I have been trying to find the following inverse Fourier transform but without success:
$$ H(\omega) = \begin{cases} e^{-j \frac{\pi}{2}} & \omega \gt 0 \\ e^{j \frac{\pi}{2}} & \omega \lt 0 \\ \end{cases} $$
I have tried using the inverse Fourier transform but of course the $e^{j \omega t}$ won't converge.
I have also tried using the Fourier's transform property of duality given that $H(\omega)$ can be expressed as a sum of 2 unit steps multiplied by the constant $e^{\pm j \frac{\pi}{2}}$. That lead me to $\frac{1}{\pi t}$ which doesn't seem correct to me.
Restrictions on how to solve it:
- Don't use any kind of sorcery math that's not taught in an engineering/CS school.
- If you use the definition of Fourier Transform I would be glad if you used this one:
$X(\omega ) = \int\limits_{ - \infty }^{ + \infty } {x(t){e^{ - j\omega t}}dt}$ and $x(t) = {1 \over {2\pi }}\int\limits_{ - \infty }^{ + \infty } {X(\omega ){e^{j\omega t}}d\omega }$
Thanks.