InverseFourierSequenceTransform
InverseFourierSequenceTransform[expr,ω,n]
gives the inverse discrete-time Fourier transform of expr.
InverseFourierSequenceTransform[expr,{ω1,ω2,…},{n1,n2,…}]
gives the multidimensional inverse Fourier sequence transform.
Details and Options
- The inverse Fourier sequence transform of
is by default defined to be 
. - The
–dimensional inverse transform is given by 
. - In the form InverseFourierSequenceTransform[expr,t,n], n can be symbolic or an integer.
- The following options can be given:
-
Assumptions $Assumptions assumptions on parameters FourierParameters {1,1} parameters to define transform GenerateConditions False whether 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
open all close allBasic Examples (2)
Find the discrete-time inverse Fourier transform of 
:
InverseFourierSequenceTransform[Abs[ω], ω, n]DiscretePlot[%, {n, -5, 5}]Find a bivariate discrete-time inverse Fourier transform:
InverseFourierSequenceTransform[Abs[ω1]I ω2, {ω1, ω2}, {n1, n2}]ListPointPlot3D[Table[%, {n1, -5, 5}, {n2, -5, 5}], Filling -> Axis, DataRange -> {{-5, 5}, {-5, 5}}, PlotRange -> All]//QuietScope (3)
Inverse transform of rational exponential function:
FourierSequenceTransform[(4 / 5) ^ n UnitStep[n], n, ω]InverseFourierSequenceTransform[%, ω, n]DiscretePlot[%, {n, 0, 10}]InverseFourierSequenceTransform[E ^ (-ω ^ 2), ω, n]DiscretePlot[%, {n, -10, 10}, Ticks -> {Automatic, None}]A constant frequency gives an impulse and vice versa:
InverseFourierSequenceTransform[1, ω, n]InverseFourierSequenceTransform[DiracDelta[ω], ω, n]InverseFourierSequenceTransform[UnitStep[ω], ω, n]InverseFourierSequenceTransform[E ^ (5 I ω), ω, n]
InverseFourierSequenceTransform[(1/1 / 5 + E^I ω), ω, n]InverseFourierSequenceTransform[(1 / 5 E^I ω/(1 / 5 + E^I ω)^2), ω, n]Options (2)
Assumptions (1)
FourierParameters (1)
Use a nondefault setting for FourierParameters:
Table[InverseFourierSequenceTransform[ω, ω, n, FourierParameters -> ps], {ps, {{1, 1}, {1, -2π}}}]Properties & Relations (6)
InverseFourierSequenceTransform is defined by an integral:
InverseFourierSequenceTransform[ω ^ 4 + 1, ω, 3]1 / (2π) Integrate[(ω ^ 4 + 1) E ^ (3 I ω), {ω, -π, π}]InverseFourierSequenceTransform and FourierSequenceTransform are inverses:
InverseFourierSequenceTransform[FourierSequenceTransform[f[n], n, ω], ω, n]FourierSequenceTransform[InverseFourierSequenceTransform[g[ω], ω, n], n, ω]FourierSequenceTransform[a ^ n UnitStep[n], n, ω]InverseFourierSequenceTransform[%, ω, n]Simplify[% - a ^ n UnitStep[n], n∈Integers]InverseFourierSequenceTransform is closely related to InverseZTransform:
{InverseFourierSequenceTransform[(E^I ω/-a + E^I ω), ω, n, Assumptions -> n ≥ 0], InverseZTransform[(E^I ω/-a + E^I ω) /. Exp[I ω] -> z, z, n]}Just as InverseFourierTransform is closely related to InverseLaplaceTransform:
{(1/Sqrt[2π])InverseFourierTransform[(1/a - I ω), ω, t, Assumptions -> t > 0 && a > 0], InverseLaplaceTransform[(1/a - I ω) /. (-I ω) -> s, s, t]}InverseFourierSequenceTransform is the same as FourierCoefficient:
FourierCoefficient[Abs[t], t, n]InverseFourierSequenceTransform[Abs[t], t, n]Inverse discrete-time Fourier transform for basis exponentials:
InverseFourierSequenceTransform[E ^ (-5 I ω), ω, n]InverseFourierSequenceTransform[E ^ (3 I ω), ω, n]InverseFourierSequenceTransform[1, ω, n]InverseFourierSequenceTransform is closely related to InverseBilateralZTransform:
{InverseBilateralZTransform[ConditionalExpression[(z^2/1 / 2 + z), Abs[z] > 1 / 2], z, n], InverseFourierSequenceTransform[(z^2/1 / 2 + z) /. z -> Exp[I ω], ω, n]//Simplify}Text
Wolfram Research (2008), InverseFourierSequenceTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseFourierSequenceTransform.html.
CMS
Wolfram Language. 2008. "InverseFourierSequenceTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InverseFourierSequenceTransform.html.
APA
Wolfram Language. (2008). InverseFourierSequenceTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseFourierSequenceTransform.html