Impulse response of integrator

Question

I want to get the impulse response of an LTI system where $$y(t) = \int_{t-2T}^{t-T} x(\alpha) d\alpha $$

To solve this I did: $$h(t) = \int_{t-2T}^{t-T} \delta(\alpha) d\alpha $$

Then you see that for the integrator to be around the impulse $$ t > T $$ $$ t < 2T $$

To me the impulse respsonse should be $$ h(t)=\pi_{3T}(t-3T/2) $$ where pi is the rectangular function. Instead in the solution the answer is $$ h(t)=\pi_{T}(t-3T/2) $$ According to me the subscript of the rectangular function is the width of it. How can this be T?

Answer 1

You showed that your impulse response is $1$ for

$$ T < t < 2T $$

meaning that the length of its support is $ 2T - T = T$, and not $T + 2T = 3T$.