Re: Fourier DFT scaling problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg126603] Re: Fourier DFT scaling problem*From*: "Kevin J. McCann" <kjm at KevinMcCann.com>*Date*: Wed, 23 May 2012 03:30:14 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <jpflom$ktd$1@smc.vnet.net>

Fourier will do the job, but I do not understand why you are doing the rotates. Here is a slightly different take on your problem with a Parseval check at the end. (Just copy and drop it into an empty notebook.) Cheers, Kevin Psi[x_] := \[Pi]^(-1/4) E^(-(x^2/2)) \[ScriptCapitalN] = 2^8; L = 32.0; \[CapitalDelta]x = L/\[ScriptCapitalN]; x = -16 + Table[i, {i, 0, \[ScriptCapitalN] - 1}]*\[CapitalDelta]x; \[CapitalDelta]k = 1/L; f = Psi[x]; k = Table[n \[CapitalDelta]k, {n, 0, \[ScriptCapitalN] - 1}]; F = \[CapitalDelta]x Fourier[f, FourierParameters -> {1, -1}]; ListPlot[Transpose[{k, Abs[F]^2}], PlotRange -> All, Joined -> True] (* This part rotates the k-spectrum so it "looks right" *) krot = Table[ n \[CapitalDelta]k, {n, -(\[ScriptCapitalN]/2), \[ScriptCapitalN]/ 2 - 1}]; Frot = RotateRight[F, \[ScriptCapitalN]/2]; ListPlot[Transpose[{krot, Abs[Frot]^2}], PlotRange -> All, Joined -> True] (* Check that Parseval is right *) {Total[Abs[f]^2] \[CapitalDelta]x, Total[Abs[F]^2] \[CapitalDelta]k} On 5/22/2012 5:20 AM, Arthur 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 >

**Follow-Ups**:**Re: Fourier DFT scaling problem***From:*W Craig Carter <ccarter@mit.edu>