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RecurrenceFilter
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RecurrenceFilter

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RecurrenceFilter[{α,β},x]

filters x using a linear recurrence equation with coefficients α and β.

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uses a discrete-time filter defined by the TransferFunctionModel tf.

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RecurrenceFilter[…,x,{y0,y-1,…}]

uses a specified list {y0,y-1,…} as the initial condition.

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RecurrenceFilter[…,image]

filters image.

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RecurrenceFilter[…,sound]

filters sampled sound object.

Details and Options

  • RecurrenceFilter[{α,β},x] gives the response y to the causal input x by solving the recurrence equation for n from 1 to Length[x], where i is Length[α] and j is Length[β].
  • RecurrenceFilter[{α,β},x] uses an initial condition .
  • RecurrenceFilter[{α,β},x] pads the input x so that . The values can be changed with the Padding option. With Padding->None, the response y effectively starts at yj, and the output dimensions are normally smaller than the input. »
  • The specified coefficients α and β are respectively the denominator and numerator polynomial coefficients of the desired transfer function model.
  • In RecurrenceFilter[tf,…], tf should be a single-input single-output (SISO) system.
  • RecurrenceFilter works with arbitrary-rank numerical arrays, 2D and 3D images, and sampled sound objects, operating separately on each channel.
  • Possible sound objects include:
  • SampledSoundList[{a1,a2,…},r]amplitude levels given in a list
    SampledSoundFunction[f,n,r]amplitude levels generated by a function
    Sound[prims,…]excluding SoundNote objects in prims
  • When applied to images and multidimensional arrays, the specified filter is applied successively to each dimension, starting at level 1.
  • For multichannel image and sound objects, RecurrenceFilter is applied to each channel separately.

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Filtering of a unit impulse sequence:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-6arz1j
Out[1]=1

Filtering of an image:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-1d6rnv
Out[1]=1

Scope  (5)Survey of the scope of standard use cases

Use symbolic values for filter coefficients:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-zk2hs6
Out[1]=1

Filter a symbolic sequence:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-vdveej
Out[1]=1

Apply a filter specified using a transfer function model:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-zh5zvi
Out[1]=1

Specify an initial condition:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-baa4q6
Out[1]=1

Filter a TimeSeries:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-62t833
In[2]:=2
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https://wolfram.com/xid/0btn4wsvu4fe-77v7p5
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0btn4wsvu4fe-18tdm0
Out[3]=3

Options  (1)Common values & functionality for each option

Padding  (1)

By default, RecurrenceFilter uses zero padding:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-dplzmb
Out[1]=1

Use nonzero constant padding:

In[2]:=2
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https://wolfram.com/xid/0btn4wsvu4fe-g2xlhs
Out[2]=2

Use periodic padding:

In[3]:=3
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https://wolfram.com/xid/0btn4wsvu4fe-efcyd5
Out[3]=3

With Padding->None, a shorter sequence is returned:

In[4]:=4
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https://wolfram.com/xid/0btn4wsvu4fe-t37jy7
Out[4]=4

Applications  (3)Sample problems that can be solved with this function

Filtering a noisy sinusoid signal with a recursive smoothing filter:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-55x29k
Out[2]=2

Filter an audio signal:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-sa7k6o
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0btn4wsvu4fe-6dlmkm
Out[2]=2

Filter an image using a lowpass Butterworth filter:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-yc21cz
Out[1]=1

Properties & Relations  (5)Properties of the function, and connections to other functions

Impulse response of first-order recursive filter:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-gue3ui
Out[1]=1

Step response of a non-recursive filter:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-f2hcnq
Out[1]=1

In 2D, the filter is applied successively to rows and columns:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-efdpyf

Use Map to apply the filter to rows only:

In[2]:=2
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https://wolfram.com/xid/0btn4wsvu4fe-8i77st

For 1D arrays, OutputResponse is equivalent to RecurrenceFilter:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-kcr0m1
Out[1]=1

Use RecurrenceTable when describing the filter using a difference equation:

In[2]:=2
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https://wolfram.com/xid/0btn4wsvu4fe-w2em3m
Out[2]=2

ListConvolve and RecurrenceFilter return equivalent results for non-recursive digital filters:

In[1]:=1
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https://wolfram.com/xid/0btn4wsvu4fe-c16hc2
Out[1]=1
Wolfram Research (2012), RecurrenceFilter, Wolfram Language function, https://reference.wolfram.com/language/ref/RecurrenceFilter.html (updated 2014).
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Wolfram Research (2012), RecurrenceFilter, Wolfram Language function, https://reference.wolfram.com/language/ref/RecurrenceFilter.html (updated 2014).

Text

Wolfram Research (2012), RecurrenceFilter, Wolfram Language function, https://reference.wolfram.com/language/ref/RecurrenceFilter.html (updated 2014).

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Wolfram Research (2012), RecurrenceFilter, Wolfram Language function, https://reference.wolfram.com/language/ref/RecurrenceFilter.html (updated 2014).

CMS

Wolfram Language. 2012. "RecurrenceFilter." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/RecurrenceFilter.html.

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Wolfram Language. 2012. "RecurrenceFilter." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/RecurrenceFilter.html.

APA

Wolfram Language. (2012). RecurrenceFilter. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RecurrenceFilter.html

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Wolfram Language. (2012). RecurrenceFilter. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RecurrenceFilter.html

BibTeX

@misc{reference.wolfram_2025_recurrencefilter, author="Wolfram Research", title="{RecurrenceFilter}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/RecurrenceFilter.html}", note=[Accessed: 08-July-2025 ]}

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@misc{reference.wolfram_2025_recurrencefilter, author="Wolfram Research", title="{RecurrenceFilter}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/RecurrenceFilter.html}", note=[Accessed: 08-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_recurrencefilter, organization={Wolfram Research}, title={RecurrenceFilter}, year={2014}, url={https://reference.wolfram.com/language/ref/RecurrenceFilter.html}, note=[Accessed: 08-July-2025 ]}

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@online{reference.wolfram_2025_recurrencefilter, organization={Wolfram Research}, title={RecurrenceFilter}, year={2014}, url={https://reference.wolfram.com/language/ref/RecurrenceFilter.html}, note=[Accessed: 08-July-2025 ]}