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FourierSequenceTransform
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FourierSequenceTransform

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gives the Fourier sequence transform of expr.

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FourierSequenceTransform[expr,{n1,n2,…},{ω1,ω2,…}]

gives the multidimensional Fourier sequence transform.

Details and Options

  • FourierSequenceTransform is also known as discrete-time Fourier transform (DTFT).
  • FourierSequenceTransform[expr,n,ω] takes a sequence whose n^(th) term is given by expr, and yields a function of the continuous parameter ω.
  • The Fourier sequence transform of is by default defined to be .
  • The multidimensional transform of is defined to be .
  • The following options can be given:
  • Assumptions$Assumptionsassumptions on parameters
    FourierParameters {1,1}parameters to definite discrete-time Fourier transform
    GenerateConditions Falsewhether to generate results that involve conditions on parameters
  • Common settings for FourierParameters include:
  • {1,1}default settings
    {1,-2Pi}period 1
    {a,b}general setting

Examples

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

Find the discrete-time Fourier transform of a simple signal:

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In[1]:=1
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https://wolfram.com/xid/0k7z1b0wab41u-b1662i
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Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0k7z1b0wab41u-kj86oh
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Out[2]=2

Find a bivariate discrete-time Fourier transform:

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In[1]:=1
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https://wolfram.com/xid/0k7z1b0wab41u-hehoex
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Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0k7z1b0wab41u-m52g20
Direct link to example
Out[2]=2

Scope  (4)Survey of the scope of standard use cases

Compute the transform for each frequency ω:

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In[1]:=1
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https://wolfram.com/xid/0k7z1b0wab41u-bary2o
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Out[1]=1

Plot the spectrum:

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In[2]:=2
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https://wolfram.com/xid/0k7z1b0wab41u-q8h2e2
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Out[2]=2

The phase:

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In[3]:=3
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https://wolfram.com/xid/0k7z1b0wab41u-b70np3
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Out[3]=3

Plot both spectrum and phase using color:

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In[4]:=4
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https://wolfram.com/xid/0k7z1b0wab41u-g234r
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Out[4]=4

Constant:

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In[1]:=1
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https://wolfram.com/xid/0k7z1b0wab41u-ffijeh
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Out[1]=1

Periodic:

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In[2]:=2
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https://wolfram.com/xid/0k7z1b0wab41u-iugvcm
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Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0k7z1b0wab41u-iuix14
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
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https://wolfram.com/xid/0k7z1b0wab41u-b1rjgw
Direct link to example
Out[4]=4
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In[5]:=5
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https://wolfram.com/xid/0k7z1b0wab41u-buel8z
Direct link to example
Out[5]=5

Impulse:

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In[6]:=6
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https://wolfram.com/xid/0k7z1b0wab41u-cbzghg
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Out[6]=6
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In[7]:=7
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https://wolfram.com/xid/0k7z1b0wab41u-cj7t8l
Direct link to example
Out[7]=7

Exponential:

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In[8]:=8
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https://wolfram.com/xid/0k7z1b0wab41u-20lof
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Out[8]=8
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In[9]:=9
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https://wolfram.com/xid/0k7z1b0wab41u-maho17
Direct link to example
Out[9]=9

Exponential polynomial:

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In[10]:=10
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https://wolfram.com/xid/0k7z1b0wab41u-ehdpui
Direct link to example
Out[10]=10
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In[11]:=11
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https://wolfram.com/xid/0k7z1b0wab41u-dkiiuy
Direct link to example
Out[11]=11

Rational sequence:

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In[1]:=1
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https://wolfram.com/xid/0k7z1b0wab41u-f9vqod
Direct link to example
Out[1]=1

Rational-trigonometric:

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In[2]:=2
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https://wolfram.com/xid/0k7z1b0wab41u-q6h6gz
Direct link to example
Out[2]=2

Hypergeometric terms:

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In[3]:=3
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https://wolfram.com/xid/0k7z1b0wab41u-cbzc1t
Direct link to example
Out[3]=3
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In[4]:=4
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https://wolfram.com/xid/0k7z1b0wab41u-iiyxcq
Direct link to example
Out[4]=4

Multivariate sequences:

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In[1]:=1
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https://wolfram.com/xid/0k7z1b0wab41u-izqxoh
Direct link to example
Out[1]=1
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In[2]:=2
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https://wolfram.com/xid/0k7z1b0wab41u-npml
Direct link to example
Out[2]=2

Options  (2)Common values & functionality for each option

FourierParameters  (1)

Use a non-default setting for FourierParameters:

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In[1]:=1
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https://wolfram.com/xid/0k7z1b0wab41u-hbsr0u
Direct link to example
Out[1]=1

GenerateConditions  (1)

Obtain conditions on parameters:

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In[1]:=1
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https://wolfram.com/xid/0k7z1b0wab41u-hkliul
Direct link to example
Out[1]=1

Properties & Relations  (5)Properties of the function, and connections to other functions

FourierSequenceTransform is defined by a doubly infinite sum:

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In[1]:=1
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https://wolfram.com/xid/0k7z1b0wab41u-gi822v
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0k7z1b0wab41u-ecyje
Direct link to example
Out[2]=2

FourierSequenceTransform and InverseFourierSequenceTransform are inverses:

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In[1]:=1
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https://wolfram.com/xid/0k7z1b0wab41u-i6nas0
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0k7z1b0wab41u-g3nh98
Direct link to example
Out[2]=2
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In[3]:=3
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https://wolfram.com/xid/0k7z1b0wab41u-xp11i
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
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https://wolfram.com/xid/0k7z1b0wab41u-0nxw6
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
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https://wolfram.com/xid/0k7z1b0wab41u-id2kzh
Direct link to example
Out[5]=5

FourierSequenceTransform is closely related to ZTransform:

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In[1]:=1
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https://wolfram.com/xid/0k7z1b0wab41u-hklar3
Direct link to example
Out[1]=1

A discrete analog of FourierTransform being closely related to LaplaceTransform:

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In[2]:=2
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https://wolfram.com/xid/0k7z1b0wab41u-cns8ua
Direct link to example
Out[2]=2

FourierSequenceTransform provides a -analog generating function:

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In[1]:=1
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https://wolfram.com/xid/0k7z1b0wab41u-bkykgh
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0k7z1b0wab41u-bbz24i
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
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https://wolfram.com/xid/0k7z1b0wab41u-dxe3y7
Direct link to example
Out[3]=3

FourierSequenceTransform is closely related to BilateralZTransform:

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In[1]:=1
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https://wolfram.com/xid/0k7z1b0wab41u-19m1r9
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
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https://wolfram.com/xid/0k7z1b0wab41u-mlx09s
Direct link to example
Out[2]=2
Wolfram Research (2008), FourierSequenceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierSequenceTransform.html.
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Wolfram Research (2008), FourierSequenceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierSequenceTransform.html.

Text

Wolfram Research (2008), FourierSequenceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierSequenceTransform.html.

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Wolfram Research (2008), FourierSequenceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierSequenceTransform.html.

CMS

Wolfram Language. 2008. "FourierSequenceTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FourierSequenceTransform.html.

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Wolfram Language. 2008. "FourierSequenceTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FourierSequenceTransform.html.

APA

Wolfram Language. (2008). FourierSequenceTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FourierSequenceTransform.html

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Wolfram Language. (2008). FourierSequenceTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FourierSequenceTransform.html

BibTeX

@misc{reference.wolfram_2025_fouriersequencetransform, author="Wolfram Research", title="{FourierSequenceTransform}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FourierSequenceTransform.html}", note=[Accessed: 21-April-2025 ]}

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@misc{reference.wolfram_2025_fouriersequencetransform, author="Wolfram Research", title="{FourierSequenceTransform}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FourierSequenceTransform.html}", note=[Accessed: 21-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_fouriersequencetransform, organization={Wolfram Research}, title={FourierSequenceTransform}, year={2008}, url={https://reference.wolfram.com/language/ref/FourierSequenceTransform.html}, note=[Accessed: 21-April-2025 ]}

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@online{reference.wolfram_2025_fouriersequencetransform, organization={Wolfram Research}, title={FourierSequenceTransform}, year={2008}, url={https://reference.wolfram.com/language/ref/FourierSequenceTransform.html}, note=[Accessed: 21-April-2025 ]}