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Re: FourierTransform[Sign[t], t, w, FourierParameters -> {1, -1}]

• To: mathgroup at smc.vnet.net
• Subject: [mg21399] Re: FourierTransform[Sign[t], t, w, FourierParameters -> {1, -1}]
• From: "Jordan Rosenthal" <jr at ece.gatech.edu>
• Date: Tue, 4 Jan 2000 02:12:32 -0500 (EST)
• Organization: Georgia Institute of Technology, Atlanta GA, USA
• References: <84eme9$60s@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Hi,

Thanks to all for the various responses both here on the newsgroup and via
private email.

Conclusion:
-------------------------------
There appears to be a bug that occurs when using the FourierParameters with
a negative value for b.

A possible workaround:
-----------------------------------
If the signal is real, use positive value for b and conjugate the result.
If the signal is complex, also conjugate the signal before taking the
transform.

Thanks again,

- Jordan


"Jordan Rosenthal" <jr at ece.gatech.edu> wrote in message
news:84eme9$60s at smc.vnet.net...
> Hi,
>
> I have only been using Mathematica for under a month and have a question.
I
> was trying to reproduce the results of some well known Fourier transforms
> (allowing distributions) in Mathematica.  For instance,
>
>     sgn(t)     <---------->     2/(j*w)
>
> where j = sqrt(-1).  (See, for example "The Fourier Integral and its
> Applications", Papoulis).  So I ran the following code
>
>   FourierTransform[Sign[t], t, w, FourierParameters -> {1, -1}]
>
> and got the following result
>
>   \!\(\(-2\)\ \[Pi]\ DiracDelta[w] +
>       2\ \((\(-\(\[ImaginaryI]\/w\)\) - \[Pi]\ DiracDelta[w])\)\)
>
> Some questions:
>
> 1) Why does the answer not match what I expect?  Am I missing an
assumption
> somewhere or using something wrong?
>
> 2) In the result given by Mathematica, two of the terms obviously combine
so
> I tried to use  / /Simplify after the input, but it didn't simplify this
> rather "simple" term.  Why?  (I know I can use / /Expand to get it to
> simplify, but was just surprised that / /Simplify didn't work).
>
> Minor question:  Is there a better way to copy an output into a posting?
> All those backslashes seem to be a bit confusing.
>
> Thanks,
>
> - Jordan
>
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