# butter

Butterworth filter design

## Description

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[b,a] = butter(n,Wn) returns the transfer function coefficients of an nth-order lowpass digital Butterworth filter with normalized cutoff frequency Wn.

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[b,a] = butter(n,Wn,ftype) designs a lowpass, highpass, bandpass, or bandstop Butterworth filter, depending on the value of ftype and the number of elements of Wn. The resulting bandpass and bandstop designs are of order 2n.

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

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[z,p,k] = butter(___) 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.

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[A,B,C,D] = butter(___) designs a lowpass, highpass, bandpass, or bandstop digital Butterworth filter and returns the matrices that specify its state-space representation.

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[___] = butter(___,'s') designs a lowpass, highpass, bandpass, or bandstop analog Butterworth filter with cutoff angular frequency Wn.

## Examples

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Design a 6th-order lowpass Butterworth filter with a cutoff frequency of 300 Hz, which, for data sampled at 1000 Hz, corresponds to $0.6\pi$ rad/sample. Plot its magnitude and phase responses. Use it to filter a 1000-sample random signal.

fc = 300;
fs = 1000;

[b,a] = butter(6,fc/(fs/2));
freqz(b,a)

dataIn = randn(1000,1);
dataOut = filter(b,a,dataIn);

Design a 6th-order Butterworth bandstop filter with normalized edge frequencies of $0.2\pi$ and $0.6\pi$ rad/sample. Plot its magnitude and phase responses. Use it to filter random data.

[b,a] = butter(3,[0.2 0.6],'stop');
freqz(b,a)

dataIn = randn(1000,1);
dataOut = filter(b,a,dataIn);

Design a 9th-order highpass Butterworth filter. Specify a cutoff frequency of 300 Hz, which, for data sampled at 1000 Hz, corresponds to $0.6\pi$ rad/sample. Plot the magnitude and phase responses. Convert the zeros, poles, and gain to second-order sections for use by fvtool.

[z,p,k] = butter(9,300/500,'high');
sos = zp2sos(z,p,k);
fvtool(sos,'Analysis','freq')

Design a 20th-order Butterworth bandpass filter with a lower cutoff frequency of 500 Hz and a higher cutoff frequency of 560 Hz. Specify a sample rate of 1500 Hz. Use the state-space representation. Design an identical filter using designfilt.

[A,B,C,D] = butter(10,[500 560]/750);
d = designfilt('bandpassiir','FilterOrder',20, ...
'HalfPowerFrequency1',500,'HalfPowerFrequency2',560, ...
'SampleRate',1500);

Convert the state-space representation to second-order sections. Visualize the frequency responses using fvtool.

sos = ss2sos(A,B,C,D);
fvt = fvtool(sos,d,'Fs',1500);
legend(fvt,'butter','designfilt')

Design a 5th-order analog Butterworth lowpass filter with a cutoff frequency of 2 GHz. Multiply by $2\pi$ to convert the frequency to radians per second. Compute the frequency response of the filter at 4096 points.

n = 5;
f = 2e9;

[zb,pb,kb] = butter(n,2*pi*f,'s');
[bb,ab] = zp2tf(zb,pb,kb);
[hb,wb] = freqs(bb,ab,4096);

Design a 5th-order Chebyshev Type I filter with the same edge frequency and 3 dB of passband ripple. Compute its frequency response.

[z1,p1,k1] = cheby1(n,3,2*pi*f,'s');
[b1,a1] = zp2tf(z1,p1,k1);
[h1,w1] = freqs(b1,a1,4096);

Design a 5th-order Chebyshev Type II filter with the same edge frequency and 30 dB of stopband attenuation. Compute its frequency response.

[z2,p2,k2] = cheby2(n,30,2*pi*f,'s');
[b2,a2] = zp2tf(z2,p2,k2);
[h2,w2] = freqs(b2,a2,4096);

Design a 5th-order elliptic filter with the same edge frequency, 3 dB of passband ripple, and 30 dB of stopband attenuation. Compute its frequency response.

[ze,pe,ke] = ellip(n,3,30,2*pi*f,'s');
[be,ae] = zp2tf(ze,pe,ke);
[he,we] = freqs(be,ae,4096);

Plot the attenuation in decibels. Express the frequency in gigahertz. Compare the filters.

plot(wb/(2e9*pi),mag2db(abs(hb)))
hold on
plot(w1/(2e9*pi),mag2db(abs(h1)))
plot(w2/(2e9*pi),mag2db(abs(h2)))
plot(we/(2e9*pi),mag2db(abs(he)))
axis([0 4 -40 5])
grid
xlabel('Frequency (GHz)')
ylabel('Attenuation (dB)')
legend('butter','cheby1','cheby2','ellip')

The Butterworth and Chebyshev Type II filters have flat passbands and wide transition bands. The Chebyshev Type I and elliptic filters roll off faster but have passband ripple. The frequency input to the Chebyshev Type II design function sets the beginning of the stopband rather than the end of the passband.

## Input Arguments

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Filter order, specified as an integer scalar. For bandpass and bandstop designs, n represents one-half the filter order.

Data Types: double

Cutoff frequency, specified as a scalar or a two-element vector. The cutoff frequency is the frequency at which the magnitude response of the filter is 1 / √2.

• If Wn is scalar, then butter designs a lowpass or highpass filter with cutoff frequency Wn.

If Wn is the two-element vector [w1 w2], where w1 < w2, then butter designs a bandpass or bandstop filter with lower cutoff frequency w1 and higher cutoff frequency w2.

• For digital filters, the cutoff frequencies must lie between 0 and 1, where 1 corresponds to the Nyquist rate—half the sample rate or π rad/sample.

For analog filters, the cutoff frequencies must be expressed in radians per second and can take on any positive value.

Data Types: double

Filter type, specified as one of the following:

• 'low' specifies a lowpass filter with cutoff frequency Wn. 'low' is the default for scalar Wn.

• 'high' specifies a highpass filter with cutoff frequency Wn.

• 'bandpass' specifies a bandpass filter of order 2n if Wn is a two-element vector. 'bandpass' is the default when Wn has two elements.

• 'stop' specifies a bandstop filter of order 2n if Wn is a two-element vector.

## Output Arguments

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Transfer function coefficients of the filter, returned as row vectors of length n + 1 for lowpass and highpass filters and 2n + 1 for bandpass and bandstop filters.

• For digital filters, the transfer function is expressed in terms of b and a as

$H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{\text{b(1)}+\text{b(2)}\text{\hspace{0.17em}}{z}^{-1}+\cdots +\text{b(n+1)}\text{\hspace{0.17em}}{z}^{-n}}{\text{a(1)}+\text{a(2)}\text{\hspace{0.17em}}{z}^{-1}+\cdots +\text{a(n+1)}\text{\hspace{0.17em}}{z}^{-n}}.$

• For analog filters, the transfer function is expressed in terms of b and a as

$H\left(s\right)=\frac{B\left(s\right)}{A\left(s\right)}=\frac{\text{b(1)}\text{\hspace{0.17em}}{s}^{n}+\text{b(2)}\text{\hspace{0.17em}}{s}^{n-1}+\cdots +\text{b(n+1)}}{\text{a(1)}\text{\hspace{0.17em}}{s}^{n}+\text{a(2)}\text{\hspace{0.17em}}{s}^{n-1}+\cdots +\text{a(n+1)}}.$

Data Types: double

Zeros, poles, and gain of the filter, returned as two column vectors of length n (2n for bandpass and bandstop designs) and a scalar.

• For digital filters, the transfer function is expressed in terms of z, p, and k as

$H\left(z\right)=\text{k}\frac{\left(1-\text{z(1)}\text{\hspace{0.17em}}{z}^{-1}\right)\text{\hspace{0.17em}}\left(1-\text{z(2)}\text{\hspace{0.17em}}{z}^{-1}\right)\cdots \left(1-\text{z(n)}\text{\hspace{0.17em}}{z}^{-1}\right)}{\left(1-\text{p(1)}\text{\hspace{0.17em}}{z}^{-1}\right)\text{\hspace{0.17em}}\left(1-\text{p(2)}\text{\hspace{0.17em}}{z}^{-1}\right)\cdots \left(1-\text{p(n)}\text{\hspace{0.17em}}{z}^{-1}\right)}.$

• For analog filters, the transfer function is expressed in terms of z, p, and k as

$H\left(s\right)=\text{k}\frac{\left(s-\text{z(1)}\right)\text{\hspace{0.17em}}\left(s-\text{z(2)}\right)\cdots \left(s-\text{z(n)}\right)}{\left(s-\text{p(1)}\right)\text{\hspace{0.17em}}\left(s-\text{p(2)}\right)\cdots \left(s-\text{p(n)}\right)}.$

Data Types: double

State-space representation of the filter, returned as matrices. If m = n for lowpass and highpass designs and m = 2n for bandpass and bandstop filters, then A is m × m, B is m × 1, C is 1 × m, and D is 1 × 1.

• For digital filters, the state-space matrices relate the state vector x, the input u, and the output y through

$\begin{array}{c}x\left(k+1\right)=\text{A}\text{\hspace{0.17em}}x\left(k\right)+\text{B}\text{\hspace{0.17em}}u\left(k\right)\\ y\left(k\right)=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{C}\text{\hspace{0.17em}}x\left(k\right)+\text{D}\text{\hspace{0.17em}}u\left(k\right).\end{array}$

• For analog filters, the state-space matrices relate the state vector x, the input u, and the output y through

$\begin{array}{l}\stackrel{˙}{x}=\text{A}\text{\hspace{0.17em}}x+\text{B}\text{\hspace{0.17em}}u\\ y=\text{C}\text{\hspace{0.17em}}x+\text{D}\text{\hspace{0.17em}}u.\end{array}$

Data Types: double

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### Limitations

Numerical Instability of Transfer Function Syntax

In general, use the [z,p,k] syntax to design IIR filters. To analyze or implement your filter, you can then use the [z,p,k] output with zp2sos. If you design the filter using the [b,a] syntax, you might encounter numerical problems. These problems are due to round-off errors and can occur for n as low as 4. The following example illustrates this limitation.

n = 6;
Wn = [2.5e6 29e6]/500e6;
ftype = 'bandpass';

% Transfer Function design
[b,a] = butter(n,Wn,ftype);      % This is an unstable filter

% Zero-Pole-Gain design
[z,p,k] = butter(n,Wn,ftype);
sos = zp2sos(z,p,k);

% Display and compare results
hfvt = fvtool(b,a,sos,'FrequencyScale','log');
legend(hfvt,'TF Design','ZPK Design')

## 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:

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

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

3. 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.

4. 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.

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