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December 16th, 2015, 02:03 PM   #1
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Joined: Oct 2014
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Math Focus: Calculus
Proof of Fourier Transform and Inverse Transform


Hi have some doubt in computing the proof of:

$\displaystyle (1)\qquad \mathcal{F}[\mathrm e^{\mathrm i\omega_0t}] = 2\pi \delta(\omega -\omega_0)$

And also its inverse:

$\displaystyle (2)\qquad\mathcal{F}[\mathrm e^{\mathrm i\omega_0}]^{-1} = \delta(t- t_0)$

Eq(2) is taken from the book by Oppenheim, Signals and Systems 2nd Ed.
Simulating the results with an online solver for the equation (2), it results that equation (2) should be:

$\displaystyle (2.1)\qquad\mathcal{F}[\mathrm e^{\mathrm i\omega_0}]^-1 = \sqrt{2 \pi} \delta(t- t_0)$

So who is right and who is wrong ?
Also, it's not clear to me where does the $\displaystyle 2 \pi$ comes from in equation (1), the same for the $\displaystyle \sqrt{2 \pi}$ in equation (2.1).

For example, I would solve equation (1) as follows:

$\displaystyle \begin{aligned}
\mathcal F[\mathrm e^{\mathrm i\omega_0 t}] &= \int_{-\infty}^{\infty}\mathrm e^{\mathrm i\omega_0 t}\mathrm e^{-\mathrm i\omega t}\,\mathrm dt\\
& = \int_{0}^{2 \pi}\mathrm e^{\mathrm it(\omega_0 - \omega)}\,\mathrm dt\\
& = \left [ {\mathrm e^{\mathrm it(\omega_0 - \omega)} \over \mathrm i (\omega_0 - \omega)}\right ]_0^{2\pi}\\
& = {\mathrm e^{\mathrm i 2 \pi (\omega_0 - \omega)} - 1 \over \mathrm i (\omega_0 - \omega)}

Which is wrong or incomplete ...
The same happens to me for the Inverse Fourier Transform:

$\displaystyle \begin{aligned}
\mathcal F[\mathrm e^{\mathrm i\omega_0}]^{-1} &= {1 \over 2\pi}\int_{-\infty}^{\infty}\mathrm e^{\mathrm i\omega_0}\mathrm e^{\mathrm i\omega t}\,\mathrm d\omega\\
& = {\mathrm e^{\mathrm i\omega_0} \over 2\pi} \int_{0}^{2\pi}\mathrm e^{\mathrm i\omega t}\,\mathrm d\omega\\
& = {\mathrm e^{\mathrm i\omega_0} \over 2\pi}\left[ {\mathrm e^{\mathrm i\omega t} \over \mathrm i t}\right ]_0^{2\pi}\\
& = {\mathrm e^{\mathrm i\omega_0} \over 2\pi}\left ( {\mathrm e^{\mathrm i 2\pi t} - 1 \over \mathrm i t}\right )

from which I am not able to determine the $\displaystyle \delta(t)$ neither the coefficient $\displaystyle \sqrt{2 \pi}$.

I don't understand where I am wrong.. If I am so..

Thank you in advance for your help.
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