From: goldstein@carafe.tay2.dec.com (Fred R. Goldstein)
Subject: Re: Hayes' New Modem
Date: 10 Jan 1994 05:41:34 GMT
Organization: Digital Equipment Corp., Littleton MA USA

In article <telecom14.19.10@eecs.nwu.edu> hummes@osf.org (Jakob
Hummes) writes:

> ...But there is an absolute limit (Shannon's Law).  The
> question was about the transmission over a *real* phone line. And that
> means there exists *noise*.  The limit of bps is proportional to the
> logarithm of the signal to noise ratio. Unfortunately I don't remember
> the constant factors.

Shannon's law is, in plaintext,
BPS(max) = Bw * log(2)((1+S)/N)

That is, take the signal-to-noise ration (adding 1 to signal, so a
negative SNR has some information present) and represent it as a power
of 2.  Multiply by bandwidth (in Hz) and you get BPS.

THus if you have a 30 dB (1000) signal to noise ratio, that's 1001/1
which is a smidgen under 2^10.  If you have 3000 Hz usable bandwidth
that's the 10 times 3000, or around 30000 bps max.

It was often said that a phone line couldn't go beyond 26000 bps or
so, based on the typical bandwidth and SNR.  Today a good clean line
is more likely to be digitally switched at 64000 bps, which is well
above the Shannon limit (digitization is lossy), but you still get a
theoretical limit closer to 40 kbps.  Thus V.34, at 28.8 kbps, is
pushing the envelope, but still possible.  But it won't work on a line
that's transcoded down to 32 kbps, or just plain noisy.  Note the 300
to 3400 Hz nominal frequency range; the 3400 is a hard filter.

Fred R. Goldstein  k1io  goldstein@carafe.tay2.dec.com
Opinions are mine alone; sharing requires permission