Nyquist Bit Rate (Noiseless Channel)

Before Shannon there was Nyquist, and his question was simpler: ignore noise entirely, how fast can a channel of bandwidth $B$ change state? The answer is $2B$ symbols per second, and if each symbol is one of $M$ distinguishable levels it carries $\log_2 M$ bits. So $C = 2B\log_2 M$, and a 3 kHz line with four levels gives $6000 \ imes 2 = 12{,}000$ bit/s.

Read the two formulas side by side and the design problem appears. Nyquist has no ceiling: raise $M$ and the rate keeps climbing, so a noiseless channel could carry infinite information. Shannon says that is nonsense, because the levels get closer together as you add them and noise eventually makes neighbours indistinguishable. Nyquist tells you the ceiling from the bandwidth, Shannon tells you the ceiling from the noise, and a real link is capped by whichever is lower. Engineers pick $M$ by working out what Shannon permits and then choosing the largest constellation that fits underneath.

The historical footnote is worth having. This relation is often credited to Hartley, who published the $\log M$ part in 1928, with Nyquist supplying the $2B$ sampling limit the same year. The two results sat separately for twenty years until Shannon combined them with a noise term and turned a pair of engineering rules into information theory.