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HilbertFilter
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HilbertFilter

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HilbertFilter[data,ωc]

applies a Hilbert filter with a cutoff frequency ωc to an array of data.

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HilbertFilter[data,ωc,n]

uses a filter kernel of length n.

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HilbertFilter[data,ωc,n,wfun]

applies a smoothing window wfun to the filter kernel.

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:
  • listarbitrary-rank numerical array
    tseriestemporal data such as TimeSeries and TemporalData
    imagearbitrary Image or Image3D object
    audioan 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 ^(th) 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:
  • BlackmanWindowsmoothing with a Blackman window
    DirichletWindowno smoothing
    HammingWindowsmoothing with a Hamming window
    {v1,v2,…}use a window with values vi
    fcreate a window by sampling f between and
  • The following options can be given:
  • Padding "Fixed"the padding value to use
    SampleRate Automaticsample 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 all

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

Hilbert filtering of a cosine sequence:

In[3]:=3
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https://wolfram.com/xid/0e5dqhq5ri8tnu-1havhv
Out[4]=4

Hilbert filtering of an image:

In[1]:=1
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https://wolfram.com/xid/0e5dqhq5ri8tnu-erf4mw
Out[1]=1

Scope  (9)Survey of the scope of standard use cases

Data  (6)

Filter a 1D pulse sequence:

In[3]:=3
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https://wolfram.com/xid/0e5dqhq5ri8tnu-b5ftuu
Out[4]=4

Filter a 2D pulse sequence:

In[1]:=1
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https://wolfram.com/xid/0e5dqhq5ri8tnu-jndczy
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-jau8ka
Out[2]=2

Filter a TimeSeries:

In[1]:=1
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https://wolfram.com/xid/0e5dqhq5ri8tnu-c6y6lt
In[2]:=2
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-h3r1ho
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-blw807
Out[3]=3

Hilbert filtering of a square wave audio signal:

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-c6vsa7
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-dbqzm5
Out[2]=2

Hilbert filtering of a 3D image:

In[1]:=1
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https://wolfram.com/xid/0e5dqhq5ri8tnu-2lmvq
Out[1]=1

Filter using exact precision:

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-ces0sy
Out[1]=1

Parameters  (3)

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

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-5w9pjk
In[2]:=2
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-zhsae7
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-uwgw39
Out[1]=1

Use a different cutoff frequency:

In[2]:=2
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-crrlvs
Out[2]=2

Use a specific window function:

In[3]:=3
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-pft8i9
Out[3]=3

Specify the window function as a numeric list:

In[4]:=4
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https://wolfram.com/xid/0e5dqhq5ri8tnu-yb4zfp
Out[4]=4

Use different cutoff frequencies in each dimension:

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-il2z2u
Out[1]=1

Options  (5)Common values & functionality for each option

Padding  (3)

By default, "Fixed" padding is used:

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-ft7zo0
Out[1]=1

Use no padding to eliminate border artifacts:

In[1]:=1
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https://wolfram.com/xid/0e5dqhq5ri8tnu-ixhhru
Out[1]=1

Different padding methods result in different edge effects:

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-7vvdo
Out[1]=1

SampleRate  (2)

Use a half-band Hilbert filter, assuming a normalized sample rate of 1:

In[1]:=1
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https://wolfram.com/xid/0e5dqhq5ri8tnu-lncjzt
Out[1]=1

Assume a sample rate of 3:

In[2]:=2
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-fzstpo
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-lbwnnj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-dsru3b
Out[2]=2

Applications  (1)Sample problems that can be solved with this function

Create an amplitude modulated signal:

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-grhz6f

Use a Hilbert filter to obtain the envelope of the modulated signal:

In[2]:=2
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-cmnhtf
Out[2]=2

Properties & Relations  (7)Properties of the function, and connections to other functions

Using a cutoff frequency of 0 returns a zero sequence:

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-b7fipj
Out[1]=1

Create a Hilbert filter using LeastSquaresFilterKernel and a Hamming window:

In[1]:=1
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https://wolfram.com/xid/0e5dqhq5ri8tnu-qpd46n
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e5dqhq5ri8tnu-9c0vud
Out[2]=2

Compare with the result of HilbertFilter:

In[3]:=3
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https://wolfram.com/xid/0e5dqhq5ri8tnu-36qnr
Out[3]=3

Impulse response of a Hilbert filter of length 21:

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-d8yiku
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-cup5u
Out[2]=2

Magnitude spectrum of filter:

In[3]:=3
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-l1j6k
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-k07uzl
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-5xus8
Out[2]=2

Magnitude spectrum of the filter:

In[3]:=3
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-j4de6a
Out[3]=3

Impulse response of even-length Hilbert filter:

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-d0xfvm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-cpu7c3
Out[2]=2

Magnitude spectrum of filter:

In[3]:=3
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-jom24
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-31f8p
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-k9xgjz
Out[1]=1

Possible Issues  (1)Common pitfalls and unexpected behavior

Filtering a short list using PaddingNone may return an empty list:

In[1]:=1
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-b3eqr2
Out[1]=1

Other paddings return same length sequence:

In[2]:=2
✖
https://wolfram.com/xid/0e5dqhq5ri8tnu-uxwfew
Out[2]=2
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).

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 ]}

✖
@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 ]}

✖
@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 ]}