## Computing the Inverse Fourier Transform

$\ X(\omega)= 8 \pi w \delta(w-9) + 2 \pi w^{3} \delta(w-4 \pi)$

The inverse Fourier transform is defined as:

$x(t) = \int_{-\infty}^{\infty} \frac{X(w)}{2 \pi} e^{jwt} dw$

Using this formula to determine the signal:

$\ x(t) = \frac{8 \pi}{2 \pi} \int_{-\infty}^{\infty} w e^{jwt} \delta(w-9) dw + \frac{2}{2 \pi} \int_{-\infty}^{\infty}w^{3} \delta(w-4 \pi) e^{jwt} dw$

Now using the sifting property of the delta function we find that the signal is

$\ x(t) = 36 e^{j9t} + 64 \pi^{2} e^{j4\pi t}$

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Ph.D. 2007, working on developing cool imaging technologies for digital cameras, camera phones, and video surveillance cameras.