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} \, $
