Fourier Transform of integral of a signal

  • #1
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Main Question or Discussion Point

Hi. I have a question regarding the continuous time Fourier Transform of an input signal:

[tex]x(t) \rightarrow X(j\omega)[/tex]

then

[tex]\int_{-\infty}^{t}x(\tau)d\tau \rightarrow \frac{X(j\omega)}{j\omega} + \pi X(0)\delta(\omega)[/tex]

but if I want to write it in terms of [itex]f = \frac{\omega}{2\pi}[/itex], should it be:

[tex]\int_{-\infty}^{t}x(\tau)d\tau \rightarrow \frac{X(j\omega)}{j\omega} + \frac{1}{2}X(0)\delta(f)[/tex]

How does the [itex]\pi[/itex] get replaced by [itex]\frac{1}{2}[/itex] here?
 

Answers and Replies

  • #2
George Jones
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Science Advisor
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How does the [itex]\pi[/itex] get replaced by [itex]\frac{1}{2}[/itex] here?[/QUOTE]

For [itex]a>0[/itex], what does [itex]\delta (ax ) [/itex] equal? Why?
 
  • #3
1,789
4
Oh ok, so

[itex]\delta(\omega) = \delta(2\pi f) = \frac{1}{2\pi}\delta(f)[/itex]

Thanks :cool:
 

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