Channel Capacity Calculator (Shannon-Hartley Theorem)
Calculate the maximum error-free data rate of a channel with the Shannon-Hartley theorem C = B·log2(1 + S/N).
Enter bandwidth and signal-to-noise ratio.
The hard ceiling on every data connection
In 1948 Claude Shannon proved something that sounds almost too strong to be true: for any communication channel with a given bandwidth and a given amount of noise, there is an absolute maximum rate at which data can be sent with arbitrarily few errors. Push below that rate and clever coding can make errors vanish. Try to push above it and no code, however ingenious, can keep the errors down. That ceiling is the channel capacity, and the Shannon-Hartley theorem puts a number on it.
The formula
C = B × log2(1 + S/N)
Where:
- C = channel capacity, the maximum error-free data rate, in bits per second
- B = bandwidth of the channel, in hertz
- S/N = the signal-to-noise ratio, as a plain power ratio (not decibels)
The logarithm is base 2 because information is measured in bits, and each bit is a binary choice.
Signal-to-noise ratio: ratio versus decibels
Engineers usually quote signal-to-noise ratio (SNR) in decibels, but the formula needs the plain power ratio. The two are linked by:
SNR(dB) = 10 × log10(S/N), and conversely S/N = 10^(SNR_dB / 10)
So 20 dB means a power ratio of 100, 30 dB means 1000, and 10 dB means 10. This calculator accepts either form and converts for you. Getting this conversion wrong (plugging decibels straight into the formula) is the single most common mistake.
Spectral efficiency: bits per second per hertz
Divide capacity by bandwidth and you get spectral efficiency, C/B = log2(1 + S/N), measured in bits per second per hertz. It tells you how much data each hertz of spectrum can carry, which is the figure of merit cellular engineers obsess over because spectrum is expensive and finite. A 30 dB link gives about log2(1001) = 9.97 bits/s/Hz, near the practical ceiling for real systems.
You can trade bandwidth for power
The formula reveals a deep trade-off. Capacity grows linearly with bandwidth but only logarithmically with signal power. Doubling your bandwidth doubles capacity. Doubling your signal power barely nudges it once the SNR is already high. This is why:
- Deep-space probes use enormous antennas and very low data rates: they cannot get more power, so they live at tiny SNR and stretch bandwidth and time instead.
- Fiber and 5G chase ever-wider bandwidth rather than just blasting more power.
- Spread-spectrum systems (GPS, military comms) deliberately spread a signal across huge bandwidth so they can operate even when the signal sits below the noise floor (S/N less than 1, where log2(1 + S/N) is still positive).
Real-world capacities
| Channel | Bandwidth | SNR | Approx capacity |
|---|---|---|---|
| Telephone modem line | 3.1 kHz | ~30 dB | ~31 kbps (hence 56k modems hit the wall) |
| Wi-Fi 20 MHz channel | 20 MHz | ~25 dB | ~166 Mbps |
| 4G LTE 20 MHz | 20 MHz | ~20 dB | ~133 Mbps per stream |
| Deep Space Network | varies | can be below 0 dB | bits to kbits per second |
The old 56k dial-up modem is a perfect illustration: a phone line is about 3.1 kHz with an SNR near 30 dB, giving a Shannon limit close to 31 kbps each way. The “56k” figure used asymmetric digital tricks on the download side, but the symmetric Shannon ceiling is exactly why dial-up could never get dramatically faster on an ordinary voice line.
Worked example
A Wi-Fi channel has 20 MHz of bandwidth and an SNR of 25 dB. First convert the SNR: S/N = 10^(25/10) = 316.2.
C = 20,000,000 × log2(1 + 316.2) = 20,000,000 × log2(317.2) = 20,000,000 × 8.31 = 166 million bits per second, about 166 Mbps.
Real Wi-Fi achieves less because of protocol overhead, imperfect coding, and interference, but it cannot exceed this number, ever.
The limit nobody beats
What makes the theorem so powerful is that it is an existence-and-impossibility proof at once. Shannon showed codes exist that approach C as closely as you like, and that no code can exceed it. Modern turbo codes and LDPC codes (used in 5G, Wi-Fi 6, and satellite links) get within a fraction of a decibel of the Shannon limit, which is why the theorem set the entire agenda for coding theory for seventy-five years.