Comparison to Optimal Chebyshev FIR Filter
It turns out that the Remez exchange algorithm has convergence problems for filters larger than a few hundred taps. Therefore, the FIR filter length was chosen above to be small enough to work out in this comparison. However, keep in mind that for very large filter orders, the Remez exchange method may not be an option. There are more recently developed methods for optimal Chebyshev FIR filter design, using ``convex optimization'' techniques, that may continue to work at very high orders [218,22,153]. The fast nonparametric methods discussed above (frequency sampling, window method) will work fine at extremely high orders.
hri = firpm(M-1, [f1,f2]/fn, [1,1], , 'Hilbert');Instead, however, we will use a more robust method  which uses the Remez exchange algorithm to design a lowpass filter, followed by modulation of the lowpass impulse-response by a complex sinusoid at frequency in order to frequency-shift the lowpass to the single-sideband filter we seek:
tic; % remember the current time hrm = firpm(M-1, [0,(f2-fs/4)/fn,0.5,1], [1,1,0,0], [1,10]); dt = toc; % design time dt can be minutes hr = hrm .* j .^ [0:M-1]; % modulate lowpass to single-sidebandThe weighting [1,10] in the call to firpm above says ``make the pass-band ripple times that of the stop-band.'' For steady-state audio spectra, pass-band ripple can be as high as dB or more without audible consequences.5.11 The result is shown in Fig.4.16 (full amplitude response) and Fig.4.17 (zoom-in on the dc transition band). By symmetry the high-frequency transition region is identical (but flipped):
More General FIR Filter Design