Function File: [b, a] = butter (n, wc)
Function File: [b, a] = butter (n, wc, filter_type)
Function File: [z, p, g] = butter (…)
Function File: [a, b, c, d] = butter (…)
Function File: […] = butter (…, "s")

Generate a Butterworth filter. Default is a discrete space (Z) filter.

The cutoff frequency, wc should be specified in radians for analog filters. For digital filters, it must be a value between zero and one. For bandpass filters, wc is a two-element vector with w(1) < w(2).

The filter type must be one of "low", "high", "bandpass", or "stop". The default is "low" if wc is a scalar and "bandpass" if wc is a two-element vector.

If the final input argument is "s" design an analog Laplace space filter.

Low pass filter with cutoff pi*Wc radians: 

[b, a] = butter (n, Wc)

High pass filter with cutoff pi*Wc radians: 

[b, a] = butter (n, Wc, "high")

Band pass filter with edges pi*Wl and pi*Wh radians: 

[b, a] = butter (n, [Wl, Wh])

Band reject filter with edges pi*Wl and pi*Wh radians: 

[b, a] = butter (n, [Wl, Wh], "stop")

Return filter as zero-pole-gain rather than coefficients of the numerator and denominator polynomials: 

[z, p, g] = butter (...)

Return a Laplace space filter, Wc can be larger than 1: 

[...] = butter (..., "s")

Return state-space matrices: 

[a, b, c, d] = butter (...)

References:

Proakis & Manolakis (1992). Digital Signal Processing. New York: Macmillan Publishing Company.

Demonstration 1

The following code

 sf = 800; sf2 = sf/2;
 data=[[1;zeros(sf-1,1)],sinetone(25,sf,1,1),sinetone(50,sf,1,1),sinetone(100,sf,1,1)];
 [b,a]=butter ( 1, 50 / sf2 );
 filtered = filter(b,a,data);

 clf
 subplot ( columns ( filtered ), 1, 1)
 plot(filtered(:,1),";Impulse response;")
 subplot ( columns ( filtered ), 1, 2 )
 plot(filtered(:,2),";25Hz response;")
 subplot ( columns ( filtered ), 1, 3 )
 plot(filtered(:,3),";50Hz response;")
 subplot ( columns ( filtered ), 1, 4 )
 plot(filtered(:,4),";100Hz response;")

Produces the following figure

Figure 1

Package: signal