ButterworthFilterModel
TransferFunctionExpand
BodePlot
OutputResponse
DiracDelta
SystemsModelOrder
StateSpaceModel
TransferFunctionPoles
ToDiscreteTimeModel

Signal Processing

Design a Lowpass Butterworth Filter

Design a Butterworth filter model and obtain the filter's transfer function and impulse response

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Create a third-order Butterworth filter model with cutoff frequency at

1

In[1]:=

tf=ButterworthFilterModel[3,s]

Out[1]=

((0.5-0.866025)+s)((0.5+0.866025)+s)(1.+s)

Use

BodePlot

to plot the magnitude response of the lowpass filter:

In[2]:=

BodePlottf,



Out[2]=

In[3]:=

OutputResponse[Rationalize[TransferFunctionExpand[tf]],DiracDelta[t],t]〚1〛//Simplify

Out[3]=

-t



HeavisideTheta[t]3-3

t/2



Cos

+

t/2



Sin



Plot the response:

In[4]:=

Plot%,{t,0,20},



Out[4]=

Poles of Butterworth Filter
(3)

Use

TransferFunctionPoles

to obtain the poles:

In[1]:=

poles=TransferFunctionPoles[tf]

Out[1]=

{{{-1.,-0.5-0.866025,-0.5+0.866025}}}

Show the results on the unit circle and, for a 3D view, plot the transfer function magnitude of the filter:

In[2]:=

GraphicsRowListPlotTranspose[ReIm/@{Flatten[poles]}],

,Plot3DEvaluate[Abs@tf[σ+ω]〚1,1〛],

,

...



Out[2]=

Plot the poles of Butterworth filters of different orders:

In[3]:=

ManipulateModule{tf,poles},tf=ButterworthFilterModel[n];poles=Flatten@TransferFunctionPoles[tf];Switchview,"3D view",Plot3DEvaluate[Abs@tf[σ+ω]〚1,1〛],

,"2D view",ListPlotTranspose[ReIm/@{Flatten[poles]}],

,{n,6,"order"},1,13,1,

...

,{{view,"3D view"},{"2D view","3D view"}}

Out[3]=

order

view

2D view

3D view

Transform the Lowpass Filter to HighPass and Other Types:
(2)



Related Symbols

ButterworthFilterModel
TransferFunctionExpand
BodePlot
OutputResponse
DiracDelta
SystemsModelOrder
StateSpaceModel
TransferFunctionPoles
ToDiscreteTimeModel

Publisher Information

Contributed by: Wolfram Staff

Design a Lowpass Butterworth Filter

Poles of Butterworth Filter (3)

Transform the Lowpass Filter to HighPass and Other Types: (2) 

Related Symbols

Publisher Information

Poles of Butterworth Filter
(3)

Transform the Lowpass Filter to HighPass and Other Types:
(2)

