Shannon-Hartley Channel Capacity
Maximum error-free bit rate for a channel given its bandwidth and signal-to-noise ratio.
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The engineering
Shannon-Hartley sets the hard ceiling: no coding scheme, no modulation, no cleverness can push error-free throughput above C for a given bandwidth and noise floor. It is the yardstick every link-budget and modem design is measured against — real systems quote how close they run to Shannon.
The SNR enters as log₂, so bandwidth is the cheaper lever: doubling B doubles capacity, but going from 30 dB to 33 dB SNR only adds about 1 bit/s/Hz. That's why wideband, low-SNR schemes (spread spectrum, LTE, Wi-Fi 6) win where power is scarce. Sanity check: 20 MHz at 30 dB SNR gives roughly 200 Mbit/s — you can never beat that on that channel.
Note the formula uses the linear power ratio, not dB — convert first (10 dB = 10×, 20 dB = 100×). The result is a theoretical maximum; expect real throughput at 50-80% of C after overhead and finite-length coding.
Where this math comes from
Claude Shannon, a Bell Labs mathematician, published "A Mathematical Theory of Communication" in 1948 — the founding document of information theory. He proved that a noisy channel has a precise, finite capacity, and that reliable communication is possible at any rate below it, an idea so counterintuitive that it took the field years to fully accept.
The 'Hartley' half honors Ralph Hartley, whose 1928 Bell Labs paper first tied information to a logarithmic measure and bandwidth. Shannon generalized Hartley's noiseless bound into the noise-limited capacity engineers still use to grade every wireless and wired link.
- 1924Harry NyquistRelates signaling rate to bandwidth, seeding the capacity idea.
- 1928Ralph HartleyDefines information logarithmically and links it to bandwidth.
- 1948Claude ShannonPublishes the channel-capacity theorem C = B·log₂(1+S/N).
- 1993Berrou, Glavieux & ThitimajshimaTurbo codes approach the Shannon limit in practice.
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