Also known as: Shannon capacity, channel capacity, Shannon limit, Shannon-Hartley theorem
Shannon capacity is the highest rate at which information can be sent over a channel with an arbitrarily small error probability: for a bandlimited channel with additive Gaussian noise, C = B · log₂(1 + SNR) bits per second, where B is the bandwidth in hertz and SNR is the linear signal-to-noise ratio.1 Proved by Claude Shannon in 1948, it is a hard ceiling — no code or modulation can exceed it — and the benchmark against which every real radio link is judged.
How it works
Two resources bound how much information a channel can carry: how wide it is and how far the signal
stands above the noise. Shannon’s theorem combines them. The bandwidth B sets how many
independent symbols per second the channel supports (via the Nyquist rate), and the SNR sets how
many distinguishable levels each symbol can reliably carry — the log₂(1 + SNR) factor is the number
of bits those levels encode. Their product is the capacity in bits per second.
Dividing through by bandwidth gives the spectral efficiency, C/B = log₂(1 + SNR) bits per second per hertz — the ceiling on how many bits each hertz of spectrum can deliver. Two behaviours follow. At high SNR capacity grows only logarithmically: every doubling of SNR (+3 dB) adds roughly one bit/s/Hz, so pushing rate up by brute-force power hits diminishing returns. At low SNR capacity is nearly linear in SNR and you can trade bandwidth for power — spreading a weak signal over more hertz still conveys the bits, which is the principle behind spread-spectrum and deep-space links.
Crucially, Shannon proved capacity is achievable — codes exist that approach C with vanishing error — but the proof is non-constructive. The gap between the limit and real systems is what decades of forward error correction work has closed: turbo codes and LDPC codes now operate within a fraction of a dB of the Shannon limit.
In practice
The theorem reframes engineering as a budget. Given a required data rate you can ask what combination of bandwidth and SNR meets it; given fixed bandwidth and power you know the maximum rate worth attempting. It also defines a related floor, the minimum energy per bit to noise density Eb/N0 of about −1.6 dB, below which reliable communication is impossible at any bandwidth. Every modulation-and-coding scheme in a modern standard is, in effect, a chosen point on the Shannon curve trading spectral efficiency against robustness.
Relevance to SDR
Shannon capacity explains the fundamental limits of the signals an SDR receives. A trunking or cellular waveform’s choice of modulation order and code rate is a point on this curve, and the SNR at your antenna decides whether the link can support that rate — below the required SNR, no amount of receiver cleverness recovers the data, because the transmitter already sent at a rate the channel cannot sustain at your noise level. This is the theoretical backing for why improving SNR (better antenna, LNA, lower noise figure) is the lever that turns a failing decode into a working one.
GopherTrunk does not compute capacity, but the principle underlies its whole decode chain: recovering bits from a waveform is only possible when the received SNR sits above what that waveform’s coding and modulation require, exactly the boundary Shannon’s formula draws.
Sources
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Shannon–Hartley theorem — Wikipedia, the C = B·log₂(1+SNR) capacity formula and its assumptions. ↩