Example of Computation of inverse Fourier transform (CT signals)
A practice problem on CT Fourier transform
Inverse Fourier Transform
$ X(\omega) = 4\delta (\omega - 3) + 5\pi \delta(\omega - 2) \! $
$ 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} 4\delta (\omega -3)e^{j\omega t} d\omega + \frac{1}{2\pi} \int_{-\infty}^{\infty} 5\pi \delta (\omega -2)e^{j\omega t} d\omega \! $
Since integrating dirac functions is extremely easy one can easily simplify to the following
$ x(t) = \frac{4}{2\pi }e^{3jt} + \frac{5\pi }{2\pi }e^{j2t} \! $ $ = \frac{2}{\pi }e^{3jt} + \frac{5 }{2 }e^{j2t} \! $
Check:
F($ \frac{2}{\pi }e^{3jt} + \frac{5 }{2 }e^{j2t} \! $) = $ X(\omega) = 4\delta (\omega - 3) + 5\pi \delta(\omega - 2) \! $
