Re: Fourier DFT scaling problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg126606] Re: Fourier DFT scaling problem*From*: Dana DeLouis <dana01 at me.com>*Date*: Fri, 25 May 2012 04:54:25 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com

> FourierTransform[Psi[x], x, k] > > OUT: (Pi^(-1/4)) (E^(-k^2/2)) Hi. I don't have a solution, but does anything here help? Does changing the default settings help in some way? xData={-1,+1}; xSignal={+1,-1}; xDefault = {0,1}; Psi[x_] := (Pi^(-1/4)) (E^(-x^2/2)) SetOptions[{FourierTransform,InverseFourierTransform},FourierParameters->xDefault]; FourierTransform[Psi[x],x,k] E^(-(k^2/2))/\[Pi]^(1/4) SetOptions[{FourierTransform,InverseFourierTransform},FourierParameters->xSignal]; FourierTransform[Psi[x],x,k] Sqrt[2] E^(-(k^2/2)) \[Pi]^(1/4) SetOptions[{FourierTransform,InverseFourierTransform},FourierParameters->xData]; FourierTransform[Psi[x],x,k] E^(-(k^2/2))/(Sqrt[2] \[Pi]^(3/4)) = = = = = Dana DeLouis Mac & Math 8 = = = = = On May 22, 5:20 am, Arthur <art... 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