Example of Computation of Fourier transform of a CT SIGNAL

A practice problem on CT Fourier transform


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} + \frac{e^{-2*\infty - \infty j \omega} - e^{0}}{-j -j\omega}\, $

$ = \frac{e^{j(15 - 5\omega )} - e^{-j(15 - 5\omega )}}{j(3-\omega )} + \frac{1}{j(1-\omega )}\, $

$ = \frac{2sin(15 - 5\omega )}{3-\omega } + \frac{1}{j(1-\omega )}\, $


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