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| − | Let <font size = '4'><math>\chi (w) = \pi \delta (w - 6) - \pi \delta (w | + | Let <font size = '4'><math>\chi (w) = \frac{\pi}{j} 4\delta (w - 6) - \frac{\pi}{j} 4\delta (w + 6)</math></font> |
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| + | Then <math>x(t) = \frac{1}{2\pi}\int^{\infty}_{-\infty} \chi (w) e^{jwt}dw</math> | ||
| + | |||
| + | <math>x(t) = \frac{1}{2\pi} [\frac{4\pi}{j}\int^{\infty}_{-\infty} \delta(w-6)e^{jwt} dw - \frac{4\pi}{j} \int_{-\infty}^{\infty} \delta(w+6)e^{jwt} dw]</math> | ||
| + | |||
| + | <math>x(t) = \frac{2}{j}e^{j6t} - \frac{2}{j}e^{-j6t} = 4[\frac{e^{j6t} - e^{-j6t}}{2j}] = 4sin(6t)</math> | ||
Revision as of 13:54, 8 October 2008
Let $ \chi (w) = \frac{\pi}{j} 4\delta (w - 6) - \frac{\pi}{j} 4\delta (w + 6) $
Then $ x(t) = \frac{1}{2\pi}\int^{\infty}_{-\infty} \chi (w) e^{jwt}dw $
$ x(t) = \frac{1}{2\pi} [\frac{4\pi}{j}\int^{\infty}_{-\infty} \delta(w-6)e^{jwt} dw - \frac{4\pi}{j} \int_{-\infty}^{\infty} \delta(w+6)e^{jwt} dw] $
$ x(t) = \frac{2}{j}e^{j6t} - \frac{2}{j}e^{-j6t} = 4[\frac{e^{j6t} - e^{-j6t}}{2j}] = 4sin(6t) $
