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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