Re: Fourier DFT scaling problem

• To: mathgroup at smc.vnet.net
• Subject: [mg126614] Re: Fourier DFT scaling problem
• From: Joseph Gwinn <joegwinn at comcast.net>
• Date: Thu, 24 May 2012 03:34:06 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jpflom\$ktd\$1@smc.vnet.net>

```In article <jpflom\$ktd\$1 at smc.vnet.net>, Arthur <arth_e at ymail.com>
wrote:

> Hi all.
>
> I hope someone can help me with what seems to be a fairly elementary problem,
> nevertheless I have spent a good deal of time trying to come to a solution
> without any success. I am fairly new to the Fourier transform.
>
> My problem is that the output of the Fourier[] command has a scaling
> dependency on my sampling parameters which I cannot figure out how to remove.
>
> In general I have a numerical function Psi[x] (a wavefunction) and I would
> like to compute an approximation to the Fourier transform F[Psi[x]] ~~
> Psihat[k].
>
> I am assuming the best way to achieve this is using the built in Fourier[]
> function.
>
> Now although I expect to have a numerical function Psi[x] my problem is
> apparent even when I define Psi[x] analytically.
>
> If I have
>
>
>
> Psi[x_] := (Pi^(-1/4)) (E^(-x^2/2))
>
> and take the continuous FT
>
> FourierTransform[Psi[x], x, k]
>
> OUT: (Pi^(-1/4)) (E^(-k^2/2))
>
> The output is as expected an identical function of k (since Psi[x] was a
> normalized eigenfunction of the FT).
>
> If I try to take the DFT of Psi[x] however:
>
> /******CODE******/
>
> n1 = 2^8.; (* sampling points *)
> L = 32.; (* domain [-L/2,L/2] *)
> dx = L/n1; (*step size*)
>
> X = Table[-dx n1/2 + (n - 1) dx, {n, n1}]; (* lattice *)
>
> rot[X_] := RotateLeft[X, n1/2]; (* rotate input/output of fourier *)
> F[X_] := rot@Fourier@rot@X; (* DFT *)
>
> SetOptions[ListLinePlot, PlotRange -> {{-L/2, L/2}, {0, 1}},
>  DataRange -> {-L/2, L/2}, Filling -> Axis];
>
> ListLinePlot[Abs[Psi[X]]^2]
> ListLinePlot[Abs[F@Psi[X]]^2]
>
>
> /******END******/
> The second graph of the squared modulus of F@Psi[X] is evidently incorrectly
> scaled. Moreover this scaling error is not improved by increasing the
> sampling frequency, and is in fact exacerbated by it.
>
> It seems reasonable that the scaling be dependent on the value dx, though I
> have done a lot of experimentation with the FourierParameters command to
> remove this dependence though I can't get it to work in a way that is
> independent of the choice of Psi[x].
>
> Any help would be *greatly* appreciated!
>
> Thanks and best regards,
>
> Arthur

In the documentation on Fourier[], the actual equation is given.  The
coefficients depend on the length of the list given to Fourier.  You
will have to divide by Sqrt[length] to have coefficients independent of
length, if I recall.  There are also a number of alternative
normalizations one can choose using options.  But you will need to
figure out the correct normalization and options for your application.

Joe Gwinn

```

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