Re: Fourier and InverseFourier

• To: mathgroup at smc.vnet.net
• Subject: [mg75589] Re: Fourier and InverseFourier
• From: rob <josh2499 at hotmail.com>
• Date: Sun, 6 May 2007 01:45:22 -0400 (EDT)
• References: <f0v61b\$8u4\$1@smc.vnet.net> <f11gpl\$kph\$1@smc.vnet.net> <200705020756.DAA05199@smc.vnet.net> <f1c3qr\$gou\$1@smc.vnet.net>

```Daniel Lichtblau wrote:
> rob wrote:
>
>>Hi, thanks for responding. No, I'm not sure it exists. I
>>tried Exp[-t^2] and it doesn't work either. I haven't yet
>>found a case where InverseFourierTransform[] works so I
>>suspect I'm still doing something wrong.
>>
>>Jens-Peer Kuska wrote:
>>
>>
>>>Hi,
>>>
>>>and you are sure that
>>>
>>>FourierTransform[Exp[-t], t, w]
>>>
>>>is exist ? Because
>>>
>>>Integrate[Exp[-t]*Exp[I*w*t], {t, -Infinity, Infinity}]/Sqrt[2Pi]
>>>
>>>gives the correct error message that the integral does not converge
>>>and in general Fourier transforms are only defined for quadratic
>>>integrable functions and Exp[-t] is not quadratic integrable.
>>>
>>>Regards
>>>  Jens
>>>
>>>rob wrote:
>>>
>>>
>>>
>>>>I kind person on this ng (Gulliet) recently contributed a
>>>>convolution scheme which works nicely to plot x2 below:
>>>>
>>>>conv[f1_, f2_] := Module[{u}, Evaluate[Integrate[f1[u] f2[#
>>>>- u], {u, 0, #}]] &]
>>>>
>>>>x2[t_] := convolve[Sin[t], Exp[-t]][t]
>>>>
>>>>Plot[x2[t], {t, 0, 15}, PlotRange -> All]
>>>>
>>>>Wondering if I could achieve the same thing in the freq.
>>>>domain, I tried what I thought should give the same result
>>>>in x3:
>>>>
>>>>fs = FourierTransform[Sin[t], t, w]
>>>>fe = FourierTransform[Exp[-t], t, w]
>>>>
>>>>x3[t_] := InverseFourierTransform[fs*fe, w, t]
>>>>
>>>>Plot[x3[t], {t, 0, 15}, PlotRange -> All]
>>>>
>>>>I find this does not work, getting this err message and Mathematica
>>>>(v.5.1) didn't stop in over 30 minutes.
>>>>
>>>>NIntegrate::ploss: Numerical integration stopping due to
>>>>loss of precision. Achieved neither the requested
>>>>PrecisionGoal nor AccuracyGoal; suspect one of the
>>>>following: highly oscillatory integrand or the true value of
>>>>the integral is 0. If your integrand is oscillatory on a
>>>>(semi-)infinite interval try using the option
>>>>Method->Oscillatory in NIntegrate.
>>>>
>>>>Since I'm using the internal integrals of
>>>>InverseFourierTransform I don't know how to try the
>>>>suggestion of Method->Oscillatory as the message suggests.
>>>>
>>>>I changed the Sin[t] to t and the process gave no err
>>>>messages and finished in just a few minutes. The plot had
>>>>axes but nothing on it.
>>>>
>>>>Can someone give me any hints as what might work?
>
>
>
> Your explicit convolution integrates from 0 to t. Your attempt with
> FT/IFT involves integrations from -infinity to infinity. In order to use
>   FT/IFT you'd need to have cutoff multipliers such as UnitStep, to get
> results comparable to the explicit code. Also for functions like Exp[-t]
> (with no cutoff) the FT does not exist because it grows too fast at
> -infinity.
>
>
> Daniel Lichtblau
> Wolfram Research
>

Sir, thank you for the suggestion. I added the UnitStep[t]
as below:

fs = FourierTransform[UnitStep[t]*Sin[t], t, w]
fe = FourierTransform[UnitStep[t]*Exp[-t], t, w]

x3[t_] := InverseFourierTransform[fs*fe, w, t]

Plot[x3[t], {t, 0, 15}, PlotRange -> All]

The x3 now computes and plots but gives a narrowband sine
output, different from the original convolution which is a
wide band pulse (looks like a Gaussian pulse with an
undershoot). So at least I'm getting an answer. Any
suggestion which might yield the same result as the original
convolution -- which I repeat below for clarity?

conv[f1_, f2_] := Module[{u}, Evaluate[Integrate[f1[u] f2[#
- u], {u, 0, #}]] &]

x2[t_] := convolve[Sin[t], Exp[-t]][t]

Plot[x2[t], {t, 0, 15}, PlotRange -> All]

Thanks again, Rob

```

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