Re: Hilbert Transform problems

• To: mathgroup at smc.vnet.net
• Subject: [mg73988] Re: Hilbert Transform problems
• From: Peter Pein <petsie at dordos.net>
• Date: Sun, 4 Mar 2007 02:04:39 -0500 (EST)
• References: <es90iu\$2id\$1@smc.vnet.net>

```rob schrieb:
...
> 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)
>

Hi Rob,

1.) Method->Oscillatory is an option for NIntegrate only (and ProncipalValue
is for Integrate only). I wonder why you didn't get error messages.

There is a numeric function CauchyPrincipalValue available:

<<NumericalMath`CauchyPrincipalValue`
HilbertTransform[f_,x_,y_,r_,(assum___)?OptionQ]:=
CauchyPrincipalValue[f/(x-y),{x,-Infinity,{y,r},Infinity},assum]/Pi;
w=5;a=1;
s[t_]:=Sin[w*t]*Exp[-(t/a)^2];
Plot[s[t],{t,-3,3},PlotRange\[Rule]All];
f[y_?NumericQ,r_?NumericQ]:=HilbertTransform[s[t],t,y,r];
Plot[f[y,1/10],{y,-3,3},PlotRange\[Rule]All];

Peter

```

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