# 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

# Details and Options • By default, the one-dimensional discrete-time Fourier transform of a list of length is computed as .
• ListFourierSequenceTransform[list,ω] is equivalent to ListFourierSequenceTransform[list,ω,0].
• ListFourierSequenceTransform takes FourierParameters option. Common settings for FourierParameters include:
•  {1,1} default settings {1,-2Pi} period 1 {a,b} general setting
• ListFourierSequenceTransform[list,ω,k] effectively computes FourierSequenceTransform[f[r],r,ω] for a sequence f with f[r-1+k]=list[[r]] for 1<=r<=Length[list] and f[r]=0 otherwise.

# 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:

Introduced in 2012
(9.0)