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FourierParameters
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FourierParameters

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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

open allclose all

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

Use a nondefault definition of the discrete Fourier transform:

In[1]:=1
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https://wolfram.com/xid/0b0kl78yxlv-bx8i6q
Out[1]=1

Use the same definition to get the inverse:

In[2]:=2
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https://wolfram.com/xid/0b0kl78yxlv-blntkx
Out[2]=2

A nondefault definition used for the continuous Fourier transform:

In[1]:=1
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https://wolfram.com/xid/0b0kl78yxlv-er3uq3
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b0kl78yxlv-nffxwk
Out[2]=2

Scope  (3)Survey of the scope of standard use cases

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

In[1]:=1
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https://wolfram.com/xid/0b0kl78yxlv-d56p7l
Out[1]=1

Use the same definition for the inverse transform:

In[2]:=2
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https://wolfram.com/xid/0b0kl78yxlv-cc82ej
Out[2]=2

A typical data-analysis definition of discrete Fourier transform:

In[1]:=1
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https://wolfram.com/xid/0b0kl78yxlv-ceciga
Out[1]=1

Use the same definition to get the correct inverse:

In[2]:=2
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https://wolfram.com/xid/0b0kl78yxlv-ltygg7
Out[2]=2

A common signal-processing definition of Fourier transform:

In[1]:=1
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https://wolfram.com/xid/0b0kl78yxlv-mdpqu8
Out[1]=1

Discrete Fourier transform:

In[2]:=2
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https://wolfram.com/xid/0b0kl78yxlv-ci12q2
Out[2]=2

Discrete-time Fourier transform:

In[3]:=3
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https://wolfram.com/xid/0b0kl78yxlv-y6msu
Out[3]=3

Possible Issues  (2)Common pitfalls and unexpected behavior

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

In[1]:=1
✖
https://wolfram.com/xid/0b0kl78yxlv-i7jjak
Out[1]=1

Here the inverse uses a different choice of FourierParameters:

In[2]:=2
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https://wolfram.com/xid/0b0kl78yxlv-dsnfhj
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0b0kl78yxlv-bmtnm4
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0b0kl78yxlv-i9xzsm
Out[2]=2
Wolfram Research (1999), FourierParameters, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierParameters.html.
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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.

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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.

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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

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

BibTeX

@misc{reference.wolfram_2025_fourierparameters, author="Wolfram Research", title="{FourierParameters}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/FourierParameters.html}", note=[Accessed: 06-June-2025 ]}

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@misc{reference.wolfram_2025_fourierparameters, author="Wolfram Research", title="{FourierParameters}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/FourierParameters.html}", note=[Accessed: 06-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_fourierparameters, organization={Wolfram Research}, title={FourierParameters}, year={1999}, url={https://reference.wolfram.com/language/ref/FourierParameters.html}, note=[Accessed: 06-June-2025 ]}

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@online{reference.wolfram_2025_fourierparameters, organization={Wolfram Research}, title={FourierParameters}, year={1999}, url={https://reference.wolfram.com/language/ref/FourierParameters.html}, note=[Accessed: 06-June-2025 ]}