Problem
Compute the convolution
$ z(t)=x(t)*y(t) \ $
between
$ x(t) = e^{jwt}u(t+2) \ $
and
$ y(t) = e^{jwt}u(t-2) \ $.
My Solution
$ \begin{align} z(t) &= e^{jwt}u(t+2) * e^{jwt}u(t-2) \\ &= \int_{-\infty}^{\infty} e^{jw\tau}u(\tau+2) e^{jw(t- \tau)}u(t - \tau -2)\mathrm{d}\tau \\ &= \int_{-2}^{t-2} e^{jw\tau} e^{jw(t- \tau)}\mathrm{d}\tau \\ &= e^{jwt} \int_{-2}^{t-2} 1 \mathrm{d}\tau \\ &= \begin{cases} t e^{jwt} &, \text{when }t > 2 \\ 0 &, \text{else}\end{cases} \end{align} $
Comments
Write them here.
