FourierSequenceTransform

FourierSequenceTransform[expr,n,ω]

gives the Fourier sequence transform of expr.

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)

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

Find a bivariate discrete-time Fourier transform:

Scope  (4)

Compute the transform for each frequency ω:

Plot the spectrum:

The phase:

Plot both spectrum and phase using color:

Constant:

Periodic:

Impulse:

Exponential:

Exponential polynomial:

Rational sequence:

Rational-trigonometric:

Hypergeometric terms:

Multivariate sequences:

Options  (2)

FourierParameters  (1)

Use a non-default setting for FourierParameters:

GenerateConditions  (1)

Obtain conditions on parameters:

Properties & Relations  (5)

FourierSequenceTransform is defined by a doubly infinite sum:

FourierSequenceTransform and InverseFourierSequenceTransform are inverses:

FourierSequenceTransform is closely related to ZTransform:

A discrete analog of FourierTransform being closely related to LaplaceTransform:

FourierSequenceTransform provides a -analog generating function:

FourierSequenceTransform is closely related to BilateralZTransform:

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.

CMS

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

BibTeX

@misc{reference.wolfram_2024_fouriersequencetransform, author="Wolfram Research", title="{FourierSequenceTransform}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FourierSequenceTransform.html}", note=[Accessed: 21-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_fouriersequencetransform, organization={Wolfram Research}, title={FourierSequenceTransform}, year={2008}, url={https://reference.wolfram.com/language/ref/FourierSequenceTransform.html}, note=[Accessed: 21-December-2024 ]}