Shannon-Hartley Channel Capacity

This is the single most consequential equation in communications: a channel of bandwidth $B$ at a signal-to-noise power ratio $S/N$ can carry $B\log_2(1 + S/N)$ bits per second with an arbitrarily small error rate, and not one bit more. A 3 kHz telephone line at a signal-to-noise ratio of 1023 gives $3000 \times \log_2(1024) = 30{,}000$ bit/s. Claude Shannon published it in 1948 and it has never been beaten, only approached.

The mistake that catches everyone is decibels. Datasheets quote signal-to-noise in dB, and this formula wants a raw power ratio. A 30 dB link has $S/N = 10^{30/10} = 1000$, not 30. Put 30 in and you will underestimate the capacity by a factor of two and wonder why your modem outperforms Shannon. There is a second trap in what "arbitrarily small error" means: Shannon proved a limit exists but said nothing about how to reach it, and it took until the 1990s and turbo codes for practical systems to get close.

Look at how the two knobs differ, because it explains most of the last thirty years of wireless. Capacity is linear in bandwidth and only logarithmic in signal power. Doubling the spectrum doubles the rate. Doubling the transmit power adds about one bit per hertz if you were already at a good SNR, and nearly nothing if you were. That asymmetry is why 5G chases millimetre-wave spectrum instead of bigger amplifiers, and why the interesting gains now come from MIMO, which effectively creates several parallel channels rather than fighting for a better ratio in one.