# butter

Create a Butterworth filter.

## Syntax

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

[b,a] = butter(n,Wp,band)

[b,a] = butter(n,Wp,domain)

[b,a] = butter(n,Wp,band,domain)

## Inputs

n
The filter order.
Type: integer
Dimension: scalar
Wp
A scalar specifying the 3dB cutoff frequency of a low or high pass filter, or a two element vector specifying the cutoff frequencies of a bandpass or bandstop filter. For a digital filter the values (in Hz) are normalized relative to the Nyquist frequency. For an analog filter the values are in radians/sec.
Type: double
Dimension: scalar | vector
band
The band type of the filter. Omit for low pass or bandpass. Use 'high' for high pass, and 'stop' for bandstop.
Type: string
domain
• Use 'z' for digital filters (default).
• Use 's' for analog filters.

## Outputs

b
The numerator polynomial coefficients of the filter.
Type: vector
a
The denominator polynomial coefficients of the filter.
Type: vector

## Example

Create a fourth order Butterworth low pass digital filter with a 300 Hz cutoff frequency and a 1000 Hz sampling frequency.

[b,a] = butter(4,300/500)
b = [Matrix] 1 x 5
0.16718  0.66872  1.00308  0.66872  0.16718
a = [Matrix] 1 x 5
1.00000  0.78210  0.67998  0.18268  0.03012

The attenuation at Wp is 20*log10(sqrt(2)), or approximately 3.0103 dB.

Filters can become unstable for high orders, and more easily so for bandpass or stopband filters.