Shannon's Law

A law/theorem put forth by Claude Shannon, a reseacher at Bell Labs, some years ago. It is the primary rule regarding the capacity of a channel to carry electrical information, and is the most important formula in telecommunications.

In 1948, Shanon proved that the maximum data rate of a 'noisy' channel whose bandwidth is B Hz, and whose signal-to-noise ratio is S/N, is given by

Channel Capacity = B log₂ (1+S/N) bps

Following this premise, we find that the bandwidth for 3.1 kHz and a signal-to-noise ratio of 30 dB (which is a ratio of 1000/1), the maximum effective data rate is 31 kbps. The maths governing this would be:

Channel Capacity = B log₂ (1+S/N) bps

= 3100 log₂ (1+1000) bps
= 3100 log₂ (1001) bps
= 3100 log₂(2^9.967) bps
= 3100 x 9.967 bps
= 30,898 bps
= 31 kbps

And so, children, this is the reason that a voice-line-to-voice-line connection cannot exceed 33.6kbps. However, the X2/k56flex/v.90 protocols get around this by having the downstream connection exist in the domain of digital, rather than analogue.

Sources: http://www.hackphreak.org/files/lectures/Introduction_to_Telecommunications/Lecture_15/
http://www.scn.org/help/modemsp.html