(New page: <math>\,\mathcal{X}(\omega)= \frac{\pi}{j} \delta (w - 2\pi) + \frac{2\pi}{j} \delta (w + 2\pi)</math>) |
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| + | Compute the Inverse Fourier Transform of: | ||
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<math>\,\mathcal{X}(\omega)= \frac{\pi}{j} \delta (w - 2\pi) + \frac{2\pi}{j} \delta (w + 2\pi)</math> | <math>\,\mathcal{X}(\omega)= \frac{\pi}{j} \delta (w - 2\pi) + \frac{2\pi}{j} \delta (w + 2\pi)</math> | ||
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| + | <math>x(t) = \frac{1}{2\pi}\int^{\infty}_{-\infty} \mathcal{X} (\omega) e^{jwt}dw</math> | ||
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| + | <math>x(t) = \frac{1}{2\pi} \frac{\pi}{j}\int^{\infty}_{-\infty} \delta(w-2\pi)e^{jwt} dw + \frac{1}{2\pi}\frac{2\pi}{j} \int_{-\infty}^{\infty} \delta(w+2\pi)e^{jwt} dw]</math> | ||
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| + | <math>x(t) = \frac{1}{2j}e^{j2\pi t} + \frac{1}{j}e^{-j2\pi t} | ||
Revision as of 16:18, 8 October 2008
Compute the Inverse Fourier Transform of:
$ \,\mathcal{X}(\omega)= \frac{\pi}{j} \delta (w - 2\pi) + \frac{2\pi}{j} \delta (w + 2\pi) $
$ x(t) = \frac{1}{2\pi}\int^{\infty}_{-\infty} \mathcal{X} (\omega) e^{jwt}dw $
$ x(t) = \frac{1}{2\pi} \frac{\pi}{j}\int^{\infty}_{-\infty} \delta(w-2\pi)e^{jwt} dw + \frac{1}{2\pi}\frac{2\pi}{j} \int_{-\infty}^{\infty} \delta(w+2\pi)e^{jwt} dw] $
$ x(t) = \frac{1}{2j}e^{j2\pi t} + \frac{1}{j}e^{-j2\pi t} $
