InverseDTFourierTransform

As of Version 7.0, InverseDTFourierTransform has been renamed to InverseFourierSequenceTransform and is part of the built-in Wolfram Language kernel.

InverseDTFourierTransform[expr,ω,n]

gives the inverse discrete time Fourier transform of expr, where expr is a periodic function of ω with period 1.

Details and Options

To use InverseDTFourierTransform, you first need to load the Fourier Series Package using Needs["FourierSeries`"].
The inverse discrete time Fourier transform of expr is by default defined to be Integrate[expr ^-2πnω,{ω,-,}].
If n is numeric, it should be an explicit integer.
Different choices for the definition of the inverse discrete time Fourier transform can be specified using the option FourierParameters.
With the setting FourierParameters->{a,b}, expr is assumed to have a period of , and the inverse discrete time Fourier transform computed by InverseDTFourierTransform is Integrate[expr ^-2πω,{ω,-,}].
In addition to the option FourierParameters, InverseDTFourierTransform can also accept the options available to Integrate. These options are passed directly to Integrate.

Wolfram Language code: Needs["FourierSeries`"]

Find a sequence with a given discrete time Fourier transform:

Wolfram Language code: InverseDTFourierTransform[Exp[-ω ^ 2], ω, n]

Compare with the answer from a numerical approximation:

Wolfram Language code: InverseDTFourierTransform[Exp[-ω ^ 2], ω, n]

Wolfram Language code: % /. {n -> 5.}

Wolfram Language code: NInverseDTFourierTransform[Exp[-ω ^ 2], ω, 5]

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