# Convolution integrals!

## Homework Statement

Right I'm having alot of problems with convolution in general. I'll give an example of a question that I understand and why I think I understand it... and then one I don't at all.

So:

Consider a system with the impulse response g(t) = 0 for t<0, $$e^{-5t}$$ for t $$\geq$$ 0.

Find the output for input f(t) = H(t) (step function).

So y(t) = $$\int g(t-\tau)f(\tau)d\tau$$ between t and $$-\infty$$

= $$\int e^{-5(t-\tau)}H(\tau)d\tau$$ between t and $$-\infty$$

= $$\int e^{-5(t-\tau)}.1.d\tau$$ between t and 0. This is because the step function takes a value of 1 for t >= 0 hence the limits change to 0 and t?

Now for the same system say you have an input of:

f(t) = (0, t<0) ..... (a) ; (v, 0<t<k) ..... (b); (0, t>k) .... (c)

To find the output you need to perform 3 integrals: 1 for (a), 1 for (b) and 1 for (c).

According to my course notes these 3 integrals are:

1. part (a):

y(t) = $$\int g(t-\tau)0d\tau$$ between t and $$-\infty$$

This sort of makes sense but why isn't the upper limit 0 as surely (a) is only for (t<0).

2. part (b):
y(t) = $$\int g(t-\tau)0d\tau$$ between 0 and $$-\infty$$
+ $$\int g(t-\tau)vd\tau$$ between t and 0.

I don't understand this bit. Firstly, why the initial bit between 0 and -infty ... surely it's unnecessary as the amplitude is 0 as in the previous example for the step input we didn't bother with t<0. Secondly for the second bit with amplitude v why on earth are the limits t and 0. Surely they should be k and 0 as the amplitude is only v between k and 0?

3. part (c):

y(t) = $$\int g(t-\tau)0d\tau$$ between 0 and $$-\infty$$
+ $$\int g(t-\tau)vd\tau$$ between k and 0.
+ $$\int g(t-\tau).0.d\tau$$ between t and k.

How are the first 2 lines in this integral even relevant to part (c) which is only for t>k where the amplitude is 0 (ie the way I see it only the third line is relevant).

Sorry for the rather unconventional post but this has been confusing me for a few months and having spent hours on it I just can't seem to understand what on earth is going on! Any help/feedback much appreciated. Cheers.