--- title: "FourierSequenceTransform" language: "en" type: "Symbol" summary: "FourierSequenceTransform[expr, n, \\[Omega]] gives the Fourier sequence transform of expr. FourierSequenceTransform[expr, {n1, n2, ...}, {\\[Omega]1, \\[Omega]2, ...}] gives the multidimensional Fourier sequence transform." keywords: - DTFT - TDFT - 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/FourierSequenceTransform.html" source: "Wolfram Language Documentation" related_guides: - title: "Fourier Analysis" link: "https://reference.wolfram.com/language/guide/FourierAnalysis.en.md" - title: "Discrete Calculus" link: "https://reference.wolfram.com/language/guide/DiscreteCalculus.en.md" - title: "Summation Transforms" link: "https://reference.wolfram.com/language/guide/SummationTransforms.en.md" - title: "Integral Transforms" link: "https://reference.wolfram.com/language/guide/IntegralTransforms.en.md" - title: "Analytic Number Theory" link: "https://reference.wolfram.com/language/guide/AnalyticNumberTheory.en.md" - title: "Signal Visualization & Analysis" link: "https://reference.wolfram.com/language/guide/SignalAnalysis.en.md" related_functions: - title: "InverseFourierSequenceTransform" link: "https://reference.wolfram.com/language/ref/InverseFourierSequenceTransform.en.md" - title: "Fourier" link: "https://reference.wolfram.com/language/ref/Fourier.en.md" - title: "FourierTransform" link: "https://reference.wolfram.com/language/ref/FourierTransform.en.md" - title: "FourierCoefficient" link: "https://reference.wolfram.com/language/ref/FourierCoefficient.en.md" - title: "ZTransform" link: "https://reference.wolfram.com/language/ref/ZTransform.en.md" - title: "BilateralZTransform" link: "https://reference.wolfram.com/language/ref/BilateralZTransform.en.md" - title: "DiscreteConvolve" link: "https://reference.wolfram.com/language/ref/DiscreteConvolve.en.md" - title: "Sum" link: "https://reference.wolfram.com/language/ref/Sum.en.md" --- # FourierSequenceTransform FourierSequenceTransform[expr, n, ω] gives the Fourier sequence transform of expr. FourierSequenceTransform[expr, {n1, n2, …}, {ω1, ω2, …}] gives the multidimensional Fourier sequence transform. ## Details and Options * ``FourierSequenceTransform`` is also known as discrete-time Fourier transform (DTFT). * ``FourierSequenceTransform[expr, n, ω]`` takes a sequence whose ``n``$$^{\text{th}}$$ term is given by ``expr``, and yields a function of the continuous parameter ``ω``. * The Fourier sequence transform of $f(n)$ is by default defined to be $\sum _{n=-\infty }^{\infty } f(n) e^{-i n \omega }$. * The Fourier sequence transform of $f(n)$ is by default periodic with a period of $2\pi$. * The multidimensional transform of $f\left(n_1,n_2,\ldots \right)$ is defined to be $\sum _{n_1=-\infty }^{\infty } \sum _{n_2=-\infty }^{\infty } \cdots f\left(n_1,n_2,\ldots \right) e^{-i \left(n_1 \omega _1+n_2 \omega _2+\cdots \right)}$. * The following options can be given: | | | | | ------------------- | ------------- | ----------------------------------------------------------------- | | Assumptions | \$Assumptions | assumptions on parameters | | FourierParameters | {1, 1} | parameters to define the discrete-time Fourier transform | | GenerateConditions | False | whether to generate results that involve conditions on parameters | * Common settings for ``FourierParameters`` include: | | | | | --- | --- | --- | | {1, 1} | $\sum _{n=-\infty }^{\infty } f(n) e^{-i n \omega }$ | default settings | | {1, 2Pi} | $\sum _{n=-\infty }^{\infty } f(n) e^{-i 2 \pi n \omega }$ | period 1 | | {a, b} | $\left\| \frac{b}{2 \pi }\right\| ^{\frac{1-a}{2}}\sum _{n=-\infty }^{\infty } f(n) e^{-i b n \omega }$ | general setting | ## Examples (13) ### Basic Examples (2) Find the discrete-time Fourier transform of a simple signal: ```wl In[1]:= FourierSequenceTransform[(1 / 2) ^ n UnitStep[n], n, ω] Out[1]= (2 E^I ω/-1 + 2 E^I ω) In[2]:= LogPlot[Abs[%], {ω, 0, 4Pi}] Out[2]= [image] ``` --- Find a bivariate discrete-time Fourier transform: ```wl In[1]:= FourierSequenceTransform[(1 / 2) ^ n1(1 / 3) ^ n2 UnitStep[n1, n2], {n1, n2}, {ω1, ω2}] Out[1]= (6 E^I (ω1 + ω2)/(-1 + 2 E^I ω1) (-1 + 3 E^I ω2)) In[2]:= Plot3D[Abs[%], {ω1, 0, 4Pi}, {ω2, 0, 4Pi}] Out[2]= [image] ``` ### Scope (4) Compute the transform for each frequency ``ω`` : ```wl In[1]:= F = FourierSequenceTransform[Sin[n 2Pi / 3](2 / 3) ^ n UnitStep[n], n, ω] Out[1]= (3 Sqrt[3] E^I ω/4 + 6 E^I ω + 9 E^2 I ω) ``` Plot the power spectrum: ```wl In[2]:= LogPlot[Abs[F] ^ 2, {ω, 0, 2π}, Ticks -> {{0, π, 2π}, Automatic}] Out[2]= [image] ``` The phase: ```wl In[3]:= Plot[Arg[F], {ω, 0, 2π}, Ticks -> {{0, π, 2π}, Automatic}] Out[3]= [image] ``` Plot both spectrum and phase using color: ```wl In[4]:= LogPlot[Abs[F] ^ 2, {ω, 0, 2π}, Ticks -> {{0, π, 2π}, Automatic}, ColorFunction -> Function[ω, Evaluate@Hue[Arg[F] / (2Pi) + 1 / 2]], ColorFunctionScaling -> False, Filling -> Axis] Out[4]= [image] ``` --- Constant: ```wl In[1]:= FourierSequenceTransform[1, n, ω] Out[1]= DiracComb[(ω/2 π)] ``` Periodic: ```wl In[2]:= FourierSequenceTransform[Exp[I n ], n, ω] Out[2]= DiracComb[(-1 + ω/2 π)] In[3]:= FourierSequenceTransform[Cos[5n], n, ω] Out[3]= (1/2) (DiracComb[(-5 + ω/2 π)] + DiracComb[(5 + ω/2 π)]) In[4]:= FourierSequenceTransform[Sin[n ω0], n, ω, Assumptions -> ω0∈Reals] Out[4]= -(1/2) I (DiracComb[(ω - ω0/2 π)] - DiracComb[(ω + ω0/2 π)]) In[5]:= FourierSequenceTransform[Mod[n, 3], n, ω] Out[5]= DiracComb[(ω/2 π)] + (1/3) (2 E^-(2 I π/3) + E^(2 I π/3)) DiracComb[(-(4 π/3) + ω/2 π)] + (1/3) (E^-(2 I π/3) + 2 E^(2 I π/3)) DiracComb[(-(2 π/3) + ω/2 π)] ``` Impulse: ```wl In[6]:= FourierSequenceTransform[DiscreteDelta[n], n, ω] Out[6]= 1 In[7]:= FourierSequenceTransform[DiscreteDelta[n - n0], n, ω] Out[7]= E^-I n0 ω ``` Exponential: ```wl In[8]:= FourierSequenceTransform[a ^ n UnitStep[n], n, ω] Out[8]= (E^I ω/-a + E^I ω) In[9]:= FourierSequenceTransform[a ^ n UnitStep[-1 - n], n, ω] Out[9]= -(E^I ω/-a + E^I ω) ``` Exponential polynomial: ```wl In[10]:= FourierSequenceTransform[(n + 1)a ^ n UnitStep[n], n, ω] Out[10]= (1/(1 - a E^-I ω)^2) In[11]:= FourierSequenceTransform[(n + 2)(n + 1) a ^ n UnitStep[n], n, ω] Out[11]= (2 E^3 I ω/(-a + E^I ω)^3) ``` --- Rational sequence: ```wl In[1]:= FourierSequenceTransform[1 / (2n + 1) ^ 2, n, ω] Out[1]= (1/4) (LerchPhi[E^-I ω, 2, (1/2)] + E^I ω LerchPhi[E^I ω, 2, (1/2)]) ``` Rational-trigonometric: ```wl In[2]:= FourierSequenceTransform[Sin[n] / (3n + 1), n, ω] Out[2]= -(1/4) I E^-I (2 E^I Hypergeometric2F1[(1/3), 1, (4/3), E^-I (-1 + ω)] - 2 E^I Hypergeometric2F1[(1/3), 1, (4/3), E^-I (1 + ω)] - E^I ω Hypergeometric2F1[(2/3), 1, (5/3), E^I (-1 + ω)] + E^2 I + I ω Hypergeometric2F1[(2/3), 1, (5/3), E^I (1 + ω)]) ``` Hypergeometric terms: ```wl In[3]:= FourierSequenceTransform[1 / n! UnitStep[n], n, ω] Out[3]= E^E^-I ω In[4]:= FourierSequenceTransform[2 ^ n / (CatalanNumber[n]n!) UnitStep[n], n, ω] Out[4]= (1/4) E^-I ω (2 + 4 E^I ω + E^(E^-I ω/2) Sqrt[E^-I ω] Sqrt[2 π] Erf[(Sqrt[E^-I ω]/Sqrt[2])] + 3 E^(E^-I ω/2) + I ω Sqrt[E^-I ω] Sqrt[2 π] Erf[(Sqrt[E^-I ω]/Sqrt[2])]) ``` --- Multivariate sequences: ```wl In[1]:= FourierSequenceTransform[E ^ (-n1)(1 / 5) ^ n2 UnitStep[n1, n2], {n1, n2}, {ω1, ω2}] Out[1]= (5 E^I (-I + ω1 + ω2)/(-1 + E^1 + I ω1) (-1 + 5 E^I ω2)) In[2]:= FourierSequenceTransform[Mod[n1 + n2, 2], {n1, n2}, {ω1, ω2}] Out[2]= (1/2) (-DiracComb[(π - ω1/2 π)] DiracComb[(π - ω2/2 π)] + DiracComb[(ω1/2 π)] DiracComb[(ω2/2 π)]) ``` ### Options (2) #### FourierParameters (1) Use a non-default setting for ``FourierParameters`` : ```wl In[1]:= FourierSequenceTransform[a ^ n UnitStep[n], n, ω, FourierParameters -> {1, -2π}] Out[1]= -(1/-1 + a E^2 I π ω) ``` #### GenerateConditions (1) Obtain conditions on parameters: ```wl In[1]:= FourierSequenceTransform[Piecewise[{{a^n, n ≥ 0}, {b^n, n < 0}}], n, ω, GenerateConditions -> True, Assumptions -> ω∈Reals] Out[1]= ConditionalExpression[((a - b)*E^(I*ω))/((-a + E^(I*ω))*(-b + E^(I*ω))), Abs[a] < 1 && Abs[b] > 1] ``` ### Properties & Relations (5) ``FourierSequenceTransform`` is defined by a doubly infinite sum: ```wl In[1]:= f = Piecewise[{{a^n, n ≥ 0}, {b^n, n < 0}}] Out[1]= Piecewise[{{a^n, n >= 0}, {b^n, n < 0}}, 0] In[2]:= {FourierSequenceTransform[f, n, ω], Sum[f Exp[-I n ω], {n, -∞, ∞}]} Out[2]= {((a - b) E^I ω/(-a + E^I ω) (-b + E^I ω)), ((a - b) E^I ω/(-a + E^I ω) (-b + E^I ω))} ``` --- ``FourierSequenceTransform`` and ``InverseFourierSequenceTransform`` 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 ``` --- ``FourierSequenceTransform`` is closely related to ``ZTransform``: ```wl In[1]:= {FourierSequenceTransform[a ^ n UnitStep[n], n, ω], ZTransform[a ^ n, n, Exp[I ω]]} Out[1]= {(E^I ω/-a + E^I ω), (E^I ω/-a + E^I ω)} ``` A discrete analog of ``FourierTransform`` being closely related to ``LaplaceTransform``: ```wl In[2]:= {Sqrt[2π]FourierTransform[Exp[-a t] UnitStep[t], t, ω], LaplaceTransform[Exp[-a t] UnitStep[t], t, -I ω]} Out[2]= {(1/a - I ω), (1/a - I ω)} ``` --- ``FourierSequenceTransform`` provides a $q$-analog generating function: ```wl In[1]:= FourierSequenceTransform[a ^ n UnitStep[n], n, ω] /. Exp[I ω] -> q Out[1]= (q/-a + q) In[2]:= GeneratingFunction[a ^ n UnitStep[n], n, 1 / q] Out[2]= (1/1 - (a/q)) In[3]:= Simplify[%% - %] Out[3]= 0 ``` --- ``FourierSequenceTransform`` is closely related to ``BilateralZTransform`` : ```wl In[1]:= {BilateralZTransform[UnitStep[n + 1]3^-n, n, Exp[I ω], Assumptions -> ω∈Reals], FourierSequenceTransform[UnitStep[n + 1]3^-n, n, ω]} Out[1]= {(9 E^2 I ω/-1 + 3 E^I ω), (9 E^2 I ω/-1 + 3 E^I ω)} In[2]:= {BilateralZTransform[5^-Abs[n + 1], n, Exp[I ω], Assumptions -> ω∈Reals], FourierSequenceTransform[5^-Abs[n + 1], n, ω]} Out[2]= {-(24 E^2 I ω/5 - 26 E^I ω + 5 E^2 I ω), -(24 E^2 I ω/5 - 26 E^I ω + 5 E^2 I ω)} ``` ## See Also * [`InverseFourierSequenceTransform`](https://reference.wolfram.com/language/ref/InverseFourierSequenceTransform.en.md) * [`Fourier`](https://reference.wolfram.com/language/ref/Fourier.en.md) * [`FourierTransform`](https://reference.wolfram.com/language/ref/FourierTransform.en.md) * [`FourierCoefficient`](https://reference.wolfram.com/language/ref/FourierCoefficient.en.md) * [`ZTransform`](https://reference.wolfram.com/language/ref/ZTransform.en.md) * [`BilateralZTransform`](https://reference.wolfram.com/language/ref/BilateralZTransform.en.md) * [`DiscreteConvolve`](https://reference.wolfram.com/language/ref/DiscreteConvolve.en.md) * [`Sum`](https://reference.wolfram.com/language/ref/Sum.en.md) ## Related Guides * [Fourier Analysis](https://reference.wolfram.com/language/guide/FourierAnalysis.en.md) * [Discrete Calculus](https://reference.wolfram.com/language/guide/DiscreteCalculus.en.md) * [Summation Transforms](https://reference.wolfram.com/language/guide/SummationTransforms.en.md) * [Integral Transforms](https://reference.wolfram.com/language/guide/IntegralTransforms.en.md) * [Analytic Number Theory](https://reference.wolfram.com/language/guide/AnalyticNumberTheory.en.md) * [Signal Visualization & Analysis](https://reference.wolfram.com/language/guide/SignalAnalysis.en.md) ## History * [Introduced in 2008 (7.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn70.en.md)