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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:

enter image description here enter image description here

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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: enter image description here

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