Question about integration

Hello everyone,

I have a question about integration and convolution. Actually this is
an example from a book that I am reading and I was not able to
understand it. I am hoping someone can shed some light on this.

I have three function definitions

Unit step function u(x) = 1 { for x > 0 and 0 when x < 0}

Rectangular window function r(x) = 1 { for |x| < 1/2 and 0 otherwise }

and we have,

delta function d(x) = 0 for (x <> 0) and whose integral is one.

Given these definitions, I want to find the convolution a rectangular
function by itself r(x) * r(x)

So, from the derivative property of convolution:

d/dx (r(x) * r(x)) = [d/dx(r(x))] * r(x)

Now, the author proceeds to write this as the following:

[d(x + 0.5) - d(x - 0.5)] * r(x)

My question is how that derivative translates to this delta function
equation...

Again, he then proceeds to write the solution as the following
integral:

r(x) * r(x) = integral {-inf, x} [r(t + 0.5) -r(t - 0.5)]dt

= 1-|x| when |x| <= 1, else 0

I am not sure how he did the substitution here and calculated the
integral range (-inf, x).

I would be really grateful if someone can help me understand this.

Thanks,

Luca

.

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