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 szz December 16th, 2015 01:03 PM

Proof of Fourier Transform and Inverse Transform

Hi,

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). :confused:

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)} \end{aligned}

Which is wrong or incomplete ... :confused:
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 ) \end{aligned}

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..