Remez Multiple Exchange Method

Let's now try the Remez multiple exchange algorithm (``Parks-McClellan algorithm'') for optimal Chebyshev (equiripple) FIR filter design, and compare it to our window-method design.

Step 1: Design an optimal lowpass filter of the desired width (takes quite a while):

  M = 257;      % FIR filter length
  fs = 22050;   % sampling rate (Hz)
  fn = fs/2;    % Nyquist limit (Hz)
  f1 = 530;     % lower passband limit (Hz)
  f2 = fn - f1; % upper passpand limit (see text, p. 136)
  hrm = firpm(M-1, [0,(f2-fs/4)/fn,0.5,1], ...
                          [1,1,0,0], [1,10]);

• The weighting [1,P] says make the passband ripple P times that of stopband

• For steady-state audio spectra, passband ripple $\approx 0.1$ dB is good
  • However, consider an FM signal in the passband $\Rightarrow$ passband ripple becomes AM sidebands

  • Here, we allow the passband ripple to be 10 times the stopband ripple, as a compromise

Step 2: Modulate the impulse response to make it a single-sideband filter. I.e., right-shift by $\pi/2$ in the frequency domain (rotate the frequency response counterclockwise along the unit circle by 90 degrees):

hr = hrm .* j .^ [0:M-1];