A LTI system takes $x(t)$ as input and $y(t)$ as output. Given $$y(t)=\int_{t-5}^{t-1}2x(\tau)d\tau$$, how to find $h(t)$, the impulse response?
Convolution $x(t)*h(t)=\int_{-\infty}^\infty x(\tau)h(t-\tau)d\tau$, so by comparing this to the given output, $h(t)$ is $2\delta(t)$, is this correct?
Best Answer
Hint: Convolution is commutative, so use the variable substitution $\tau = t-\tau'$. Don't forget to change the bounds, too. Can you see what function is integrated against $x$ now?