studying
Miguel Bazdresch
miguel at thewizardstower.org
Wed Feb 19 09:20:09 PST 2003
* Csaba Henk <tphhec01 at degas.ceu.hu> [03-0218 13:40]:
> On Mon, 17 Feb 2003, Miguel Bazdresch wrote:
> > > E.g., do you know what is conditional exceptable value in probability
> > > theory? I don't mean the elementary concept of conditional exceptable
> > > value of event A wrt. event B (E(A|B)), but the highly complex concept of
> > > conditional exceptable value of the random variable X wrt. the random
> > > variable Y (E(X|Y)), or still more purely, wrt. the sigma algebra S
> > > (E(X|S)). Or do you know what a martingal is? I had to learn these notions
> > > without any hindsight given.
> >
> > You'd be surprised but I do know them;
>
> Why would I be surprised? I can imagine you're a clever guy :-)
More than anything because it gave me the impression that you somehow
have heard rumors that they're useful concepts, in a practical sense,
but never heard an engineer actually say they're essential to understand
digital telecommunications.
> > I'm doing a PhD in
> > Telecommunications and believe me, there's no way to understand telecom
> > systems without them.
>
> I know this. I wrote in my previous mail the I know that these are useful.
> I just do not see the reality to which these correspond. The problem is
> with my education or maybe with Hungarian mathematical culture or maybe
> with today's mathematical culture. (Don't know which one.)
But that's easily fixed, at least in my field, which is digital
communications. Believe me, the hardest part of it is the math. If
you're good at it, the concepts will come to you easily. You can even
find you can make contributions that somebody with an engineering
background just isn't able to imagine.
Our bible is: J. Proakis, Digital Communications, McGraw-Hill. If it's
not in your library and you're really interested in taking a look at a
book on the subject, let me know and I'll suggest a cheaper one; Proakis
costs more than 100 euros.
> > What is sad is that no one could explain to you
> > that these concepts are useful, and give examples.
>
> Or would you say that the problem roots in the nature of these notions?
> Concerning those abstract notions for which I managed to find their base
> intuition, I think they could be teached in a setup which tells about
> their origin.
*IF* you had a professor that understands the whole thing. Most don't.
There are subjects, like probability, geometry, and linear algebra that
originated as purely mathematical constructs with no useful practical
purposes (when I say probability I guess I mean more things like
stochastic processes than means and standard deviations). These were
later adopted by engineers when they were seen as good tools to model
and solve problems.
There are other, like queue theory, that originated as engineering ideas
and were later refined in mathematics.
There are other variants (Fourier for example). The thing is, it's a
rare person that really understands the whole context and history of an
idea, and then is motivated enough to teach them that way (in my
experience of course). BTW, I'm not (yet) like that; I've just had
the luck of having two or three profs in my life that are.
It's enlightening.
> > You might like to know that if the sphere-packing problem was solved,
> > then we telecom people could design the best error-correcting codes
> > possible. There's a problem you might want to tackle for your postdoc :)
>
> Could you tell it in more details? I had one idea what kind of
> sphere-packing problem you are speaking of, but then I found a paper on
> the net which said that that one is solved ("Kepler's Sphere Packing
> Problem Solved", http://www.maa.org/devlin/devlin_9_98.html). And how does
> it relate to error-correcting codes?
I'll try. One of the interesting things about digital comm. is that many
problems have parallels in geometry. This means that the set of signals
that are transmitted can be regarded as points in n-dimensional space.
Noise will move the points, so that the receiver can see any point in
the space, and its job is to decide which point was actually
transmitted. Bear in mind that the set of points that are transmitted is
finite.
The best receiver possible picks as its choice the point that, out of
the set of transmit points, is closer to the received point. So every
point has a sphere around it, and the receiver associates all points
within the sphere with its center.
Now the distance between points is related to the energy being fed to
the antenna, which one wants to minimise. This means that one must find
the best packing of spheres, in order to have the transmit points as
separate as possible with the least amount of energy expenditure.
Whew! I hope you get the general idea. Other "sphere" problems related
to digital comm. are the kissing number, and the less dense packing of
spheres that fills the space.
The proof you refer to says that the maximum
density of spheres in space is around 0.74. The problem, unsolved
for many dimensions, is to find an actual packing with that density.
Also, in order to simplifly the receiver, many times one wants to have a
packing with certain order. Of particular convenience are so-called
lattice packings. The best lattice packings known are very far from the
theoretical densities.
The classic paper on the subject is: N.J.A. Sloane, The Packing of
Spheres. Unfortunately I don't have right now the year and magazine
name.
--
Miguel Bazdresch
http://thewizardstower.org/
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