Butterworth Lowpass Filter Example

This example illustrates the design of a 5th-order Butterworth lowpass filter, implementing it using second-order sections. Since all three sections contribute to the same passband and stopband, it is numerically advisable to choose a series second-order-section implementation, so that their passbands and stopbands will multiply together instead of add.

fc = 1000; % Cut-off frequency (Hz)
fs = 8192; % Sampling rate (Hz)
order = 5; % Filter order
[B,A] = butter(order,2*fc/fs); % [0:pi] maps to [0:1] here
[sos,g] = tf2sos(B,A)
% sos =
%  1.00000  2.00080   1.00080  1.00000  -0.92223  0.28087
%  1.00000  1.99791   0.99791  1.00000  -1.18573  0.64684
%  1.00000  1.00129  -0.00000  1.00000  -0.42504  0.00000
%
% g = 0.0029714
%
% Compute and display the amplitude response
Bs = sos(:,1:3); % Section numerator polynomials
As = sos(:,4:6); % Section denominator polynomials
[nsec,temp] = size(sos);
nsamps = 256; % Number of impulse-response samples
% Note use of input scale-factor g here:
x = g*[1,zeros(1,nsamps-1)]; % SCALED impulse signal
for i=1:nsec
  x = filter(Bs(i,:),As(i,:),x); % Series sections
end
%
%plot(x); % Plot impulse response to make sure
          % it has decayed to zero (numerically)
%
% Plot amplitude response
% (in Octave - Matlab slightly different):
figure(2);
X=fft(x); % sampled frequency response
f = [0:nsamps-1]*fs/nsamps; grid('on');
axis([0 fs/2 -100 5]); legend('off');
plot(f(1:nsamps/2),20*log10(X(1:nsamps/2)));

The final plot appears in Fig.9.7. A Matlab function for frequency response plots is given in §J.4. (Of course, one can also use freqz in either Matlab or Octave, but that function uses subplots which are not easily printable in Octave.)

Note that the Matlab Signal Processing Toolbox has a function called sosfilt so that ``y=sosfilt(sos,x)'' will implement an array of second-order sections without having to unpack them first as in the example above.

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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.