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


uses a filter kernel of length n.


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×.


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Basic Examples  (2)

Hilbert filtering of a cosine sequence:

Hilbert filtering of an image:

Scope  (9)

Data  (6)

Filter a 1D pulse sequence:

Filter a 2D pulse sequence:

Filter a TimeSeries:

Hilbert filtering of a square wave audio signal:

Hilbert filtering of a 3D image:

Filter using exact precision:

Parameters  (3)

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

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

Use a different cutoff frequency:

Use a specific window function:

Specify the window function as a numeric list:

Use different cutoff frequencies in each dimension:

Options  (5)

Padding  (3)

By default, "Fixed" padding is used:

Use no padding to eliminate border artifacts:

Different padding methods result in different edge effects:

SampleRate  (2)

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

Assume a sample rate of 3:

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

Applications  (1)

Create an amplitude modulated signal:

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

Properties & Relations  (7)

Using a cutoff frequency of 0 returns a zero sequence:

Create a Hilbert filter using LeastSquaresFilterKernel and a Hamming window:

Compare with the result of HilbertFilter:

Impulse response of a Hilbert filter of length 21:

Magnitude spectrum of filter:

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

Magnitude spectrum of the filter:

Impulse response of even-length Hilbert filter:

Magnitude spectrum of filter:

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

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

Possible Issues  (1)

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

Other paddings return same length sequence:

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).


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. Retrieved from https://reference.wolfram.com/language/ref/HilbertFilter.html


@misc{reference.wolfram_2024_hilbertfilter, author="Wolfram Research", title="{HilbertFilter}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/HilbertFilter.html}", note=[Accessed: 15-June-2024 ]}


@online{reference.wolfram_2024_hilbertfilter, organization={Wolfram Research}, title={HilbertFilter}, year={2016}, url={https://reference.wolfram.com/language/ref/HilbertFilter.html}, note=[Accessed: 15-June-2024 ]}