Hello all I am having some trouble with the following convolution integral
$$h(t) = e^{-t} u(t) ~,~x(t) = e^{-2t}u(t-3)$$
I see that to evaluate that I need to first shift x(t) by t-3
$$e^{-2(t-3)}u(t-3) $$
After which I can perform the convolution of the by modifying h(t) since it is a simpler function
$$\int_{3}^{t} e^{-2(\tau-3)}u(\tau-3)~e^{\tau+t}u(t-\tau)d\tau$$
After integrating I end up with the following answer:
$$(e^{9-t}-e^{6})u(t-3)$$
I dont feel that this answer is correct, can someone point out where my mistake is? Thanks in advanced
Best Answer
$$\int_{-\infty}^{\infty} x(\tau) h(t-\tau)d\tau$$
$$=\int_{-\infty}^{\infty} e^{-2\tau}u(\tau-3)~e^{-t+\tau}u(t-\tau)d\tau$$
$$=\int_{-\infty}^{\infty} e^{-(t+\tau)}u(\tau-3)u(t-\tau)d\tau$$
There are two cases:
Case $1$: If $t < 3$, then the answer is $0$.
Case $2$: If $t \geq 3$, then:
$$=\int_{3}^{t} e^{-(t+\tau)}d\tau=e^{-t}(e^{-3}-e^{-t})$$
In a simple way, the answer is $e^{-t}(e^{-3}-e^{-t})u(t-3).$