--- title: "InverseFourierSequenceTransform" language: "en" type: "Symbol" summary: "InverseFourierSequenceTransform[expr, \\[Omega], n] gives the inverse discrete-time Fourier transform of expr. InverseFourierSequenceTransform[expr, {\\[Omega]1, \\[Omega]2, \\ ...}, {n1, n2, ...}] gives the multidimensional inverse Fourier sequence transform." keywords: - inverse fourier sequence transform - TDFT - DTFT - IDTFT - discrete-time Fourier transform - sampled Fourier transform - sequence Fourier transform - discrete Fourier transform - discrete-time signal - sequence transform - summation transform canonical_url: "https://reference.wolfram.com/language/ref/InverseFourierSequenceTransform.html" source: "Wolfram Language Documentation" related_guides: - title: "Fourier Analysis" link: "https://reference.wolfram.com/language/guide/FourierAnalysis.en.md" - title: "Summation Transforms" link: "https://reference.wolfram.com/language/guide/SummationTransforms.en.md" related_functions: - title: "FourierSequenceTransform" link: "https://reference.wolfram.com/language/ref/FourierSequenceTransform.en.md" - title: "InverseFourier" link: "https://reference.wolfram.com/language/ref/InverseFourier.en.md" - title: "InverseFourierTransform" link: "https://reference.wolfram.com/language/ref/InverseFourierTransform.en.md" - title: "InverseZTransform" link: "https://reference.wolfram.com/language/ref/InverseZTransform.en.md" - title: "Integrate" link: "https://reference.wolfram.com/language/ref/Integrate.en.md" --- # 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 $f(\omega )$ is by default defined to be $\frac{1}{2 \pi }\int _{-\pi }^{\pi }f(\omega ) e^{i n \omega }d\omega$. * The $m$–dimensional inverse transform is given by $\frac{1}{(2\pi )^m}\int _{-\pi }^{\pi }\int _{-\pi }^{\pi }\cdots f\left(\omega _1,\omega _2,\ldots \right) e^{i \left(n_1 \omega _1+n_2 \omega _2+\cdots \right)}d\omega _1d\omega _2\cdots$. * 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} | $\frac{1}{2\pi }\int _{-\pi }^{\pi }f(\omega ) e^{i n \omega }d\omega$ | default settings | | {1, -2Pi} | $\int _{-\frac{1}{2}}^{\frac{1}{2}}f(\omega ) e^{-i 2\pi n \omega }d\omega$ | period 1 | | {a, b} | $\left\| \frac{b}{2 \pi }\right\| ^{\frac{a+1}{2}}\int _{-\frac{\pi }{\| b\| }}^{\frac{\pi }{\| b\| }}f(\omega ) e^{i b n \omega }d\omega$ | general setting | --- ## Examples (13) ### Basic Examples (2) Find the discrete-time inverse Fourier transform of $| \omega |$ : ```wl In[1]:= InverseFourierSequenceTransform[Abs[ω], ω, n] Out[1]= Piecewise[{{Pi/2, n == 0}}, -((-1 + (-1)^n)^2/(2*n^2*Pi))] In[2]:= DiscretePlot[%, {n, -5, 5}] Out[2]= [image] ``` --- Find a bivariate discrete-time inverse Fourier transform: ```wl In[1]:= InverseFourierSequenceTransform[Abs[ω1]I ω2, {ω1, ω2}, {n1, n2}] Out[1]= ((-1)^n1 + n2 (-1 + (-1)^n1) (-1 + (-1)^n1 - I (1 + (-1)^n1) n1 π)/2 n1^2 n2 π) In[2]:= ListPointPlot3D[Table[%, {n1, -5, 5}, {n2, -5, 5}], Filling -> Axis, DataRange -> {{-5, 5}, {-5, 5}}, PlotRange -> All]//Quiet Out[2]= [image] ``` ### Scope (3) Inverse transform of rational exponential function: ```wl In[1]:= FourierSequenceTransform[(4 / 5) ^ n UnitStep[n], n, ω] Out[1]= (5 E^I ω/-4 + 5 E^I ω) In[2]:= InverseFourierSequenceTransform[%, ω, n] Out[2]= Piecewise[{{(5/4)^(-n), n >= 0}}, 0] In[3]:= DiscretePlot[%, {n, 0, 10}] Out[3]= [image] ``` --- Gaussian function: ```wl In[1]:= InverseFourierSequenceTransform[E ^ (-ω ^ 2), ω, n] Out[1]= (I E^-(n^2/4) (Erfi[(n/2) - I π] - Erfi[(n/2) + I π])/4 Sqrt[π]) In[2]:= DiscretePlot[%, {n, -10, 10}, Ticks -> {Automatic, None}] Out[2]= [image] ``` --- A constant frequency gives an impulse and vice versa: ```wl In[1]:= InverseFourierSequenceTransform[1, ω, n] Out[1]= DiscreteDelta[n] In[2]:= InverseFourierSequenceTransform[DiracDelta[ω], ω, n] Out[2]= (1/2 π) In[3]:= InverseFourierSequenceTransform[UnitStep[ω], ω, n] Out[3]= -(I (-1 + (-1)^n)/2 n π) In[4]:= InverseFourierSequenceTransform[E ^ (5 I ω), ω, n] Out[4]= DiscreteDelta[5 + n] ``` Rational function in $\exp (i \omega )$ : ```wl In[5]:= InverseFourierSequenceTransform[(1/1 / 5 + E^I ω), ω, n] Out[5]= \[Piecewise]| | | | :-------------- | :---- | | -(-1)^n 5^1 - n | n > 0 | | 0 | True | In[6]:= InverseFourierSequenceTransform[(1 / 5 E^I ω/(1 / 5 + E^I ω)^2), ω, n] Out[6]= \[Piecewise]| | | | :------------ | :---- | | -(-(1/5))^n n | n ≥ 0 | | 0 | True | ``` ### Options (2) #### Assumptions (1) Specify assumptions on a parameter: ```wl In[1]:= InverseFourierSequenceTransform[E ^ (I a ω), ω, n, Assumptions -> a∈Integers] Out[1]= DiscreteDelta[a + n] ``` #### FourierParameters (1) Use a nondefault setting for ``FourierParameters`` : ```wl In[1]:= Table[InverseFourierSequenceTransform[ω, ω, n, FourierParameters -> ps], {ps, {{1, 1}, {1, -2π}}}] Out[1]= {-(I (-1)^n/n), (I (-1)^n/2 n π)} ``` ### Properties & Relations (6) ``InverseFourierSequenceTransform`` is defined by an integral: ```wl In[1]:= InverseFourierSequenceTransform[ω ^ 4 + 1, ω, 3] Out[1]= (4/27) (2 - 3 π^2) In[2]:= 1 / (2π) Integrate[(ω ^ 4 + 1) E ^ (3 I ω), {ω, -π, π}] Out[2]= (4/27) (2 - 3 π^2) ``` --- ``InverseFourierSequenceTransform`` and ``FourierSequenceTransform`` are inverses: ```wl In[1]:= InverseFourierSequenceTransform[FourierSequenceTransform[f[n], n, ω], ω, n] Out[1]= f[n] In[2]:= FourierSequenceTransform[InverseFourierSequenceTransform[g[ω], ω, n], n, ω] Out[2]= g[ω] In[3]:= FourierSequenceTransform[a ^ n UnitStep[n], n, ω] Out[3]= (E^I ω/-a + E^I ω) In[4]:= InverseFourierSequenceTransform[%, ω, n] Out[4]= Piecewise[{{a^n, n > -1}}, 0] In[5]:= Simplify[% - a ^ n UnitStep[n], n∈Integers] Out[5]= 0 ``` --- ``InverseFourierSequenceTransform`` is closely related to ``InverseZTransform``: ```wl In[1]:= {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]} Out[1]= {a^n, a^n} ``` Just as ``InverseFourierTransform`` is closely related to ``InverseLaplaceTransform``: ```wl In[2]:= {(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]} Out[2]= {E^-a t, E^-a t} ``` --- ``InverseFourierSequenceTransform`` is the same as ``FourierCoefficient``: ```wl In[1]:= FourierCoefficient[Abs[t], t, n] Out[1]= \[Piecewise]| | | | :------------------------- | :---- | | (π/2) | n == 0 | | -((-1 + (-1)^n)^2/2 n^2 π) | True | In[2]:= InverseFourierSequenceTransform[Abs[t], t, n] Out[2]= \[Piecewise]| | | | :------------------------- | :---- | | (π/2) | n == 0 | | -((-1 + (-1)^n)^2/2 n^2 π) | True | ``` --- Inverse discrete-time Fourier transform for basis exponentials: ```wl In[1]:= InverseFourierSequenceTransform[E ^ (-5 I ω), ω, n] Out[1]= DiscreteDelta[-5 + n] In[2]:= InverseFourierSequenceTransform[E ^ (3 I ω), ω, n] Out[2]= DiscreteDelta[3 + n] In[3]:= InverseFourierSequenceTransform[1, ω, n] Out[3]= DiscreteDelta[n] ``` --- ``InverseFourierSequenceTransform`` is closely related to ``InverseBilateralZTransform`` : ```wl In[1]:= {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} Out[1]= {Piecewise[{{(-(1/2))^(1 + n), n == -1 || n >= 0}}, 0], Piecewise[{{(-(1/2))^(1 + n), n == -1 || n >= 0}}, 0]} ``` ## See Also * [`FourierSequenceTransform`](https://reference.wolfram.com/language/ref/FourierSequenceTransform.en.md) * [`InverseFourier`](https://reference.wolfram.com/language/ref/InverseFourier.en.md) * [`InverseFourierTransform`](https://reference.wolfram.com/language/ref/InverseFourierTransform.en.md) * [`InverseZTransform`](https://reference.wolfram.com/language/ref/InverseZTransform.en.md) * [`Integrate`](https://reference.wolfram.com/language/ref/Integrate.en.md) ## Related Guides * [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.en.md) * [Summation Transforms](https://reference.wolfram.com/language/guide/SummationTransforms.en.md) ## History * [Introduced in 2008 (7.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn70.en.md)