ButterworthFilterModel

ButterworthFilterModel[n]

creates a lowpass Butterworth filter of order n and cutoff frequency of 1.

ButterworthFilterModel[{n,ωc}]

uses the cutoff frequency ωc.

ButterworthFilterModel[{"type",spec}]

creates a filter of a given "type" using the specified parameters spec.

ButterworthFilterModel[{"type",spec},var]

expresses the model in terms of the variable var.

Details

  • ButterworthFilterModel returns the filter as a TransferFunctionModel.
  • ButterworthFilterModel[{n,ω}] returns a lowpass filter with attenuation of (approximately 3 dB) at frequency ω.
  • ButterworthFilterModel[n] uses the cutoff frequency of 1.
  • Lowpass filter specification {"type",spec} can be any of the following:
  • {"Lowpass",n}lowpass filter of order n and cutoff frequency 1
    {"Lowpass",n,ωp}use cutoff frequency ωp
    {"Lowpass",{ωp,ωs},{ap,as}}use full filter specification giving passband and stopband frequencies and attenuations
  • Highpass filter specifications:
  • {"Highpass",n}highpass filter with cutoff frequency 1
    {"Highpass",n,ωp}use cutoff frequency ωp
    {"Highpass",{ωs,ωp},{as,ap}}full filter specification
  • Bandpass filter specifications:
  • {"Bandpass",n,{ωp1,ωp2}}bandpass filter with passband frequencies ωp1 and ωp2
    {"Bandpass",n,{{ω,q}}}use center frequency ω and quality factor q
    {"Bandpass",{ωs1,ωp1,ωp2,ωs2},{as,ap}}full filter specification
  • Bandstop filter specifications:
  • {"Bandstop",n,{ωp1,ωp2}}bandstop filter with passband frequencies ωp1 and ωp2
    {"Bandstop",n,{{ω,q}}}use center frequency ω and quality factor q
    {"Bandstop",{ωp1,ωs1,ωs2,ωp2},{ap,as}}full filter specification
  • Values ap and as are, respectively, absolute values of passband and stopband attenuations.
  • Given a gain fraction , the attenuation is .
  • The quality factor q is defined as , with being the center frequency of a bandpass or bandstop filter. Higher values of q give narrower filters.

Examples

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Basic Examples  (2)

A third-order Butterworth filter model with cutoff frequency at :

Bode plot of the filter:

A lowpass Butterworth filter using the full specification:

Magnitude response of the filter showing the ideal filter characteristics:

Scope  (6)

A symbolic lowpass Butterworth filter of order 3 with a cutoff frequency ω:

Use cutoff frequency :

Same filter using the full specification:

Create a highpass Butterworth filter of order 3 with a cutoff frequency of 10:

Same filter using the full specification:

Create a "Bandpass" filter with passband frequencies and and attenuation of order 3:

Same filter using center frequency and quality factor specification {{ω,q}}:

Same filter using the full specification:

Create a bandstop Butterworth filter:

Exact computation of the model:

Computation of the model with precision 24:

Create a filter model using the variable s:

Applications  (6)

Create a lowpass Butterworth filter:

Filter out high-frequency noise from a sinusoidal signal:

Butterworth filter phase shifts the response by Arg[tf[ω ]], where ω is the frequency of the input sinusoid:

Correct for the phase shift:

Create a highpass Butterworth filter from the lowpass prototype:

Filter out low-frequency sinusoid from the input:

Design a digital FIR lowpass filter using the Butterworth approximation that satisfies the following passband and stopband frequencies and attenuations:

Obtain the equivalent analog frequencies assuming a sampling period of 1:

Compute the analog Butterworth transfer function:

Convert to discrete-time model:

Create an FIR approximation of a discrete-time Butterworth IIR filter.

Implement a lowpass digital Butterworth filter:

Obtain the desired number of FIR samples from the impulse response of the discrete-time Butterworth filter:

Plot the FIR filter:

Smooth financial data using an FIR approximation of a Butterworth filter:

Filter an image using a discrete-time lowpass Butterworth filter:

Filter an image using a highpass Butterworth filter:

Properties & Relations  (9)

Stopband attenuation increases by a factor of per decade as order increases:

Passband width of "Bandpass" filter decreases with increasing quality factor q:

Phase response of a third-order lowpass Butterworth filter:

Compare phase responses for different filter orders:

Phase response of a "Bandpass" filter for several quality factors:

Extract the order of the Butterworth polynomial:

The order of the Butterworth polynomial for lowpass and highpass is the same as the specified order:

The filter order for bandpass and bandstop is twice the given order:

Show the Butterworth polynomial in the denominator of the transfer function:

Find the poles of a Butterworth filter by solving for the roots of the denominator:

Extract poles using TransferFunctionPoles:

Plot poles of the Butterworth filter:

Implement a lowpass digital Butterworth filter:

Plot poles of the digital Butterworth filter:

Create a highpass filter using a lowpass prototype:

Introduced in 2012
 (9.0)
 |
Updated in 2014
 (10.0)
2015
 (10.2)
2016
 (10.4)