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A Fourier transform

  • Thread starter WarnK
  • Start date
  • #1
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Hi!

I want to find the Fourier transform of

[tex] \int_{-\infty}^t f(s-t)g(s) ds [/tex].

The FT

[tex] \int_{-\infty}^t h(s) ds \rightarrow H(\omega)/i\omega + \pi H(0) \delta(\omega)[/tex]

is found in lots of textbooks. So if I let h(s) = f(s-t)g(s), I need to find the FT of h(s)

[tex] H(\omega) = \int_{-\infty}^{\infty} h(s) \frac{e^{-i\omega s}}{\sqrt(2\pi)}ds = \int_{-\infty}^{\infty} f(s-t)g(s) \frac{e^{-i\omega s}}{\sqrt(2\pi)} ds [/tex].

But there I'm stumped, what can I do? I can do FT of products like f(s)g(s), but this isn't exactly like that.
 
Last edited:

Answers and Replies

  • #2
Defennder
Homework Helper
2,591
5
Did you try the Fourier transform of a convolution?
 
  • #3
rbj
2,226
7
Did you try the Fourier transform of a convolution?
yeah, but the OP needs to reverse the "time" argument in f(.).

WarnK, where did you get that icky [itex]1/\sqrt{2 \pi}[/itex] definition for the F.T.?

i really recommend this definition:

[tex] X(f) = \int_{-\infty}^{+\infty} x(t) e^{-i 2 \pi f t} dt [/tex]

with inverse

[tex] x(t) = \int_{-\infty}^{+\infty} X(f) e^{i 2 \pi f t} df [/tex]

while remembering that [itex] \omega \equiv 2 \pi f [/itex], and when comparing to the double-sided Laplace Transform to substitute [itex] i \omega \leftarrow s [/itex].

it will make your life much easier.
 
  • #4
Defennder
Homework Helper
2,591
5
I'm rather new to this myself. So the answer differs by a minus sign?
 

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