Noisy-channel capacity theorem
Shannon-Hartley Theorem
A band-limited Gaussian channel has a finite information capacity set by bandwidth and signal-to-noise ratio, with only logarithmic returns from added signal power.
C = B log2(1 + S/N)
C is capacity in bits per second, B is bandwidth in hertz, and S/N is the average received signal-to-noise power ratio. The expression applies to an ideal additive white Gaussian noise channel.
The oscilloscope converts dB to a linear power ratio and lets bandwidth vary from 0.5 to 5 MHz. Protocol overhead, finite blocks, fading, interference, implementation loss, and latency are excluded.
(Mbit/s)
Lower SNR makes the received trace visibly less distinguishable. Bandwidth adds more signaling space; power improves capacity only through the logarithm.
- CHANGE
- Received signal-to-noise ratio
- WATCH
- channel capacity
- MEANING
- The oscilloscope converts dB to a linear power ratio and lets bandwidth vary from 0.5 to 5 MHz. Protocol overhead, finite blocks, fading, interference, implementation loss, and latency are excluded.
Bandwidth scales capacity; power buys logarithmic gains.
The capacity curve rises quickly at low SNR and then bends. Doubling bandwidth doubles the bound, while doubling signal power adds progressively less capacity.
What it actually says
The Shannon-Hartley expression gives the supremum of reliable communication rate for a continuous-time channel with limited bandwidth and additive white Gaussian noise. Rates below capacity can, in principle, achieve arbitrarily small error with sufficiently long and well-designed codes; rates above capacity cannot.
Capacity is not a modulation setting and not the throughput a device will automatically deliver. It is a boundary derived from the probability model. Practical systems spend part of the theoretical margin on finite delay, synchronization, estimation, coding gaps, control traffic, and changing channel conditions.
"A useful law compresses a pattern. It does not erase the conditions that make the pattern true."
How the idea developed
The modern form emerged through observation, argument, and later refinement. The timeline separates the first insight from the version now used in textbooks and practice.[1]
Harry Nyquist and Ralph Hartley relate signaling rate, bandwidth, and distinguishable levels.
Claude Shannon publishes A Mathematical Theory of Communication and proves noisy-channel coding results.
Coding theory develops practical block, convolutional, Reed-Solomon, turbo, and related codes.
LDPC, polar, MIMO, and adaptive systems operate increasingly close to model-specific capacity limits.
How the pattern works
The relation becomes useful only when its mechanism, measurement process, and operating range are visible.
More independent channel degrees of freedom per second increase capacity in direct proportion to B.
Higher received signal power makes more message distinctions reliable, but the logarithm creates diminishing returns.
Long codewords distribute information so noise-corrupted observations can still identify the intended message.
White Gaussian noise, average power, and ideal bandwidth define the mathematical channel being bounded.
C is capacity in bits per second, B is bandwidth in hertz, and S/N is the average received signal-to-noise power ratio. The expression applies to an ideal additive white Gaussian noise channel.
Where it earns its keep
Applications are strongest when the law changes a decision, measurement, model, or experiment rather than merely providing an analogy.
Set a link-budget ceiling
ApplicationEngineers compare required rate with bandwidth, received power, interference, and coding performance.
Use effective SINR and measured implementation loss, not transmitter power alone.
Trade bandwidth against energy
ApplicationPower-constrained links can use wider bandwidth and stronger coding to approach the wideband regime.
Antenna gain, atmospheric loss, Doppler, and regulatory bandwidth remain part of the design.
Distinguish physical and application throughput
ApplicationThe theorem bounds one channel layer before framing, retransmission, contention, and application overhead.
Do not advertise Shannon capacity as end-user speed.
Where it stops working
The compact formula is for an AWGN channel. Fading, colored noise, interference, multiple antennas, feedback, peak-power limits, nonlinearity, and finite alphabets lead to different capacity problems or require additional assumptions.
The classical theorem is asymptotic: arbitrarily low error may require codewords and decoding complexity incompatible with real-time latency, memory, power, or hardware constraints. Finite-blocklength information theory quantifies that gap.
"Capacity is the speed a modem will achieve"
Better: It is an ideal upper bound under a stated channel model."More transmit power gives linear throughput"
Better: Capacity grows logarithmically with SNR."Bandwidth and SNR are interchangeable without cost"
Better: They have different physical, regulatory, and energy consequences."Shannon says error-free communication is easy"
Better: Approaching capacity can demand long codes and substantial complexity.Sources and further reading
Original publications and serious secondary scholarship are prioritized over summaries.
- Shannon - A Mathematical Theory of Communication, Part IThe 1948 primary paper establishing modern information theory and noisy-channel capacity.https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
- IEEE REACH - A Mathematical Theory of CommunicationIEEE-hosted primary-source edition and historical context.https://reach.ieee.org/primary-sources/a-mathematical-theory-of-communication/
- Hartley - Transmission of InformationThe 1928 bandwidth and signaling precursor cited by Shannon.https://doi.org/10.1002/j.1538-7305.1928.tb01236.x
- Cover and Thomas - Elements of Information TheoryStandard rigorous reference on entropy, channel capacity, and coding theorems.https://onlinelibrary.wiley.com/doi/book/10.1002/047174882X