Revision as of 17:49, 8 October 2008 by Drecord (Talk)

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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) \! $

Alumni Liaison

BSEE 2004, current Ph.D. student researching signal and image processing.

Landis Huffman