       Hilbert Transform problems

• To: mathgroup at smc.vnet.net
• Subject: [mg73851] Hilbert Transform problems
• From: rob <robIV at piovere.com>
• 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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