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.)

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

Figure 9.7: Measured amplitude response of three second-order sections (in series) implementing a 5th order Butterworth lowpass filter with half-power point at $f_c=1$ kHz.