HilbertFilter
✖
HilbertFilter
Details and Options


- HilbertFilter is a finite-impulse response (FIR) discrete-time filter typically used to obtain an approximation of data with a 90-degree phase shift.
- The data can be any of the following:
-
list arbitrary-rank numerical array tseries temporal data such as TimeSeries and TemporalData image arbitrary Image or Image3D object audio an Audio or Sound object - Data smoothing with cutoff frequency ωc reduces the susceptibility of the evaluation to signal noise with the amount of smoothing dependent on the value of the cutoff frequency ωc.
- The cutoff frequency ωc should be between 0 and
. Smaller values of ωc result in greater smoothing.
- When applied to images and multidimensional arrays, filtering is applied successively to each dimension, starting at level 1. HilbertFilter[data,{ωc1,ωc2,…}] uses the frequency ωci for the
dimension.
- HilbertFilter[data,ωc] uses a filter kernel length and smoothing window suitable for the cutoff frequency ωc and the input data.
- Typical smoothing windows wfun include:
-
BlackmanWindow smoothing with a Blackman window DirichletWindow no smoothing HammingWindow smoothing with a Hamming window {v1,v2,…} use a window with values vi f create a window by sampling f between and
- The following options can be given:
-
Padding "Fixed" the padding value to use SampleRate Automatic sample rate assumed for the input - By default, SampleRate->1 is assumed for images as well as data. For a sampled sound object of sample rate of r, SampleRate->r is used.
- With SampleRate->r, the cutoff frequency ωc should be between 0 and r×
.

Examples
open allclose allBasic Examples (2)Summary of the most common use cases
Scope (9)Survey of the scope of standard use cases
Data (6)

https://wolfram.com/xid/0e5dqhq5ri8tnu-b5ftuu


https://wolfram.com/xid/0e5dqhq5ri8tnu-jndczy


https://wolfram.com/xid/0e5dqhq5ri8tnu-jau8ka

Filter a TimeSeries:

https://wolfram.com/xid/0e5dqhq5ri8tnu-c6y6lt

https://wolfram.com/xid/0e5dqhq5ri8tnu-h3r1ho


https://wolfram.com/xid/0e5dqhq5ri8tnu-blw807

Hilbert filtering of a square wave audio signal:

https://wolfram.com/xid/0e5dqhq5ri8tnu-c6vsa7


https://wolfram.com/xid/0e5dqhq5ri8tnu-dbqzm5

Hilbert filtering of a 3D image:

https://wolfram.com/xid/0e5dqhq5ri8tnu-2lmvq


https://wolfram.com/xid/0e5dqhq5ri8tnu-ces0sy

Parameters (3)
With an audio signal, a numeric cutoff frequency is interpreted as radians per second:

https://wolfram.com/xid/0e5dqhq5ri8tnu-5w9pjk

https://wolfram.com/xid/0e5dqhq5ri8tnu-zhsae7

Hilbert transform of a unit step sequence using a filter of length 5:

https://wolfram.com/xid/0e5dqhq5ri8tnu-uwgw39

Use a different cutoff frequency:

https://wolfram.com/xid/0e5dqhq5ri8tnu-crrlvs

Use a specific window function:

https://wolfram.com/xid/0e5dqhq5ri8tnu-pft8i9

Specify the window function as a numeric list:

https://wolfram.com/xid/0e5dqhq5ri8tnu-yb4zfp

Use different cutoff frequencies in each dimension:

https://wolfram.com/xid/0e5dqhq5ri8tnu-il2z2u

Options (5)Common values & functionality for each option
Padding (3)
By default, "Fixed" padding is used:

https://wolfram.com/xid/0e5dqhq5ri8tnu-ft7zo0

Use no padding to eliminate border artifacts:

https://wolfram.com/xid/0e5dqhq5ri8tnu-ixhhru

Different padding methods result in different edge effects:

https://wolfram.com/xid/0e5dqhq5ri8tnu-7vvdo

SampleRate (2)
Use a half-band Hilbert filter, assuming a normalized sample rate of 1:

https://wolfram.com/xid/0e5dqhq5ri8tnu-lncjzt


https://wolfram.com/xid/0e5dqhq5ri8tnu-fzstpo

Apply a half-band Hilbert filter to audio sampled at a rate of :

https://wolfram.com/xid/0e5dqhq5ri8tnu-lbwnnj


https://wolfram.com/xid/0e5dqhq5ri8tnu-dsru3b

Applications (1)Sample problems that can be solved with this function
Properties & Relations (7)Properties of the function, and connections to other functions
Using a cutoff frequency of 0 returns a zero sequence:

https://wolfram.com/xid/0e5dqhq5ri8tnu-b7fipj

Create a Hilbert filter using LeastSquaresFilterKernel and a Hamming window:

https://wolfram.com/xid/0e5dqhq5ri8tnu-qpd46n


https://wolfram.com/xid/0e5dqhq5ri8tnu-9c0vud

Compare with the result of HilbertFilter:

https://wolfram.com/xid/0e5dqhq5ri8tnu-36qnr

Impulse response of a Hilbert filter of length 21:

https://wolfram.com/xid/0e5dqhq5ri8tnu-d8yiku


https://wolfram.com/xid/0e5dqhq5ri8tnu-cup5u


https://wolfram.com/xid/0e5dqhq5ri8tnu-l1j6k

Impulse response of a Hilbert filter of length 21 without a smoothing window:

https://wolfram.com/xid/0e5dqhq5ri8tnu-k07uzl


https://wolfram.com/xid/0e5dqhq5ri8tnu-5xus8

Magnitude spectrum of the filter:

https://wolfram.com/xid/0e5dqhq5ri8tnu-j4de6a

Impulse response of even-length Hilbert filter:

https://wolfram.com/xid/0e5dqhq5ri8tnu-d0xfvm


https://wolfram.com/xid/0e5dqhq5ri8tnu-cpu7c3


https://wolfram.com/xid/0e5dqhq5ri8tnu-jom24

The magnitude response of the Hilbert filter improves as the length of the filter is increased:

https://wolfram.com/xid/0e5dqhq5ri8tnu-31f8p

The magnitude response of a half-band Hilbert filter of length 21:

https://wolfram.com/xid/0e5dqhq5ri8tnu-k9xgjz

Possible Issues (1)Common pitfalls and unexpected behavior
Wolfram Research (2012), HilbertFilter, Wolfram Language function, https://reference.wolfram.com/language/ref/HilbertFilter.html (updated 2016).
Text
Wolfram Research (2012), HilbertFilter, Wolfram Language function, https://reference.wolfram.com/language/ref/HilbertFilter.html (updated 2016).
Wolfram Research (2012), HilbertFilter, Wolfram Language function, https://reference.wolfram.com/language/ref/HilbertFilter.html (updated 2016).
CMS
Wolfram Language. 2012. "HilbertFilter." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/HilbertFilter.html.
Wolfram Language. 2012. "HilbertFilter." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/HilbertFilter.html.
APA
Wolfram Language. (2012). HilbertFilter. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HilbertFilter.html
Wolfram Language. (2012). HilbertFilter. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HilbertFilter.html
BibTeX
@misc{reference.wolfram_2025_hilbertfilter, author="Wolfram Research", title="{HilbertFilter}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/HilbertFilter.html}", note=[Accessed: 05-June-2025
]}
BibLaTeX
@online{reference.wolfram_2025_hilbertfilter, organization={Wolfram Research}, title={HilbertFilter}, year={2016}, url={https://reference.wolfram.com/language/ref/HilbertFilter.html}, note=[Accessed: 05-June-2025
]}