the Nyquist theorem
/ Nyquist -> 'NY-kwist' /
Suppose you could send a signal over a perfect wire with no noise at all. How fast could you go? You might think 'infinitely fast', but there is still a speed limit, and it comes from the channel's bandwidth. The Nyquist theorem says: a channel of bandwidth B hertz can carry at most 2B symbols per second cleanly. Squeeze symbols closer than that and they smear into each other so badly that no receiver can tell them apart — like trying to flash your torch faster than your friend's eyes can follow distinct flashes.
Turning symbols into bits gives the famous formula for a noiseless channel: maximum data rate = 2B times log2(M) bits per second, where B is the bandwidth in hertz and M is the number of distinct signal levels per symbol. The 2B part is the symbol-per-second ceiling that bandwidth alone fixes. The log2(M) part is how many bits you stuff into each symbol by using more levels. For example, a 3,000 Hz channel with M = 2 levels tops out at 2 times 3,000 times log2(2) = 6,000 bps; bump to M = 4 levels and it doubles to 12,000 bps — for free, in a perfect channel.
The catch — and Nyquist's deep lesson — is that the formula has no noise in it, so it suggests you can keep raising M forever and get unlimited data rate, which is absurd in the real world. In reality every channel has noise, and packing more levels makes them sit closer together until noise flips them. That is exactly the gap Shannon's capacity formula fills: Nyquist gives the bandwidth-imposed symbol limit, while Shannon gives the noise-imposed bit limit, and the real maximum is whichever is lower.
An old telephone line offers about 3,100 Hz of usable bandwidth. By Nyquist alone, with 2 levels it caps at 2 times 3,100 = 6,200 bps; modems beat that only by using many levels per symbol (large M) — but how large is then capped by noise, via Shannon.
Nyquist sets the symbol-rate ceiling from bandwidth; it ignores noise, so it is only half the story.
Beware: the Nyquist bit-rate formula here (about channel capacity) is related to but distinct from the Nyquist–Shannon sampling theorem (sample at twice the highest frequency). Both come from Nyquist; do not conflate the two.