Re: Beginner questions on distributions
- From: W^3 <aderamey.addw@xxxxxxxxxxx>
- Date: Fri, 27 Feb 2009 15:16:57 -0800
In article
<e25cd254-3feb-480d-bda2-2865adb223ea@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Achava Nakhash, the Loving Snake" <achava@xxxxxxxxxxx> wrote:
Hi Group,
On and off, I have been trying to pick up some knowledge of EE. One
of the first weirdities they hit you with is the idea of the unit
impulse function, aka the Dirac delta function which is has the
properties that delta(0) = 1, d(x) = 0 for x not equal to 1, and the
integral form minus infinity to infinity of delta(x) dx = 1. When I
first learned of it I concluded that Electrical Engineers and
Physicists were all insane.
I later mellowed in my views (at least as they pertain to the unit
impulse function) as it is on the same level of whackiness as idea of
a point mass, which I have always managed to swallow whole without
gagging, and in fact, it even took me a while to realize that there
was an actual problem here.
Anyway, I have been given to understand that Distributions, aka
Generalized Functions, come to the rescue as far as the unit impulse
and other so-called singularity functions are concerned.
As I understand it, a distribution D is actually a bounded linear
functional on the space of all C^infinity functions from reals to
reals with compact support. The analog of integrating D times a
regular function from minus inf to inf is then given as D(g), or some
appropriate limit if the g doesn't happen to be of the above type. I
don't actually know what this appropriate limit might be, but I can at
least guess.
Now EE types have a compulsion to take convolutions and to do Fourier
analysis with all the functions that they use, and so it is going to
be necessary to do that with distributions as well, not to mention
differentiating. Other arithmetic operations are a little bit more
obvious. It seems to me that, in order to do these things, we need
the following definitions: What is the integral of D times g(x) from
a to b where a and b are arbitrary real numbers or plus or minus
infinitsy? What is the analog of the integral for D(t - x) times g(x)
over any interval, or even only over the interval from minus infinity
to infinity? What is the derivative of D? I can play a game with
integration by parts to come to the conclusion that the derivative of
D takes g to the value you get when D is applied to the derivative of
g. Is that correct?
No, you're missing a minus sign.
What else do I need to know how to do in order
to be able to do the usual analytic tricks using distributions instead
of functions? I only really care about going from R to R at the
moment .
Just pretend D is a test function, do the formal manipulations, and
the definitions usually suggest themselves.
Now I realize that I can probably find the answers to these questions
in a variety of textbooks, and I do want the mathematician version of
othis so that I can deal better with the EE version of this, but my
analysis background is fairly weak, and I simply don't have the time
to learn enough to make this a viable option. So please help.
That's asking a lot. Why not post when you have a specific question?
.
Thanks,
Achava
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- Beginner questions on distributions
- From: Achava Nakhash, the Loving Snake
- Beginner questions on distributions
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