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MathGroup Archive 2004

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Re: Problem with function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48318] Re: [mg48306] Problem with function
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Mon, 24 May 2004 00:45:12 -0400 (EDT)
  • Reply-to: hanlonr at cox.net
  • Sender: owner-wri-mathgroup at wolfram.com

I assume that the function is intended to be zero outside of the defined 
region.  Then using UnitStep to define g[t]

g[t_] := t*UnitStep[t]+3(1-t)/2*UnitStep[t-1]-(3-t)/2*UnitStep[t-3];

Plot[g[t], {t,-1,4}, PlotRange->All];

Simplify[g[t], t<0]

0

Simplify[g[t], 0<=t<1]

t

Simplify[g[t], 1<=t<3]//Expand//N

1.5 - 0.5*t

Simplify[g[t], 3<=t]

0

Taking the inverse transform of the Fourier transform will provide an 
equivalent but simpler expression for g[t]

g2[t_] := Evaluate[
      InverseFourierTransform[
        FourierTransform[g[t], t, w],w,t]];

g2[t]

(1/4)*((t - 3)*Sign[t - 3] - 3*(t - 1)*Sign[t - 1] + 
   2*t*Sign[t])

Plot[g2[t],{t,-1,4}, PlotRange->All];

Simplify[g2[t], t<0]

0

FullSimplify[g2[t], 0<=t<1]

t

FullSimplify[g2[t], 1<=t<3]//Expand//N

1.5 - 0.5*t

FullSimplify[g2[t], 3<=t]

0

The Fourier transform is then

FourierTransform[g2[t], t, w]//InputForm

-((-1 + E^(I*w))^2*(2 + E^(I*w)))/(2*Sqrt[2*Pi]*w^2)

InverseFourierTransform[%,w,t]==g2[t]

True


Bob Hanlon

> 
> From: "DJkapi" <djkapi at poczta.onet.pl>
To: mathgroup at smc.vnet.net
> Date: 2004/05/23 Sun AM 06:15:34 EDT
> To: mathgroup at smc.vnet.net
> Subject: [mg48318] [mg48306] Problem with function
> 
> How to compute in Mathematica  fourier transform  of function:
> 
>             t     ,0=< t =<1
> g(t)=
>             1.5-0.5t , 1< t =<3
> 
> to tell the truth i dont even know how to make Mathematica print that
> function.
> 
> Regards,
> DJKapi
> 
> 
> 

Bob Hanlon
Chantilly, VA


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