Example of Computation of Fourier transform of a CT SIGNAL

A practice problem on CT Fourier transform


Specify a signal x(t) and compute its Fourier transform using the integral formula.( Make a hard one)

$ e^{-2(t-1)}u(t-1)\, $

$ \,\mathcal{X}(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt\, $

$ \,\mathcal{X}(\omega)= \int_{1}^{ \infty} e^{2-t(2+jw)}dt\, $

integrating and putting in limits

$ \,\mathcal{X}(\omega)= \frac{e^{2-(2+jw)}}{2+jw} \, $

$ \,\mathcal{X}(\omega)= \frac{e^{2-2-jw}}{2+jw} \, $

$ \,\mathcal{X}(\omega)= \frac{e^{-jw}}{2+jw} \, $


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