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MathGroup Archive 2002

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Re: Numerical Differentiation using Fourier Transform

  • To: mathgroup at smc.vnet.net
  • Subject: [mg33011] Re: Numerical Differentiation using Fourier Transform
  • From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
  • Date: Tue, 26 Feb 2002 04:34:51 -0500 (EST)
  • Organization: Universitaet Leipzig
  • References: <a4vgmt$r4b$1@smc.vnet.net> <200202210707.CAA02160@smc.vnet.net> <a57gs9$2dk$1@smc.vnet.net>
  • Reply-to: kuska at informatik.uni-leipzig.de
  • Sender: owner-wri-mathgroup at wolfram.com

Hi,


it he would use a Fourier approximation to it's discrete sampled points
he would use the NumericalMath`TrigFit` package. *This* calculates
a function expansion with continuous time and discrete frequencies!

But he asked how to use a discret to discret Fourier transform to
compute
the derivative and I told him that a discrete derivative does not
exist (what a surprise). In addition to this bad news, I pointed out
that he can use a discrete approximation of a derivative and use it
with a DFT.

> I don't think you're right about the previous question by Mike Croucher.
> His practical problem is to use an approximation for a function given a
> finite set of sampling points. Tha'ts quite common thing !
> And he clearly wants to use a Fourier approximation (instead of
> interpolation polynomials, or whatever). This has decisive advantages if you
> want to remove noise for example from an experimental array of values.

TrigFit[] has a complete different error than a Fourier expansion,
because
it make a least square fit. So it is not realy  a Fourier series.

> One point, however, is very important to have in mind. This is the so called
> 'Gibbs' phenomenon arising because of discontinuity between the first point
> and the last point of the array. If the function does not take the same
> value, then you will get strange things due to non absolute convergence of
> the Fourier approximation. Just use the following code by replacing the
> function f[x] by a function taking different values at f[0] and f[xmax]. You
> will see what I mean by very strange (and yet perfectly well understood)
> phenomenon §
> 
> **************************
> Clear[test, xt, Foutest, n, nHalf, xmax, ismooth]
> xmax = 6.
> nHalf = 200
> n = 2*nHalf + 1
> xt = Table[i*xmax/n , {i, 0, n - 1}];
> f[x_] := E^(-(x - xmax/2.)^2)
> test = Table[f[i*xmax/n] , {i, 0, n - 1}];
> Foutest = Chop[Fourier[test]];
> FouDer = Foutest;
> FouDer[[1]] = 0.;
> ismooth = 1000000;
> freq1 = -I*Table[i*Exp[-(i/ismooth)^2], {i, 1, nHalf}]; (* Here is one half
> of  the frequency terms *)
> freq2 = I*Table[i*Exp[-(i/ismooth)^2], {i, 1, nHalf}]; (* Here is the other
> half . Notice the Reverse order below*)
> 

We must have different analysis books. I always thought that
Limit[(f[x+h]-f[x])/h,h->0] is the derivative and not


-(s^2)^(3/2)*
  Integrate[
      t*E^(-(s^2*t^2)/4)*f[x - t], {t, -Infinity, Infinity}]/(4*
        Sqrt[Pi])

Using your new definition one can express an approximation of
the new "Dumas derivative" in terms of the old definition

DDumas[f[x],x] approx f'[x]+x*f''[x]


did the original poster want this ? really ? I must have overseen
that !

If you want reduce the noise, the approximation of the
derivative should have more points like my suggestion
(f[i-2] - 8*f[i-1] + 8*f[i+1] - f[i+2])/12 but TrigFit[] should
also smooth the data. But TrigFit[] has nothing to do with DFT.

Regards
  Jens


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