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Hilbert Transform problems

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  • Subject: [mg73851] Hilbert Transform problems
  • From: rob <robIV at>
  • Date: Fri, 2 Mar 2007 06:26:07 -0500 (EST)

Hi, I'm wondering if someone can help me produce a useful 
result on the Hilbert transform of what I call a wideband 
wavelet. I got the basic HilbertTransform definition off the 
Wolfram site. It seems to me the transform of this wavelet 
exists (it becomes a symmetrical wavelet) but I can only get 
a very spiky result where the true result can barely be seen 
under the noise. The procedure seems to be having all kinds 
of problems dealing with singularities. I added the 
Method->Oscillatory because without it, the result appears 
to be only noise. I've tried all kinds of Methods and other 
options but none have helped - most make it worse.

I'd like to say that in the past I have received many very 
helpful things here but when I try to repost a reply with 
thanks and explanations, I never see it posted. But 
sometimes a new posting does make it. Please let me post my 
thanks here in advance.

My current best code with all converted to "Input Form" is 
below. (Why my notebook does not put my entries in Input 
Form -they are in Standard Form --I've never figured out.)

5.1 for Microsoft Windows (January 27, 2005)

HilbertTransform[f_, x_, y_, (assum___)?OptionQ] :=
   Integrate[f/(x - y), {x, -Infinity, Infinity}, Method -> 
Oscillatory, PrincipalValue -> True, assum]/Pi

w = 5.; a = 1.;
s[t_] := Sin[w*t]*Exp[-(t/a)^2];
Plot[s[t], {t, -10, 10},PlotRange->All]

f[y_] = HilbertTransform[s[t], t, y]

Plot[f[x], {x, -3, 3}]

(for email, remove the IV)

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