Let h[n] be the optimal type I equiripple lowpass filter shown in Figure, designed with weighting function W(e^j?) and desired frequency response H_d(e^j?). 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^j?) and a piecewise-constant W(e^j?), where F is a union of disjoint closed intervals on 0 ? ? ? ?. If h_i[n] is optimal, determine the corresponding H_d(e^j?) and W(e^j?). If h_i[n] is not optimal, explain why.

hi[n] = h[-n]. %3D ha[n] = (-1)"h[n]. %3D ha[n] = h[n] + h[n]. haln] = h[n] K8[n], where K is a constant, h[n/2] for n even, hs[r] = {0/21 otherwise. h,[n] = min max (W(ei")Ha(ei)- H,(e") h,n] weF H(e ) 1+ 81 1-8, 82 Wp