Let h[n] be the optimal type I equiripple lowpass filter shown in Figure P7.61, designed...

Let h[n] be the optimal type I equiripple lowpass filter shown in Figure P7.61, designed with weighting function W(e^jω) and desired frequency response H_d (e^jω). For simplicity, assume that the filter is zero phase (i.e., noncausal).We will use h[n] to design five different FIR filters as follows:

For each filter h_i [n], determine whether h_i [n] is optimal in the minimax sense. That is, determine whether

for some choices of a piecewise-constant H_d (e^jω) and a piecewise-constant W(e^jω), where F is a union of disjoint closed intervals on 0 ≤ ω ≤ π. If h_i [n] is optimal, determine the corresponding H_d (e^jω) and W(e^jω). If h_i [n] is not optimal, explain why.