Generates the lowest filter order and the corresponding corner frequencies of a Butterworth digital or analog filter meeting a given set of filter specifications.  The filter type (high-pass, low-pass, band- pass, or band-stop) is specified by the relative values of the arguments.

Syntax

Order, Wn = ButterOrd(Wp, Ws, Rp, Rs, Type)

Arguments

Name

Description

Wp

Passband corner frequency normalized by the Nyquist frequency.  Ranges between 0 and 1 in the digital case and positive in the analog case.  For low-pass and high-pass filters, this is a scalar.  For band-pass and band-stop, it's a two element vector, with the second element being larger of the two.

Ws

Stopband corner frequency normalized by the Nyquist frequency.  Ranges between 0 and 1 in the digital case and positive in the analog case.  For low-pass and high-pass filters, this is a scalar.  For band-pass and band-stop, it's a two element vector, with the second element being larger of the two.

Rp

Maximum permissible passband attenuation (loss) in dB.  Must be a negative scalar.

Rs

Minimum permissible stopband attenuation (loss) in dB, relative to the maximum passband magnitude.  Must be a  negative scalar and greater than Rp in magnitude.

Type

(optional)

The digital/analog flag. For analog, set to ‘s.’ For digital, set to ‘z’ (default).

Outputs

Name

Description

Order

The minimum filter order that meets the above specifications.

Wn

Alternate passband corner frequencies that will meet the above specifications.

Example 1

A low-pass Butterworth filter design with a passband of 0-150 Hz, with attenuation of no more than 3 dB, and a minimum of 20 dB attenuation in the stopband defined from 250 Hz to the Nyquist frequency.  Data sampling frequency is 1000 Hz:

Syntax

n, Wn = ButterOrd(150/500,250/500,-3,-20)

Result

n = 4

Wn = 0.33

Example 2

A band-pass Butterworth filter with a passband of 150-250 Hz, with maximum attenuation of no more than 3 dB and a minimum of 20 dB attenuation in stopbands that have corner frequencies 50 Hz away from the passband corners.  Data sampling frequency is 1000 Hz:

Syntax

n, Wn = ButterOrd([150/500 250/500],[100/500 300/500],-3,-20)

Result

n = 4

Wn = 0.29 0.52

Comments

The Nyquist frequency is half the sampling frequency.  All frequency arguments are to be normalized with this.

The guidelines for specifying different filter types are listed below:

Type

Stopband and Passband Conditions

Stopband

Passband

Lowpass

Wp < Ws where both are scalars

(Ws, 1)

(0, Wp)

Highpass

Wp > Ws where both are scalars

(0, Ws)

(Wp, 1)

Bandpass

Ws(1) < Wp(1) < Wp(2) < Ws(2)

(0, Ws(1)) and (Ws(2), 1)

(Wp(1), Wp(2))

Bandstop

Wp(1) < Ws(1) < Ws(2) < Wp(2)

(Ws(1), Ws(2))

(0, Wp(1)) and (Ws(2), 1)

See Also:

Cheby1Ord

Cheby2Ord

EllipOrd