Response to David Reed

[David Farber <dave () farber net>]

Begin forwarded message:

From: Brett Glass <brett () lariat net>
Date: September 7, 2006 10:35:25 AM EDT
To: David Farber <dave () farber net>
Subject: Re: Response to David Reed

At 07:44 AM 9/7/2006, David Farber wrote:

> I never got it, send again

Here goes....

-- Brett Glass

------------------

David Reed writes:

> The Shannon-Hartley Theorem does not provide a limit to the
> communications capacity of the electromagnetic field, as Brett would
> have it.

No one, to my knowledge, has made a claim that "the electromagnetic
field" has a communications capacity. Such a claim simply doesn't make
sense, since there are many, many ways of communicating via
electromagnetic fields -- from twisted pairs of wire to coaxial
cables to smoke signals. (Smoke signals would qualify, come to think
of it, since after all one detects them visually -- via the
electromagnetic radiation known as light.) Even speaking may be said
to be a form of communication via electromagnetic fields, since
electrostatic forces are at least partially responsible for the
effects when air molecules jostle one another to produce sound.

However, every method of telecommunication -- including tin cans and
string -- is subject to the Shannon-Hartley theorem. That is its
beauty; it is a general principle of information theory and is
independent of the physical nature of the communications medium.

> The Shannon-Hartley theorem describes a "channel" which is
> an abstraction for a single receiver which is receiving the
> superposition of a single signal and a gaussian noise source with
> bounded energy.

...which is an excellent abstraction for describing the behavior of
a radio receiver. After all, the functions of a radio receiver are:

a) To convert changing electromagnetic fields into a voltage or
   current which it can process further;

b) To distinguish the signal of interest from the noise (which may
   include other intentionally transmitted signals as well as random
   or naturally generated noise); and

c) To extract the information to be received from the signal of
   interest (a process usually called "demodulation").

All radio receivers must accomplish these functions; there's no
getting past it.

The fact that the model covers a single receiver and a single signal
presents no problem, because it is trivial to apply the theorem to
multiple receivers and signals via linear combination.

David also writes:

> 1) the physical universe is not a gaussian process.   Any assumption
> of that sort should be tested against reality - otherwise it is
> merely a statement of rhetorical faith, or merely an assumption made
> to make analysis possible.   Lightning strikes, for example, are not
> obviously gaussian in their effect on radios - and meteor trails,
> used practically to assist radio communications in some applications,
> are not gaussian.

Shannon's Law does not pertain only to situations where noise or
interference is Gaussian. The formula can easily be modified to
take into account the characteristics of other types of noise
(for example, "pink noise" or regular pulses of interference).

Shannon chooses a Gaussian noise source for his paper not only to
make the demonstration straightforward because most real world noise
(especially thermal noise) either is Gaussian or can be decomposed
into a linear combination of noise sources with Gaussian  
characteristics.
What's more, as Shannon notes in his paper, Gaussian noise is the worst
case for a bandwidth-limited communications channel. Since the airwaves
are currently allocated by frequency (which is, of course, what creates
the whole issue of spectrum and the "spectrum gold rush"), all legal
radio communications are bandwidth-limited -- including experimental
"ultrawideband" systems. Therefore, Gaussian noise is a good place to
start when developing the theory.

Yes, some noise -- such as impulse noise from lightning -- is a poor
fit because it occurs at random and is rare. But this sort of noise
isn't what is at issue when we talk about sharing spectrum. Optimal
use of spectrum requires that one use one's full allocation and
saturate it with nonredundant transmissions. Therefore, the manmade
noise which is at issue when we discuss spectrum allocation will be
Gaussian or close to it.

Here's a real life example. The image at

http://www.brettglass.com/Laramie900.jpg

shows the levels of noise on the 900 MHz unlicensed radio band in
the city of Laramie, Wyoming. (I took this measurement from the
bed of my pickup truck with an Avcom spectrum analyzer from a
hilltop overlooking the city.) Note that, within the band (902 to
928 MHz), the noise level is nearly constant -- making it,
effectively, Gaussian "white noise." This even though it is manmade
and not the result of a natural or stochastic process.

> It doesn't have any general application beyond a
> radio that exists at a single point in space, under that limited set
> of assumptions.

That set of assumptions covers virtually all real life cases. To my
knowledge, no radio receiver exists which can be in two places at one
time. Some radios do have multiple receiving antennas ("antenna
diversity") or have clever ways of combining redundant signals to
cancel noise (e.g. Orthogon Systems' "time/space multiplexing", which
sends the same signal twice at different times and with different
antenna polarizations). But this, again, can be subsumed into the
model of the Shannon-Hartley Theorem as an adjustment to the signal
to noise ratio.

> 2) Channels other than the simple channel described in the S-H
> theorem (such as so-called mult-terminal systems) have very different
> communications capacities.  The simplest such channel is the  
> multiple- access channel, which happens when 2 or more signals are  
> superposed
> on a noise source (which may or may not be gaussian).

 From the perspective of the receiver, the undesired signals are simply
noise and must be distinguished from the signal of interest just like
any other noise. What's more, due to the nature of spectrum regulation,
these signals will be bandwidth limited and thus will likely average
out to a Gaussian "buzz." If they do not, however, it does not negate
the Shannon-Hartley Theorem; it merely requires a calculation for that
special case. The basic principle, however, will still apply.

> These
> channels seem more appropriate for modeling large-scale networked
> communications.   For example, one somewhat clear analysis done by
> Xie and Kumar under plausible assumptions (of the gaussian sort
> challenged above, caveat lector) concludes that the limit of capacity
> increases linearly with the number of transceivers in a shared medium.
>
> The last point suggests that it may be the case that under suitable
> cooperative behavior (rules) radios can be designed to communicate
> without very serious limits.

Allocation of the airwaves by frequency (that is, assignment of  
spectrum)
is exactly that: a set of rules which can be used to limit interference.
There are an unlimited number of others which may, potentially, be
better and may eliminate the "beachfront property" problem in which
certain swatches of spectrum are more desirable than others. See my
article and slides at http://www.brettglass.com/ISART for a discussion
of this.

I am a strong advocate of such measures as spectrum etiquettes which
reduce the waste caused by unused spectrum and allow the efficient
sharing of spectrum. I have also advocated allocation schemes based on
signal characteristics other than frequency. (My article at the link
above discusses this.)

However, to suggest that any technology -- however sophisticated --  
could
remove all constraints on electromagnetic communications is either naive
or "hand waving." As Mankind has learned quite painfully from his  
experience
with depleted reserves of fossil fuels, pollution of the oceans, and  
global
warming, we must be highly skeptical of any claim that a resource is
inexhaustible. I do not know of any such claim which has ever proven  
to be
correct.

--Brett Glass



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