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
- ListFourierSequenceTransform gives the discrete-time Fourier transform (DTFT) of a numeric array, typically used to obtain the frequency response of an FIR filter.
- 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
open all close allBasic Examples (2)
Discrete-time Fourier transform (DTFT) of a constant vector:
ListFourierSequenceTransform[{1, 1, 1} / 3, ω]dtft = ListFourierSequenceTransform[(1/9)(| | | |
| - | - | - |
| 1 | 1 | 1 |
| 1 | 1 | 1 |
| 1 | 1 | 1 |), {u, v}]Plot3D[Abs[dtft], {u, -π, π}, {v, -π, π}, PlotRange -> {0, 1}]Scope (2)
DTFT of a left-aligned vector is returned by default:
ListFourierSequenceTransform[{1, 0, -1}, ω]Plot[Arg[%], {ω, 0, π}]DTFT of a center-aligned vector:
ListFourierSequenceTransform[{1, 0, -1}, ω, -1]Plot[Arg[%], {ω, 0, π}]DTFT of a centered 3x3 constant array:
ListFourierSequenceTransform[(1/9)(| | | |
| - | - | - |
| 1 | 1 | 1 |
| 1 | 1 | 1 |
| 1 | 1 | 1 |), {u, v}, {-1, -1}]Options (1)
FourierParameters (1)
ListFourierSequenceTransform[{1, 1, 1} / 3, ω, FourierParameters -> {1, 1}]
ListFourierSequenceTransform[{1, 1, 1} / 3, ω, FourierParameters -> {-1, 1}]Switch to a fundamental frequency interval of 
, or a period of 1:
ListFourierSequenceTransform[{1, 1, 1} / 3, f, FourierParameters -> {1, 2π}]Plot[Abs[%], {f, -1 / 2, 1 / 2}]Applications (1)
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:
f[n_] := (1/5)Sum[DiscreteDelta[n - k], {k, 0, 4}]FourierSequenceTransform[f[n], n, ω] == Together@ListFourierSequenceTransform[{(1/5), (1/5), (1/5), (1/5), (1/5)}, ω]Inverse of a discrete-time Fourier transform of a list:
dtft = ListFourierSequenceTransform[{(1/5), (1/4), (1/3), (1/2), 1}, ω]Table[InverseFourierSequenceTransform[dtft, ω, n], {n, 0, Length[dtft] - 1}]Fourier of a length-
list returns samples of the ListFourierSequenceTransform at frequencies that are multiples of 
:
dft = Fourier[{(1/5), (1/4), (1/3), (1/2), 1}, FourierParameters -> {1, -1}]//Chopdtft = ListFourierSequenceTransform[{(1/5), (1/4), (1/3), (1/2), 1}, ω]Table[dtft, {ω, 0, 6, 2Pi / 5}]//NListFourierSequenceTransform is equivalent to computing ListZTransform on the unit circle:
H[z_] = ListZTransform[{(1/5), (1/4), (1/3), (1/2), 1}, z];
dtft = ListFourierSequenceTransform[{(1/5), (1/4), (1/3), (1/2), 1}, ω];
H[E ^ (I ω)] === dtftText
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