Example of Computation of inverse Fourier transform (CT signals)
A practice problem on CT Fourier transform
Specify a Fourier transform X(w) and compute its inverse Fourier transform using the integral formula. (Make sure your signal is not trivial to transform; it should be hard enough to be on a test).
Define X(w):
$ \mathcal{X}(\omega) = 4 \pi \delta(\omega - 3) + 4 \pi \delta(\omega + 3) - 8 \pi \delta(\omega - 7) $
By the integral formula:
$ x(t)= \frac{1}{2 \pi} \int_{-\infty}^{\infty} \mathcal{X}(\omega) e^{-j\omega t}\,d \omega $
Therefore:
$ x(t)= \frac{1}{2 \pi} \int_{-\infty}^{\infty} (4 \pi \delta(\omega - 3) + 4 \pi \delta(\omega + 3) - 8 \pi \delta(\omega - 7)) e^{-j\omega t}\,d \omega $
$ x(t)= \frac{1}{2 \pi} \int_{-\infty}^{\infty} 4 \pi \delta(\omega - 3) e^{-j\omega t}\,d \omega + \frac{1}{2 \pi} \int_{-\infty}^{\infty} 4 \pi \delta(\omega + 3) e^{-j\omega t}\,d \omega - \frac{1}{2 \pi} \int_{-\infty}^{\infty} 8 \pi \delta(\omega - 7) e^{-j\omega t}\,d \omega $
Pull out the constants:
$ x(t)= \frac{4 \pi}{2 \pi} \int_{-\infty}^{\infty} \delta(\omega - 3) e^{-j\omega t}\,d \omega + \frac{4 \pi}{2 \pi} \int_{-\infty}^{\infty} \delta(\omega + 3) e^{-j\omega t}\,d \omega - \frac{8 \pi}{2 \pi} \int_{-\infty}^{\infty} \delta(\omega - 7) e^{-j\omega t}\,d \omega $
Simplifying:
$ x(t)= 2 \int_{-\infty}^{\infty} \delta(\omega - 3) e^{-j\omega t}\,d \omega + 2 \int_{-\infty}^{\infty} \delta(\omega + 3) e^{-j\omega t}\,d \omega - 4 \int_{-\infty}^{\infty} \delta(\omega - 7) e^{-j\omega t}\,d \omega $
Remember that:
$ \int_{-\infty}^{\infty} \delta(\omega - T_0) e^{-j\omega t}\,d \omega = e^{-j T_0 t} $
Therefore:
$ \ x(t)= 2e^{-j3t} + 2e^{j3t} -4e^{-j7t} $
Recall that:
$ Cos(\omega t) = \frac{e^{j \omega t} + e^{-j \omega t}}{2} $
Therefore:
$ \ x(t) = 4cos(3t) - 4e^{-j7t} $
