the Shannon capacity
/ Shannon -> 'SHAN-un' /
Imagine talking to someone across a noisy room. The wider the range of pitches your voice can use (bandwidth) and the louder you are compared with the background din (signal-to-noise ratio), the more words per second you can get across reliably. Shannon's capacity is the precise, provable speed limit on a real, noisy channel: the absolute maximum rate at which you can send bits with arbitrarily few errors, no matter how clever your coding.
The formula is C = B times log2(1 + S/N), where C is the capacity in bits per second, B is the bandwidth in hertz, and S/N is the signal-to-noise ratio (signal power divided by noise power, a plain ratio, not in decibels here). Two levers raise capacity: more bandwidth, and a better signal-to-noise ratio. For example, a 3,000 Hz line with S/N = 1,000 gives C = 3,000 times log2(1,001), which is about 3,000 times 9.97, roughly 29,900 bps. Crucially, doubling bandwidth doubles capacity, but doubling S/N only adds about one bit per symbol — so beyond a point, throwing more power at noise gives sharply diminishing returns.
Shannon's result is profound and a little eerie: it proves a hard ceiling that no encoding scheme can ever exceed, yet it also proves that codes exist which approach that ceiling with vanishing error — it just does not tell you how to build them (modern turbo and LDPC codes get close). Pair it with Nyquist: Nyquist says how many symbols the bandwidth allows, Shannon says how many bits the noise allows. The true data rate of a channel is the smaller of the two, and Shannon is why you cannot 'just add more signal levels' forever.
Wi-Fi gets faster when you stand near the router not because the bits travel faster, but because a stronger signal raises S/N, lifting the Shannon ceiling so the radio can safely switch to a denser modulation (more bits per symbol). Walk away and S/N falls, capacity drops, and it downshifts to a slower, more robust scheme.
Shannon: C = B log2(1 + S/N). More bandwidth scales capacity; more power only adds bits logarithmically.
Shannon capacity is a ceiling, not a recipe — it proves how fast you could go, not how to achieve it, and real systems run somewhat below it. It also assumes a specific noise model (additive white Gaussian noise); messier noise lowers the practical limit.