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Chapters

Introduction to Digital Filters
    Frequency Response Analysis
       Frequency Response as a Ratio of DTFTs
          Example LPF Frequency Response Using freqz

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Example LPF Frequency Response Using freqz

Figure 7.2 lists a short matlab program illustrating usage of freqz in Octave (as found in the octave-forge package). The same code should also run in Matlab, provided the Signal Processing Toolbox is available. The lines of code not pertaining to plots are the following:

  [B,A] = ellip(4,1,20,0.5); % Design lowpass filter B(z)/A(z)
  [H,w] = freqz(B,A);        % Compute frequency response H(w)

The filter example is a recursive fourth-order elliptic function lowpass filter cutting off at half the Nyquist limit (`` $0.5\pi$ '' in the fourth argument to ellip). The maximum passband ripple^8.2is 1 dB (2nd argument), and the maximum stopband ripple is 20 dB (3rd arg). The sampled frequency response is returned in the H array, and the specific radian frequency samples corresponding to H are returned in the w (``omega'') array. An immediate plot can be obtained in either Matlab or Octave by simply typing

  plot(w,abs(H));
  plot(w,angle(H));

However, the example of Fig.7.2 uses more detailed ``compatibility'' functions listed in Appendix J. In particular, the freqplot utility is a simple compatibility wrapper for plot with label and title support (see §J.2 for Octave and Matlab version listings), and saveplot is a trivial compatibility wrapper for the print function, which saves the current plot to a disk file (§J.3). The saved freqplot plots are shown in Fig.7.3(a) and Fig.7.3(b).^8.3

**Figure:** Illustration of using `freqz` from `octave-forge` (version 2006-07-09) to produce Fig.7.3 (using Octave version = 2.9.7). Also illustrated are a few plotting and plot-saving utilities from Appendix J.
[B,A] = ellip(4,1,20,0.5); % Design the lowpass filter [H,w] = freqz(B,A); % Compute its frequency response % Plot the frequency response H(w): % figure(1); freqplot(w,abs(H),'-k','Amplitude Response',... 'Frequency (rad/sample)', 'Gain'); saveplot('../eps/freqzdemo1.eps'); figure(2); freqplot(w,angle(H),'-k','Phase Response',... 'Frequency (rad/sample)', 'Phase (rad)'); saveplot('../eps/freqzdemo2.eps'); % Plot frequency response in a "multiplot" like Matlab uses: % figure(3); plotfr(H,w/(2*pi)); if exist('OCTAVE_VERSION') disp('Cannot save multiplots to disk in Octave') else saveplot('../eps/freqzdemo3.eps'); end

[B,A] = ellip(4,1,20,0.5); % Design the lowpass filter
[H,w] = freqz(B,A);        % Compute its frequency response

% Plot the frequency response H(w):
%
figure(1);
freqplot(w,abs(H),'-k','Amplitude Response',...
         'Frequency (rad/sample)', 'Gain');
saveplot('../eps/freqzdemo1.eps');

figure(2);
freqplot(w,angle(H),'-k','Phase Response',...
         'Frequency (rad/sample)', 'Phase (rad)');
saveplot('../eps/freqzdemo2.eps');

% Plot frequency response in a "multiplot" like Matlab uses:
%
figure(3);
plotfr(H,w/(2*pi));
if exist('OCTAVE_VERSION')
  disp('Cannot save multiplots to disk in Octave')
else
  saveplot('../eps/freqzdemo3.eps');
end

**Figure:** Frequency response of an order 4 elliptic function lowpass filter computed using the matlab listing in Fig.7.2.
$\includegraphics{eps/freqzdemo1}$ Amplitude Response $\includegraphics{eps/freqzdemo2}$ Phase Response

Previous: Frequency Response in Matlab
Next: Phase and Group Delay

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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