# butter

Butterworth filter design

## Syntax

## Description

`[`

designs
a lowpass, highpass, bandpass, or bandstop Butterworth filter, depending
on the value of `b,a`

] = butter(`n`

,`Wn`

,`ftype`

)`ftype`

and the number of elements
of `Wn`

. The resulting bandpass and bandstop designs
are of order 2`n`

.

**Note:** See Limitations for information about numerical issues that affect
forming the transfer function.

`[`

designs
a lowpass, highpass, bandpass, or bandstop digital Butterworth filter
and returns its zeros, poles, and gain. This syntax can include any
of the input arguments in previous syntaxes.`z,p,k`

] = butter(___)

## Examples

## Input Arguments

## Output Arguments

## More About

## Algorithms

Butterworth filters have a magnitude response that is maximally flat in the passband and monotonic overall. This smoothness comes at the price of decreased rolloff steepness. Elliptic and Chebyshev filters generally provide steeper rolloff for a given filter order.

`butter`

uses a five-step algorithm:

It finds the lowpass analog prototype poles, zeros, and gain using the function

`buttap`

.It converts the poles, zeros, and gain into state-space form.

If required, it uses a state-space transformation to convert the lowpass filter into a bandpass, highpass, or bandstop filter with the desired frequency constraints.

For digital filter design, it uses

`bilinear`

to convert the analog filter into a digital filter through a bilinear transformation with frequency prewarping. Careful frequency adjustment enables the analog filters and the digital filters to have the same frequency response magnitude at`Wn`

or at`w1`

and`w2`

.It converts the state-space filter back to its transfer function or zero-pole-gain form, as required.

## Extended Capabilities

## See Also

`besself`

| `buttap`

| `buttord`

| `cheby1`

| `cheby2`

| `designfilt`

| `ellip`

| `filter`

| `maxflat`

| `sosfilt`

**Introduced before R2006a**