Example of Computation of Fourier transform of a CT SIGNAL
A practice problem on CT Fourier transform
Let $ x(t) = e^{-a(t+1)} u(t + 1) $
$ \chi(w) = \mathcal{F} (x(t)) = \int^{\infty}_{-\infty} e^{-at}e^{-a} u(t + 1) e^{-jwt} dt $
$ = e^{-a} \int^{\infty}_{-1} e^{-at}.e^{-jwt} dt $
$ = e^{-a} \int^{\infty}_{-1}e^{-(a+jw)t} dt $
$ = -\frac{e^{-a}}{a+jw} [e^{-(a+jw)t}]^{\infty}_{1} $
$ = -\frac{e^{-a}}{a+jw} [-e^{-(a+jw)}] $
$ =\frac{e^{-(2a+jw)}}{a+jw} $
