Example of Computation of inverse Fourier transform (CT signals)
A practice problem on CT Fourier transform
Compute the Inverse Fourier transform of:
$ \mathcal{X}(\omega)=3\pi \delta (\omega-\pi) $
Using the Formula for Inverse Fourier Transforms:
$ \mathcal{F}^{-1}(\mathcal{X}(\omega))= x(t)= \frac{1}{2\pi}\int_{-\infty}^{\infty}\mathcal{X}(\omega)e^{j\omega t} \,d\omega $
So:
$ x(t)= \frac{1}{2\pi}\int_{-\infty}^{\infty}3\pi \delta (\omega-\pi)e^{j\omega t} \,d\omega $ Using sifting property:
$ =\frac{3\pi}{2\pi}e^{j\pi t}=1.5e^{j\pi t} $
