω₃) is increased. Will the optimum filter for these new specifications necessarily have a smaller value of the maximum approximation error than the optimum filter associated with the original specifications? Clearly show your reasoning.

(g) In the lowpass filter case, all local minima and maxima of A_e(e^jω) must occur in the approximation bands ω ∈ F; they cannot occur in the “don’t care” bands. Also, in the lowpass case, the local minima and maxima that occur in the approximation bands must be alternations of the error. Show that this is not necessarily true in the bandpass filter case. Specifically, use the alternation theorem to show (i) that local maxima and minima of A_e(e^jω) are not restricted to the approximation bands and (ii) that local maxima and minima in the approximation bands need not be alternations.