Convolution of two rectangular pulses 
Determine the shape of the following function$$\int^\infty_{-\infty} \Pi(4\tau) \Pi(t-\tau) d\tau$$

Attempt:
This function is a convolution of two rectangular functions. I know that the result has to be a triangular pulse, but how do we determine the width and the height of this triangle?
I know that the first term is just a rectangular pulse compressed by a factor of 4. Here's a picture showing $\Pi(4\tau)$ (magenta) compared to $\Pi(\tau)$ (blue):

But what about $\Pi(t-\tau)$? And how do I work out the width of the convolution of the two?
 A: We define the rectangular pulses as follows
$$p_1 (t) := u \left(t + \frac{T}{2}\right) - u \left(t - \frac{T}{2}\right)$$
$$p_2 (t) := u \left(t + \frac{T}{8}\right) - u \left(t - \frac{T}{8}\right)$$
where $u$ is the Heaviside step. Let $x = p_1 * p_2$. When convolving piecewise constant functions, a useful "trick" is to differentiate
$$\dot x (t) = (\dot p_1 * p_2) (t) = p_2 \left(t + \frac{T}{2}\right) - p_2 \left(t - \frac{T}{2}\right)$$
and then integrate
$$x (t) = r \left(t + \frac{5T}{8}\right) - r \left(t + \frac{3T}{8}\right) - r \left(t - \frac{3T}{8}\right) + r \left(t - \frac{5T}{8}\right)$$
where
$$r (t) := \begin{cases} t & \text{if } t \geq 0 \\ 0 & \text{otherwise}\end{cases}$$
is the ramp function.
A: Some hints:

*

*Since the pulses have different widths you don't get a triangle but a trapezoid


*Take for example $t = 100$, then $\Pi(4\tau) \Pi(t-\tau)$ is zero for all $\tau$. What's the smallest and the largest $t$ such that the integral is non-zero? This gives you the width of the resulting pulse.


*For what range of $t$ do the two pulses overlap completely (like in your plot)?
