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<math> = \frac{e^{3jt - j\omega t}}{3j-j\omega}]_{-5}^{5} + \frac{e^{-2t - j\omega t}}{-2 -j\omega}]_{0}^{\infty}\,</math> | <math> = \frac{e^{3jt - j\omega t}}{3j-j\omega}]_{-5}^{5} + \frac{e^{-2t - j\omega t}}{-2 -j\omega}]_{0}^{\infty}\,</math> | ||
| + | |||
| + | <math> = \frac{e^{15j - 5j\omega} - e^{-15j + 5j\omega}}{3j-j\omega}\,</math> | ||
Revision as of 11:28, 7 October 2008
Signal
$ x(t) = e^{3jt}*(u(t+5) - u(t-5)) + e^{-2t}*u(t)\, $
Transformed
$ X(\omega) = \int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt\, $
$ = \int_{-\infty}^{\infty}e^{3jt}*(u(t+5) - u(t-5))e^{-j\omega t}dt + \int_{-\infty}^{\infty}e^{-2t}u(t)e^{-j\omega t}dt\, $
$ = \int_{-5}^{5}e^{3jt}e^{-j\omega t}dt + \int_{0}^{\infty}e^{-2t}e^{-j\omega t}dt\, $
$ = \int_{-5}^{5}e^{3jt -j\omega t}dt + \int_{0}^{\infty}e^{-2t -j\omega t}dt\, $
$ = \int_{-5}^{5}e^{t*(3j -j\omega )}dt + \int_{0}^{\infty}e^{t*(-2 -j\omega )}dt\, $
$ = \frac{e^{3jt - j\omega t}}{3j-j\omega}]_{-5}^{5} + \frac{e^{-2t - j\omega t}}{-2 -j\omega}]_{0}^{\infty}\, $
$ = \frac{e^{15j - 5j\omega} - e^{-15j + 5j\omega}}{3j-j\omega}\, $
