buttord

Signal Processing Toolbox

Butterworth filter order selection.

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

Description

buttord selects the minimum order digital or analog Butterworth filter required to meet a set of filter design specifications:


`Wp`	Passband corner frequency. `Wp`, the cutoff frequency, has a value between 0 and 1, where 1 corresponds to half the sampling frequency (the Nyquist frequency).
`Ws`	Stopband corner frequency. `Ws` is in the same units as `Wp`; it has a value between 0 and 1, where 1 corresponds to half the sampling frequency (the Nyquist frequency).
`Rp`	Passband ripple, in decibels. This value is the maximum permissible passband loss in decibels. The passband is `0` `<` `w` `<` `Wp`.
`Rs`	Stopband attenuation, in decibels. This value is the number of decibels the stopband is down from the passband. The stopband is `Ws` `<` `w <` `1`.

Digital Domain

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

returns the order n of the lowest order digital Butterworth filter that loses no more than Rp dB in the passband and has at least Rs dB of attenuation in the stopband. The passband runs from 0 to Wp and the stopband runs from Ws to 1, the Nyquist frequency. buttord also returns Wn, the Butterworth cutoff frequency that allows butter to achieve the given specifications (the "3 dB" frequency).

Use buttord for highpass, bandpass, and bandstop filters. For highpass filters, Wp is greater than Ws. For bandpass and bandstop filters, Wp and Ws are two-element vectors that specify the corner frequencies at both edges of the filter, lower frequency edge first. For the band filters, buttord returns Wn as a two-element row vector for input to butter.

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

Analog Domain

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

finds the minimum order n and cutoff frequencies Wn for an analog filter. In this case the frequencies in Wp and Ws are in radians per second and may be greater than 1.

Use buttord for highpass, bandpass, and bandstop filters, as described under "Digital Domain."

Examples

For 1000 Hz data, design a lowpass filter with less than 3 dB of attenuation from 0 to 100 Hz, and attenuation at least 15 dB from 150 Hz to the Nyquist frequency. Plot the filter's frequency response:

Wp = 100/500; Ws = 150/500;
[n,Wn] = buttord(Wp,Ws,3,15)
n =
     4
Wn =
    0.2042
[b,a] = butter(n,Wn);
freqz(b,a,512,1000)

Next design a bandpass filter with passband of 100 Hz to 200 Hz, less than 3 dB of attenuation at the passband corners, and attenuation down 30 dB by 50 Hz out on both sides of the passband:

Wp = [100 200]/500; Ws = [50 250]/500;
Rp = 3; Rs = 30;
[n,Wn] = buttord(Wp,Ws,Rp,Rs); [b,a] = butter(n,Wn);
freqz(b,a,128,1000)

Algorithm

buttord's order prediction formula is described in [1]. It 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, 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/sec (for low- and highpass filters) and to -1 and 1 rad/sec (for bandpass and bandstop filters). It then computes the minimum order required for a lowpass filter to meet the stopband specification.

`butter`	Butterworth analog and digital filter design.
`cheb1ord`	Chebyshev type I filter order selection.
`cheb2ord`	Chebyshev type II filter order selection.
`ellipord`	Elliptic filter order selection.