Re: Differentiation question...
- From: ... <...@xxxxxxxxxxxxxx>
- Date: Mon, 22 Oct 2007 01:24:07 -0400
On Sun, 21 Oct 2007 17:27:36 -0700, luca.pamparana@xxxxxxxxx wrote:
Hello all,
Been trying to figure some equations out... actually owe a lot to this
group for all the help people have given me. So, thanks!
I have a rectangular function with the jump function at + 1/2 and -1/2
and as someone pointed out the differentiation of this would have a
delta function at the same points. So, the books right the
differentiation as follows:
d/dx (rect(x)) = delta(x+ 1/2) - delta(x -1/2).
My question is when you do the differentiation, why do you have the
minus sign between the two terms....
Thanks,
Luca
Surely the material you're looking at talks about the unit step
function U(x), also known as the Heaviside step function H(x).
The notation U(x) and H(x) is common but not universal.
Simplest, I think, is to consider rect(x) to be the sum of two
shifted unit step functions.
H(x + 1/2) is a unit step that is +1 for x >= -1/2
-H(x - 1/2) is a unit step that is -1 for x >= 1/2
Add these two and you have rect(x),
rect(x) = H(x + 1/2) - H(x - 1/2)
The derivative of H(x) is your delta(x), that is
d/dx H(x) = delta(x)
Hence
d/dx rect(x) = d/dx H(x + 1/2) - d/dx H(x - 1/2)
d/dx rect(x) = delta(x + 1/2) - delta(x - 1/2)
.
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