On 12/08/2022 7:34 am, David P. Reed via Starlink wrote:
I'll give you another example of a serious misuse of a theorem outside
its range of applicability:
Shannon proved a channel capacity theorem: C = W log(S / N). The proof
is mathematical, and correct.
Indeed.
But hiding in the assumptions are some very strong and rarely
applicable conditions. It was a very useful result in founding
information theory.
But... it is now called "Shannon's Law" and asserted to be true and
applicable to ALL communications systems.
...and it is. But it needs to be applied correctly.
This turns out not to be correct. And it is hardly ever correct in
practice.
Ahem ... if it's proven, it's correct, even in practice ;-)
An example of non-correct application turns out to be when multiple
transmissions of electromagnetic waves occur at the same time. EE
practice is to treat "all other signals" as Gaussian Noise. They are
not - they never are
Therein lies the problem. Correct theorem, incorrectly applied.
.
So, later information theorists discovered that where there are
multiple signals received by a single receiving antenna, and only a
little noise (usually from the RF Front End of the receiver, not the
environment) the Slepian-Wolf capacity theorem applies C = W
log(\sum(S[i]. i=1,N) /W).
Note: N here isn't the noise power (just the number of signals).
That's a LOT more capacity than Shannon's Law predicts, especially in
narrowband signalling.
Only if you lump in correlated signals with noise, which is an incorrect
(or rather, over-simplified) application of the Shannon-Hartley theorem.
And noise itself is actually "measurement error" at the receiver,
which is rarely Gaussian, in fact it really is quite predictable
and/or removable.
Noise in the Shannon sense is random and therefore not predictable or
correlated. Interference can be both predictable and correlated, and
therefore can sometimes be removed / to an extent. Modelling
interference as noise means not exploiting its inherent properties, and
yes that means ending lower capacity. But that doesn't mean that either
theorem is inapplicable - Shannon's fundamental limit still holds, even
in the multi-user case, as long as the noise you plug in is the "little
noise" from the RF front end and leave the interference out.
The point I guess is that models are just models, and the more you know
about what it is that you are dealing with, the better you can model.
Which, I suppose, applies to managing queues also. The more you know
what's in them and how it'll respond when you manage it, the better.
--
****************************************************************
Dr. Ulrich Speidel
School of Computer Science
Room 303S.594 (City Campus)
The University of Auckland
[email protected]
http://www.cs.auckland.ac.nz/~ulrich/
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