Example of Computation of inverse Fourier transform (CT signals)
A practice problem on CT Fourier transform
$ X(\omega) = 2\delta (\omega - 3) + 7\pi \delta(\omega - 1) \! $ ,IFT: $ x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega )e^{j\omega t} d\omega \! $
$ x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} 2\delta (\omega -3)e^{j\omega t} d\omega + \frac{1}{2\pi} \int_{-\infty}^{\infty} 7\pi \delta (\omega -1)e^{j\omega t} d\omega \! $
$ \int_{-\infty}^{\infty} \delta (\omega -t_0) e^{jwt} d\omega = e^{jt_0 t} \! $
$ x(t) = \frac{2}{2\pi }e^{3jt} + \frac{7\pi }{2\pi }e^{jt} \! $
$ x(t) = \frac{1}{\pi }e^{3jt} + \frac{7}{2}e^{jt} \! $
