Help with Inverse Fourier Transform Integral

  • Thread starter eloso
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Hi, I am taking a random process class and I came across a problem that has stumped me. I believe I know the end result but I would like to know how it is solved. I have been out of college for a while and I am a little rusty with integration.

Homework Statement



What I need is to find out the inverse Fourier transform of S(ω)=1/((1+ω^2)^2)


Homework Equations



I know that F^-1(S(ω)) = 1/(2∏) * ∫(S(ω)*e^(j*ω*t), ω,-∞,∞) where j is an imaginary number.

→ F^-1(S(ω)) = ∫(1/((1+ω^2)^2)*e^(j*ω*t), ω,-∞,∞)

The Attempt at a Solution



I tried to solve it using by parts, but it got very messy and I may be heading the wrong path. This is what I did:

u = (1+ω^2)^-2 du = -4*ω*(1+ω^2)^-3
dv = e^(j*ω*t) v = e^(j*ω*t)/(j*t)


I plugged it in to the by parts equation and I felt like going on wouldn't help. I would appreciate any help. Thanks.
 
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Answers and Replies

  • #2
Dick
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Homework Helper
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Hi, I am taking a random process class and I came across a problem that has stumped me. I believe I know the end result but I would like to know how it is solved. I have been out of college for a while and I am a little rusty with integration.

Homework Statement



What I need is to find out the inverse Fourier transform of S(ω)=1/((1+ω^2)^2)


Homework Equations



I know that F^-1(S(ω)) = 1/(2∏) * ∫(S(ω)*e^(j*ω*t), ω,-∞,∞) where j is an imaginary number.

→ F^-1(S(ω)) = ∫(1/((1+ω^2)^2)*e^(j*ω*t), ω,-∞,∞)

The Attempt at a Solution



I tried to solve it using by parts, but it got very messy and I may be heading the wrong path. This is what I did:

u = (1+ω^2)^-2 du = -4*ω*(1+ω^2)^-3
dv = e^(j*ω*t) v = e^(j*ω*t)/(j*t)


I plugged it in to the by parts equation and I felt like going on wouldn't help. I would appreciate any help. Thanks.
Mmm. Elementary integration techniques like integration by parts are going to get you nowhere here. You need to use complex analysis for integrals like this (which is pretty typical for Fourier integrals). (1+w^2)=(1+jw)(1-jw). So you have second order poles at +j and -j. You need to use a contour integral and the residue theorem. Any experience with that??
 
  • #3
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Unfortunately I don't, but I can definitely research the theorems. I appreciate the response.
 

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