Example of Computation of Fourier transform of a CT SIGNAL

A practice problem on CT Fourier transform


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} \, $


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