Channel capacity

The formula

C = B · log₂(1 + S/N)

Capacity in bits per second equals bandwidth in hertz times the log of one plus the linear signal-to-noise ratio. That is the Shannon–Hartley theorem, and it holds for a channel corrupted by additive white Gaussian noise.

C = max I(X; Y)

The general definition underneath it: capacity is the mutual information between what you send and what arrives, maximized over every possible way of using the input. Shannon–Hartley is what this maximization works out to for the band-limited Gaussian channel.

N = N₀ · B        (noise power = noise density × bandwidth)
C = B · log₂(1 + S/(N₀·B))

The version worth memorizing. Bandwidth shows up twice — once helping you, once hurting you — which is where most of the intuition (and most of the mistakes) live.

Eb/N₀ ≥ ln 2 ≈ 0.693   →   −1.59 dB

The ultimate limit as bandwidth goes to infinity: reliable communication below −1.59 dB of energy-per-bit over noise density is impossible. Not hard. Impossible.

Where you meet it

Link budget review. You close the budget at 6 dB margin, someone on the board divides the required data rate by the allocated bandwidth and checks the spectral efficiency against log₂(1 + SNR). If the waveform needs 6 bits/s/Hz and the budgeted SNR supports 4, the review is over. Capacity is the sanity check that catches a link budget lying to itself.
On the bench with a signal analyzer. A 20 MHz channel at 30 dB SNR caps at 199 Mbit/s — measured, not aspirational. When the radio under test delivers 40 Mbit/s in that channel, capacity tells you there's 5× on the table and the limitation is the modem, not physics. When it delivers 195, stop tuning; you're done.
Telemetry from a range asset or a deep-space bird. Power-limited links live on the other side of the curve, where SNR is below 0 dB and capacity is nearly linear in power and almost indifferent to bandwidth. That's why spread-spectrum telemetry burns huge bandwidth at tiny SNR and still closes: at S/N₀ of 30 dB-Hz the infinite-bandwidth ceiling is about 1,443 bit/s no matter how wide you spread.
Vendor claims. A datasheet promising a throughput that works out past B·log₂(1+SNR) for the stated conditions is wrong. Not optimistic — wrong. This check takes thirty seconds and has ended more than one procurement conversation.

How it works

Capacity is a theorem, not a rule of thumb. Shannon proved two directions at once: below C there exists a coding scheme with error probability as small as you like, and above C no scheme can be reliable — errors are guaranteed regardless of cleverness. That two-sided structure is what makes it a ceiling rather than a benchmark.

What the log does to your options:

In the bandwidth-limited regime (high SNR), capacity grows linearly with bandwidth but only logarithmically with power. At 20 dB SNR, doubling transmit power buys you about 15% more capacity (log₂(201)/log₂(101) ≈ 1.15). Doubling bandwidth at the same SNR doubles it. When spectrum is available, spectrum beats watts every time.
In the power-limited regime (low SNR), the roles flip: capacity tracks power almost linearly and extra bandwidth barely helps. log₂(1 + x) ≈ 1.44·x for small x, so at −10 dB SNR you get about 0.14 bits/s/Hz and the only real lever left is energy per bit.
The classic double-count. Widening the channel also widens the noise, because N = N₀·B. Double B at fixed transmit power and SNR halves; capacity still goes up (a 1 Hz channel at SNR 100 gives 6.66 bits, a 2 Hz channel at SNR 50 gives 11.3), but it does not double, and it converges to the 1.44·S/N₀ ceiling. Engineers who treat B and SNR as independent knobs write link budgets that can't be built.

The mistakes people actually make:

Plugging dB into the formula. S/N in Shannon–Hartley is a linear ratio. log₂(1 + 30) is not the capacity of a 30 dB channel; log₂(1 + 1000) is. This error survives in briefing charts more often than anyone admits.
Treating capacity as a design target for a specific modem. Capacity assumes coding with unbounded block length and latency. An uncoded QPSK link needs roughly 9–10 dB more SNR than capacity suggests for the same error rate. Modern LDPC and turbo codes close to within fractions of a dB of the limit — which is exactly why the DVB-S2 and CCSDS committees adopted them — but the gap you run at is a waveform property, not a physics one.
Forgetting the fine print. The theorem as written is for additive white Gaussian noise. Multipath fading, co-channel interference, phase noise, and nonlinear amplifiers all change the answer — usually downward, occasionally (as with MIMO exploiting multipath) upward. Quoting Shannon–Hartley for a fading tactical channel without an outage analysis is quoting the wrong theorem.
Using the wrong bandwidth. B is the noise bandwidth the receiver actually integrates, not the allocated channel spacing and not the −3 dB filter width. On the bench these differ by enough to matter.

Two numbers make the theorem concrete. A 3.1 kHz voice telephone channel on a good local loop runs about 33–38 dB SNR, which caps capacity near 34–39 kbit/s — which is why dial-up modems fought their way to 33.6 kbit/s and no coding trick pushed a V.34 connection past it; the last analog generation was riding the Shannon limit of the local loop. The 56k modems that followed didn't break the theorem — they changed the channel, exploiting the all-digital trunk to eliminate one analog-to-digital conversion and its quantization noise. And an 80 MHz Wi-Fi channel at 25 dB SNR caps near 665 Mbit/s per spatial stream — which is why every rate past that on the box came from MIMO streams, not from better modulation.

History

The ceiling was assembled in three moves, all inside the Bell System orbit. In April 1928, Harry Nyquist published "Certain Topics in Telegraph Transmission Theory" in the Transactions of the AIEE, showing that a channel of bandwidth B can carry at most 2B independent pulses per second ^[1]^[9]. Three months later, in the July 1928 Bell System Technical Journal, Ralph Hartley's "Transmission of Information" gave information a quantitative measure — logarithmic in the number of distinguishable levels — so that the two ideas together said rate scales as bandwidth times a log ^[1]^[2]^[5]^[6]. What neither man had was noise: nothing in 1928 said how many levels a real channel could actually distinguish.

Claude Shannon supplied that. Working at Bell Labs, where he'd been since 1941, he published "A Mathematical Theory of Communication" in the Bell System Technical Journal in 1948 ^[3]^[4], defining capacity in full generality and proving the noisy-channel coding theorem — the startling half being that noise limits rate, not reliability. The following January, "Communication in the Presence of Noise" in the Proceedings of the IRE worked the band-limited Gaussian case and produced the formula engineers now carry around ^[7]^[8]. The result carries both surnames because Shannon's theorem lands exactly where Hartley's 1928 expression pointed, with 1 + S/N doing the work Hartley's level count could only gesture at ^[1].

The strangest part of the story is the fifty-year gap that followed: the theorem guaranteed near-capacity codes existed but gave no recipe, and it took until the 1990s — turbo codes, then the rediscovery of LDPC — for practice to reach what 1948 had promised.

Related tools

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →