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Function Repository Resource:

NFourierSequenceTransform

Source Notebook

Find a numerical approximation to the Fourier sequence transform

Contributed by: Wolfram Research

ResourceFunction["NFourierSequenceTransform"][expr,n,ω]

gives a numerical approximation to the Fourier sequence transform of expr evaluated at the numerical value ω, where expr is a function of n.

Details and Options

Fourier sequence transform is also known as discrete-time Fourier transform (DTFT).
ResourceFunction["NFourierSequenceTransform"][expr,n,ω] takes a sequence whose nth term is given by expr.
The numerical approximation to the Fourier sequence transform of expr is by default defined to be NSum[expr-nω,{n,-∞,}].
Different choices for the definition of the Fourier sequence transform can be specified using the option FourierParameters.
With the setting FourierParameters{a,b}, the Fourier sequence transform computed by ResourceFunction["NFourierSequenceTransform"] is .
The parameter b in the setting FourierParameters{a,b} must be numeric.
In addition to the option FourierParameters, ResourceFunction["NFourierSequenceTransform"] can also accept the options available to NSum. These options are passed directly to NSum.

Examples

Basic Examples

Different definitions for numerical approximation of a Fourier sequence transform:

In[1]:=
ResourceFunction["NFourierSequenceTransform"][1/(3 n + 5)^2, n, 0.7]
Out[1]=
In[2]:=
ResourceFunction["NFourierSequenceTransform"][1/(3 n + 5)^2, n, 0.7, FourierParameters -> {1, -2 \[Pi]}]
Out[2]=

Compare with the answer from symbolic evaluation with FourierSequenceTransform:

In[3]:=
ResourceFunction["NFourierSequenceTransform"][1/(3 n + 5)^2, n, 0.7]
Out[3]=
In[4]:=
FourierSequenceTransform[1/(3 n + 5)^2, n, \[Omega]]
Out[4]=
In[5]:=
% /. {\[Omega] -> 0.7}
Out[5]=

Requirements

Wolfram Language 11.3 (March 2018) or above

Resource History

See Also

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