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FourierParameters

is an option to Fourier and related functions that specifies the conventions to use in computing Fourier transforms.

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FourierParameters

FourierParameters

is an option to Fourier and related functions that specifies the conventions to use in computing Fourier transforms.

Details

  • A typical setting is FourierParameters->{a,b}.
  • Some common choices for {a,b} are {0,1} (default), {-1,1} (data analysis), {1,-1} (signal processing).

Examples

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Basic Examples  (2)

Use a nondefault definition of the discrete Fourier transform:

Wolfram Language code: Fourier[{1, 2, 3, 4, 5, 6}, FourierParameters -> {1, 1}]

Use the same definition to get the inverse:

Wolfram Language code: InverseFourier[%, FourierParameters -> {1, 1}]

A nondefault definition used for the continuous Fourier transform:

Wolfram Language code: FourierTransform[Exp[-t ^ 2], t, ω, FourierParameters -> {1, 1}]
Wolfram Language code: InverseFourierTransform[%, ω, t, FourierParameters -> {1, 1}]

Scope  (3)

A typical pure mathematics or systems-engineering definition of Fourier transform:

Wolfram Language code: FourierTransform[Exp[-Abs[t]], t, ω, FourierParameters -> {1, -1}]

Use the same definition for the inverse transform:

Wolfram Language code: InverseFourierTransform[%, ω, t, FourierParameters -> {1, -1}]

A typical data-analysis definition of discrete Fourier transform:

Wolfram Language code: Fourier[Range[6], FourierParameters -> {-1, 1}]

Use the same definition to get the correct inverse:

Wolfram Language code: InverseFourier[%, FourierParameters -> {-1, 1}]

A common signal-processing definition of Fourier transform:

Wolfram Language code: FourierTransform[Exp[-Abs[t]], t, ω, FourierParameters -> {1, -1}]

Discrete Fourier transform:

Wolfram Language code: Fourier[Range[6], FourierParameters -> {1, -1}]

Discrete-time Fourier transform:

Wolfram Language code: FourierSequenceTransform[(1 / 2) ^ n UnitStep[n], n, ω, FourierParameters -> {1, 1}]

Possible Issues  (2)

The same FourierParameters values need to be used for both forward and inverse transforms:

Wolfram Language code: Fourier[{1, 2, 3, 4, 5, 6}, FourierParameters -> {-1, 1}]

Here the inverse uses a different choice of FourierParameters:

Wolfram Language code: InverseFourier[%]

The second parameter needs to be relatively prime to the data length to guarantee invertibility:

Wolfram Language code: Fourier[{1, 0, 0, 0, 0, 0}, FourierParameters -> {-1, 3}]
Wolfram Language code: InverseFourier[%, FourierParameters -> {-1, 3}]//Chop
Wolfram Research (1999), FourierParameters, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierParameters.html.

Text

Wolfram Research (1999), FourierParameters, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierParameters.html.

CMS

Wolfram Language. 1999. "FourierParameters." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FourierParameters.html.

APA

Wolfram Language. (1999). FourierParameters. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FourierParameters.html

BibTeX

@misc{reference.wolfram_2026_fourierparameters, author="Wolfram Research", title="{FourierParameters}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/FourierParameters.html}", note=[Accessed: 25-October-2026]}

BibLaTeX

@online{reference.wolfram_2026_fourierparameters, organization={Wolfram Research}, title={FourierParameters}, year={1999}, url={https://reference.wolfram.com/language/ref/FourierParameters.html}, note=[Accessed: 25-October-2026]}