# Signal convolution: continuous signals

I really didn't understand how to do it. Please, can anyone help me??

Determine h (t) * x (t) when h (t) and x (t) are the rectangular pulses shown in Fig. A. Plot the result of this convolution.

I answered below, but I am in doubt if the path was correct:

Convolution is comprised of three steps:

• Introduce a dummy variable $$\tau$$ and use it to represent our functions. Also, reflect a function ($$x(\tau)$$) about $$x=0$$ with our dummy variable: $$x(-\tau)$$.
• Introduce a time offset for that function ($$t$$) allowing us to 'slide' $$x(-\tau)$$ along the $$x$$ axis.
• Find the integral of the product of our two functions at key values of $$t$$.

So, let's reflect $$x(t)$$ by making it $$x(-\tau)$$. You'll note that at this point, neither of the functions are overlapping and therefore the integral of their product is 0. Now, we're going to use $$t$$ to slide $$x(-\tau)$$ toward $$x\rightarrow \infty$$ and it will begin to overlap with $$h(\tau)$$: $$x(t-\tau)$$.

At $$t=4$$ the two rectangular pulses will be half-overlapping eachother. Therefore, the integral of their product is going to be $$4\times20=80$$.

At $$t=8$$ the two rectangular pulses will be completely overlapping eachother. They're symmetrical so the integral of the product is now going to be $$8\times20=160$$.

For $$t>8$$ the two rectangular pulses will begin to move away from one another and thus the integral of their product will begin to decline.

A plotted result of this convolution would look like: