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**halfnormalled** Hello, I've once again run into a small hurdle in my quest to learn maths on my own, with no one to talk to about it except for you fine folk! So If you would, please lend a hand again, I really do appreciate it!

The question is:

A linear, time invariant system has the impulse response $\displaystyle h(t) = e^{-et}u(t)$ find the system response to the input $\displaystyle x(t) = u(t) - u(t-3)$

Now, I know the answer is

$\displaystyle \frac{e^{-3t}}{3}\left(e^{9}-1\right)$

My answer is

$\displaystyle \frac{e^{-3t}}{3}\left(e^{3t}-1\right)$

Now I can see how to get to that answer. I'm taking the limits of the integral as follows:

$\displaystyle \int_0^t x(\tau)h(t-\tau) \, d\tau$

But I guess it should be

$\displaystyle \int_0^3 x(\tau)h(t-\tau) \, d\tau$

Can anyone help to explain why you need the upper limit to be three? Surely when $\displaystyle t$ is less than 3, you don't want $\displaystyle \tau$ to go beyond 3? But then in my example, you don't want $\displaystyle \tau$ to go beyond 3 when $\displaystyle t$ is more than 3... So that doesn't make sense really either...

(I do grasp the logic behind convolution, can visualise what is happening, and can implement it in some basic signal processing. It's just parts of the mathematical proof that get to me!)