RecurrenceFilter
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RecurrenceFilter
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 allBasic Examples (2)Summary of the most common use cases
Scope (5)Survey of the scope of standard use cases
Use symbolic values for filter coefficients:

https://wolfram.com/xid/0btn4wsvu4fe-zk2hs6


https://wolfram.com/xid/0btn4wsvu4fe-vdveej

Apply a filter specified using a transfer function model:

https://wolfram.com/xid/0btn4wsvu4fe-zh5zvi


https://wolfram.com/xid/0btn4wsvu4fe-baa4q6

Filter a TimeSeries:

https://wolfram.com/xid/0btn4wsvu4fe-62t833

https://wolfram.com/xid/0btn4wsvu4fe-77v7p5


https://wolfram.com/xid/0btn4wsvu4fe-18tdm0

Options (1)Common values & functionality for each option
Padding (1)
By default, RecurrenceFilter uses zero padding:

https://wolfram.com/xid/0btn4wsvu4fe-dplzmb


https://wolfram.com/xid/0btn4wsvu4fe-g2xlhs


https://wolfram.com/xid/0btn4wsvu4fe-efcyd5

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

https://wolfram.com/xid/0btn4wsvu4fe-t37jy7

Applications (3)Sample problems that can be solved with this function
Filtering a noisy sinusoid signal with a recursive smoothing filter:

https://wolfram.com/xid/0btn4wsvu4fe-55x29k


https://wolfram.com/xid/0btn4wsvu4fe-sa7k6o


https://wolfram.com/xid/0btn4wsvu4fe-6dlmkm

Filter an image using a lowpass Butterworth filter:

https://wolfram.com/xid/0btn4wsvu4fe-yc21cz

Properties & Relations (5)Properties of the function, and connections to other functions
Impulse response of first-order recursive filter:

https://wolfram.com/xid/0btn4wsvu4fe-gue3ui

Step response of a non-recursive filter:

https://wolfram.com/xid/0btn4wsvu4fe-f2hcnq

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

https://wolfram.com/xid/0btn4wsvu4fe-efdpyf

Use Map to apply the filter to rows only:

https://wolfram.com/xid/0btn4wsvu4fe-8i77st

For 1D arrays, OutputResponse is equivalent to RecurrenceFilter:

https://wolfram.com/xid/0btn4wsvu4fe-kcr0m1

Use RecurrenceTable when describing the filter using a difference equation:

https://wolfram.com/xid/0btn4wsvu4fe-w2em3m

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

https://wolfram.com/xid/0btn4wsvu4fe-c16hc2

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).
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.
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
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
]}
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
]}