Example of Computation of inverse Fourier transform (CT signals)
A practice problem on CT Fourier transform
Compute the inverse fourier transform of the fourier transform below:
$ \,\mathcal{X}(\omega)= \delta(\omega - 3\pi) e^{-t}\, $
$ \,x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t}\,d\omega \, $
$ \,x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} \delta(\omega - 3\pi) e^{-t} e^{j\omega t}\,d\omega \, $
$ \,x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty} \delta(\omega - 3\pi) e^{(j\omega - 1)t}\,d\omega \, $
$ \,x(t)=\frac{1}{2\pi} e^{(j(-3\pi) - 1)t}\, $
