# ListFourierSequenceTransform

ListFourierSequenceTransform[list,ω]

gives the discrete-time Fourier transform (DTFT) of a list as a function of the parameter ω.

ListFourierSequenceTransform[list,ω,k]

places the first element of list at integer time k on the infinite time axis.

ListFourierSequenceTransform[list,{ω1,ω2,},{k1,k2,}]

gives the multidimensional discrete-time Fourier transform

# Examples

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## Basic Examples(2)

Discrete-time Fourier transform of a constant vector:

Two-dimensional DTFT:

## Applications(1)

Create a lowpass filter:

Visualize the frequency response:

Apply to a noisy signal:

## Properties & Relations(4)

Discrete-time Fourier transform of a numeric list is equal to the Fourier sequence transform of a sum of shifted unit samples:

Inverse of a discrete-time Fourier transform of a list:

Fourier of a length- list returns samples of the ListFourierSequenceTransform at frequencies that are multiples of :

ListFourierSequenceTransform is equivalent to computing ListZTransform on the unit circle:

Wolfram Research (2012), ListFourierSequenceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/ListFourierSequenceTransform.html.

#### Text

Wolfram Research (2012), ListFourierSequenceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/ListFourierSequenceTransform.html.

#### CMS

Wolfram Language. 2012. "ListFourierSequenceTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ListFourierSequenceTransform.html.

#### APA

Wolfram Language. (2012). ListFourierSequenceTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ListFourierSequenceTransform.html

#### BibTeX

@misc{reference.wolfram_2023_listfouriersequencetransform, author="Wolfram Research", title="{ListFourierSequenceTransform}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ListFourierSequenceTransform.html}", note=[Accessed: 25-February-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_listfouriersequencetransform, organization={Wolfram Research}, title={ListFourierSequenceTransform}, year={2012}, url={https://reference.wolfram.com/language/ref/ListFourierSequenceTransform.html}, note=[Accessed: 25-February-2024 ]}