buttord

Butterworth filter order and cutoff frequency

Syntax

[n,Wn] = buttord(Wp,Ws,Rp,Rs)

[n,Wn] = buttord(Wp,Ws,Rp,Rs,'s')

Description

[n,Wn] = buttord(Wp,Ws,Rp,Rs) returns the lowest order, n, of the digital Butterworth filter with no more than Rp dB of passband ripple and at least Rs dB of attenuation in the stopband. Wp and Ws are respectively the passband and stopband edge frequencies of the filter, normalized from 0 to 1, where 1 corresponds to π rad/sample. The scalar (or vector) of corresponding cutoff frequencies, Wn, is also returned. To design a Butterworth filter, use the output arguments n and Wn as inputs to butter.

[n,Wn] = buttord(Wp,Ws,Rp,Rs,'s') finds the minimum order n and cutoff frequencies Wn for an analog Butterworth filter. Specify the frequencies Wp and Ws in radians per second. The passband or the stopband can be infinite.

Examples

Lowpass Butterworth Filter

For data sampled at 1000 Hz, design a lowpass filter with no more than 3 dB of ripple in a passband from 0 to 40 Hz, and at least 60 dB of attenuation in the stopband. Find the filter order and cutoff frequency.

Wp = 40/500;
Ws = 150/500;

[n,Wn] = buttord(Wp,Ws,3,60)

n = 
5

Wn = 
0.0810

Specify the filter in terms of second-order sections and plot the frequency response.

[z,p,k] = butter(n,Wn);
sos = zp2sos(z,p,k);

freqz(sos,512,1000)
title(sprintf('n = %d Butterworth Lowpass Filter',n))

Bandpass Butterworth Filter

Design a bandpass filter with a passband from 100 to 200 Hz with at most 3 dB of passband ripple and at least 40 dB attenuation in the stopbands. Specify a sample rate of 1 kHz. Set the stopband width to 50 Hz on both sides of the passband. Find the filter order and cutoff frequencies.

Wp = [100 200]/500;
Ws = [50 250]/500;
Rp = 3;
Rs = 40;
[n,Wn] = buttord(Wp,Ws,Rp,Rs)

n = 
8

Wn = 1×2

    0.1951    0.4080

Specify the filter in terms of second-order sections and plot the frequency response.

[z,p,k] = butter(n,Wn);
sos = zp2sos(z,p,k);

freqz(sos,128,1000)
title(sprintf('n = %d Butterworth Bandpass Filter',n))

Input Arguments

Wp — Passband corner (cutoff) frequency

scalar | two-element vector

Passband corner (cutoff) frequency, specified as a scalar or a two-element vector with values between 0 and 1, with 1 corresponding to the normalized Nyquist frequency, π rad/sample.

• If Wp and Ws are both scalars and Wp < Ws, then buttord returns the order and cutoff frequency of a lowpass filter. The stopband of the filter ranges from Ws to 1 and the passband ranges from 0 to Wp.

• If Wp and Ws are both scalars and Wp > Ws, then buttord returns the order and cutoff frequency of a highpass filter. The stopband of the filter ranges from 0 to Ws and the passband ranges from Wp to 1.

• If Wp and Ws are both vectors and the interval specified by Ws contains the one specified by Wp (Ws(1) < Wp(1) < Wp(2) < Ws(2)), then buttord returns the order and cutoff frequencies of a bandpass filter. The stopband of the filter ranges from 0 to Ws(1) and from Ws(2) to 1. The passband ranges from Wp(1) to Wp(2).

• If Wp and Ws are both vectors and the interval specified by Wp contains the one specified by Ws (Wp(1) < Ws(1) < Ws(2) < Wp(2)), then buttord returns the order and cutoff frequencies of a bandstop filter. The stopband of the filter ranges from Ws(1) to Ws(2). The passband ranges from 0 to Wp(1) and from Wp(2) to 1.

Data Types: single | double

Note

If your filter specifications call for a bandpass or bandstop filter with unequal ripple in each of the passbands or stopbands, design separate lowpass and highpass filters and cascade the two filters together.

Ws — Stopband corner frequency

scalar | two-element vector

Stopband corner frequency, specified as a scalar or a two-element vector with values between 0 and 1, with 1 corresponding to the normalized Nyquist frequency, π rad/sample.

Data Types: single | double

Rp — Passband ripple

scalar

Passband ripple, specified as a scalar expressed in dB.

Data Types: single | double

Rs — Stopband attenuation

scalar

Stopband attenuation, specified as a scalar expressed in dB.

Data Types: single | double

Output Arguments

n — Lowest filter order

integer scalar

Lowest filter order, returned as an integer scalar.

Wn — Cutoff frequencies

scalar | vector

Cutoff frequencies, returned as a scalar or vector.

Algorithms

buttord’s order prediction formula operates in the analog domain for both analog and digital cases. For the digital case, it converts the frequency parameters to the s-domain before estimating the order and natural frequency. The function then converts back to the z-domain.

buttord initially develops a lowpass filter prototype by transforming the passband frequencies of the desired filter to 1 rad/second (for lowpass and highpass filters) and to –1 and 1 rad/second (for bandpass and bandstop filters). It then computes the minimum order required for a lowpass filter to meet the stopband specification.

References

[1] Rabiner, Lawrence R., and Bernard Gold. Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

All inputs must be constants. Expressions or variables are allowed if their values do not change.

Version History

Introduced before R2006a