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MathGroup Archive 2005

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Re: FourierTransform

  • To: mathgroup at smc.vnet.net
  • Subject: [mg57614] Re: FourierTransform
  • From: "Mariusz Jankowski" <mjankowski at usm.maine.edu>
  • Date: Thu, 2 Jun 2005 05:16:51 -0400 (EDT)
  • Organization: University of Southern Maine
  • References: <d7k36h$oi1$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Panie Marcinie, 

The two forms of evaluation give different results because the Fourier
integral does not converge for the case of the unit step function (violates
the so-called Dirichlet conditions) and thus cannot be obtained by direct
evaluation of the integral. However, using the integration property of the
FT you can, at least indirectly, obtain the result you show below (Oppenheim
and Willsky, "Signals and Systems," page 307).


Pozdrawiam, Mariusz





>>> Marcin Rak<umrakmm at cc.umanitoba.ca> 06/01/05 6:38 AM >>>

Hi,

I was wondering what the exact mathematica equation of the following 
Mathematica Command was:

FourierTransform[UnitStep[t],t,w,FourierParameters->{1,-1}]

This gives -i/w + pie*DiracDelta[w]

which is correct.  However, when I substitute my UnitStep[t] function 
into the direct definition employed by FourierTransform, I don't get the 
same result.  ie

Integrate[UnitStep[t]*Exp[-i*w*t],{t,-infinity,infinity}] Doesn't give 
the same result, despite the fact that in section 3.5.11 of the 
Mathematica book it is defined as such?

Thanks
MR





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