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| − | <math> x(t)={e^{- | + | Computer the Fourier Transform of the Following: |
| − | + | <math>\, x(t)={e^{-2|t|}, |t|<1}\,</math> | |
| + | <math>\, x(t)=0, |t|>1 \,</math> | ||
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| + | <math>\,\mathcal{X}(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt\,</math> | ||
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| + | <math>\,\mathcal{X}(\omega)=\int_{-1}^{0}e^{2t}e^{-j\omega t}\,dt\ + \int_{0}^{1}e^{-2t}e^{-j\omega t}\,dt\,</math> | ||
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| + | <math>\,\mathcal{X}(\omega)=\int_{-1}^{0}e^{t(2-j\omega)}\,dt\ + \int_{0}^{1}e^{-t(2+j\omega)}\,dt\,</math> | ||
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| + | <math>\,\mathcal{X}(\omega)={\left. \frac{e^{t(2-j\omega )}}{(2-j\omega )}\right]_{-1}^{0} + {\left. \frac{e^{-t(2+j\omega )}}{(2+j\omega )}\right]_0^{1}}}\,</math> | ||
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| + | <math>\,\mathcal{X}(\omega)=\frac{1}{2-j\omega} - \frac{e^{-(2-j\omega)}}{2-j\omega} + \frac{e^{-(2+j\omega)}}{2+j\omega} - \frac{1}{2+j\omega} \,</math> | ||
Revision as of 15:44, 8 October 2008
Computer the Fourier Transform of the Following:
$ \, x(t)={e^{-2|t|}, |t|<1}\, $ $ \, x(t)=0, |t|>1 \, $
$ \,\mathcal{X}(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt\, $
$ \,\mathcal{X}(\omega)=\int_{-1}^{0}e^{2t}e^{-j\omega t}\,dt\ + \int_{0}^{1}e^{-2t}e^{-j\omega t}\,dt\, $
$ \,\mathcal{X}(\omega)=\int_{-1}^{0}e^{t(2-j\omega)}\,dt\ + \int_{0}^{1}e^{-t(2+j\omega)}\,dt\, $
$ \,\mathcal{X}(\omega)={\left. \frac{e^{t(2-j\omega )}}{(2-j\omega )}\right]_{-1}^{0} + {\left. \frac{e^{-t(2+j\omega )}}{(2+j\omega )}\right]_0^{1}}}\, $
$ \,\mathcal{X}(\omega)=\frac{1}{2-j\omega} - \frac{e^{-(2-j\omega)}}{2-j\omega} + \frac{e^{-(2+j\omega)}}{2+j\omega} - \frac{1}{2+j\omega} \, $
