Communication Systems
Shannon Channel Capacity Calculator
Shannon–Hartley capacity C = B·log₂(1+SNR) — solve for capacity, required SNR, or required bandwidth.
About this calculator
The Shannon–Hartley theorem sets the hard ceiling on communication: over a channel of bandwidth B corrupted by additive white Gaussian noise at signal-to-noise ratio SNR, information can be sent with arbitrarily low error at any rate below C = B·log₂(1 + SNR) — and at no rate above it, ever, with any modulation or code. It is not a performance estimate; it is a law about what is achievable.
The formula rewards reading in both directions. Capacity grows only logarithmically with SNR — every doubling of signal power adds the same ~1 bit/s/Hz once SNR is high — but linearly with bandwidth, which is why modern systems chase spectrum before power. Solved for SNR, it gives the minimum signal-to-noise ratio any scheme needs to carry a given spectral efficiency C/B: 1 bit/s/Hz needs at least 0 dB; 10 bit/s/Hz needs at least 30.1 dB, no matter how clever the modem.
This calculator solves the theorem for any one of the three quantities given the other two, reports the spectral efficiency, and plots capacity against SNR for your bandwidth so the logarithmic flattening is visible. Real systems operate below the curve: the gap between a practical modem's required SNR and the Shannon minimum (typically 1–3 dB for modern codes near their design point) is the coding gap. Use the BER calculator to see what uncoded modulation needs, and the SNR / Eb/N0 converter to move between the SNR here and the Eb/N0 of modem datasheets.
Assumptions
- Additive white Gaussian noise channel with flat noise across the bandwidth B — colored noise or interference changes the achievable rate.
- SNR is the average signal power over average noise power in B; peak-power limits and fading are not modeled.
- Capacity is the information-theoretic supremum over all possible modulations and codes, with unconstrained complexity and delay.
When to use this calculator
Appropriate for
- Finding the theoretical channel capacity of an AWGN channel from bandwidth and SNR
- Establishing an upper bound to compare a real modulation-and-coding scheme against
- Teaching the bandwidth–SNR tradeoff of the Shannon–Hartley limit
Not suitable for
- Predicting a real system's throughput — practical schemes operate below capacity with finite coding gain and overhead
- Non-AWGN channels (fading, interference-limited, or bursty), where the Gaussian-capacity formula does not apply
- Fading or MIMO channels, which require their own capacity models
What this calculator does not cover
- An upper bound, not a design value — practical modems need SNR above the Shannon minimum, with modern codes closing to within roughly 1–3 dB near their design point.
- Single AWGN link: fading channels, interference-limited cells, and MIMO systems have their own capacity formulas that this one does not cover.
- No account of implementation constraints — peak-to-average power, synchronization, and finite block length all reduce what is achievable at a given error rate.
- As with every calculator on this site: results are preliminary and educational, are not verified for any specific installation, and must be reviewed against the applicable code edition and stamped by a licensed Professional Engineer before real-world use.
Frequently asked questions
Can a real system ever reach the Shannon capacity?
Not exactly reach it — the theorem allows rates arbitrarily close to C with arbitrarily low error, but at the cost of unbounded coding complexity and delay. Modern LDPC and polar codes operate within a fraction of a dB to a few dB of the limit, which is why capacity is the standard benchmark for modem performance.
Why does doubling bandwidth beat doubling power?
Capacity is linear in B but logarithmic in SNR. Doubling B doubles capacity outright (if the extra band has the same SNR); doubling power adds only one bit per symbol's worth once SNR is already high — about 3 dB for ~1 bit/s/Hz.
What happens as bandwidth goes to infinity?
Capacity does not grow without bound — with fixed signal power, SNR falls as B rises, and capacity approaches the limit S/(N₀·ln 2). This wideband limit is where the famous −1.59 dB minimum Eb/N₀ comes from.
References
- Shannon, C. E., “A Mathematical Theory of Communication,” Bell System Technical Journal, vol. 27, 1948
- Proakis, J. & Salehi, M., Digital Communications, 5th ed. (channel capacity)
- Haykin, S., Communication Systems, 4th ed. (Shannon–Hartley law)
Related calculators
BER vs Eb/N0 Calculator (AWGN)
Theoretical bit error rate for BPSK, QPSK, and coherent BFSK over an AWGN channel, with the waterfall curve.
SNR / Eb/N0 / Es/N0 Converter
Convert between SNR, Eb/N0, and Es/N0 given bits per symbol, symbol rate, and bandwidth — the classic source of dB bookkeeping errors.
Thermal Noise & Noise Figure Calculator
kTB thermal noise power, noise power spectral density, noise temperature from noise figure, and the receiver noise floor.
Get the ComputeVerse.ai quick reference
A free engineering quick-reference PDF, plus a note when new calculators launch across the library — one or two emails a month, nothing else.