Shannon's Channel Capacity Theorem | WRL-101

Module 5 built a ladder of schemes for packing more bits into every symbol — and each rung cost noise margin. That raises the question the whole ladder is climbing toward: for a given channel, how many bits per second is it possible to send, by any scheme whatsoever? In 1948 Claude Shannon answered it exactly, and the answer — a single, short equation — founded the entire field of information theory. It sets a hard ceiling no clever modulation or coding can beat.

The Formula

Shannon’s capacity theorem gives the channel capacity C — the maximum rate at which information can be sent with arbitrarily small error — for a channel of bandwidth B corrupted by additive white Gaussian noise:

Shannon–Hartley Theorem

C = B\,\log_2\!\left(1 + \mathrm{SNR}\right)

C is capacity in bits per second, B is bandwidth in hertz, and SNR is the linear signal-to-noise ratio (power ratio, not decibels). To use a dB figure, convert first: SNR = 10^(SNR_dB/10).

What each variable means

C — the ceiling on error-free bits per second. Not a rate you achieve automatically; a rate you cannot exceed.
B — the width of the channel in hertz. Wider channel, more capacity, in direct proportion.
SNR — signal power divided by noise power (Module 9), as a linear ratio. It enters through a logarithm, so its effect is far weaker than bandwidth’s.

Worked Examples

The formula is quick to apply. A classic case is the old telephone line: about 3.1 kHz of bandwidth at roughly 30 dB SNR (a ratio of 1000), giving C = 3100 × log₂(1001) ≈ 30.9 kbps — which is exactly why dial-up modems stalled near 33.6 kbps. Modern channels are wider and cleaner:

Channel	Bandwidth B	SNR	Capacity C
Telephone line	3.1 kHz	30 dB (×1000)	≈ 31 kbps
WiFi channel (1 stream)	20 MHz	25 dB (×316)	≈ 166 Mbps
5G NR carrier	100 MHz	20 dB (×100)	≈ 666 Mbps
Deep-space link	1 MHz	0 dB (×1)	1 Mbps

Notice the deep-space row: even when the signal and noise are equal (0 dB), the channel still carries one bit per second per hertz. Capacity never drops to zero until the signal does.

Two Knobs, Very Different Returns

There are only two ways to raise C, and they are not equal:

Bandwidth is linear. Double B and you double C. This is why every generation of wireless chases wider channels — 20 → 40 → 80 → 160 MHz in WiFi, and the jump to millimetre-wave in 5G for gigahertz of bandwidth.
SNR is logarithmic. Inside the log, so its returns diminish sharply. Once SNR is large, doubling the signal power (a 3 dB gain) adds only about B more bits per second — one extra bit per symbol. To double the rate through power alone you must roughly square the SNR.

This split defines two regimes. When bandwidth is scarce and SNR is high, you are bandwidth-limited and reach for higher-order QAM (Module 5). When power is scarce and bandwidth is plentiful — a deep-space probe — you are power-limited and spread the signal over more bandwidth instead.

Why You Cannot Beat It

Shannon proved two halves. The achievability half says that for any rate below C, a code exists that drives the error rate as close to zero as you like — given enough coding and delay. The converse half says that above C, no code can keep the error rate from rising; errors become unavoidable. Together they make C a genuine wall, not merely the best anyone has managed. What Shannon did not do was tell us how to build the codes — only that they exist.

The Ultimate Floor: −1.59 dB

Push the bandwidth-limited logic to its extreme. If bandwidth were truly unlimited, how little energy per bit could a link survive on? Taking B → ∞ in the theorem yields a hard floor on the energy-per-bit to noise ratio:

The Shannon Limit on E_b/N₀

\frac{E_b}{N_0} \;\ge\; \ln 2 \;\approx\; -1.59\ \text{dB}

Below E_b/N₀ = ln 2 ≈ −1.59 dB, reliable communication is impossible at any rate, with any code, over any bandwidth. It is the most fundamental limit in the subject.

Approaching the Limit — Modern Codes

For fifty years Shannon’s ceiling stood far above what real systems reached. Then two families of error-correcting codes closed the gap almost entirely. Turbo codes (1993) and LDPC codes (invented by Gallager in 1962, forgotten, and rediscovered in the 1990s) get within a fraction of a dB of capacity by using long blocks and iterative decoding. They are now everywhere: LDPC in WiFi, DVB-S2 satellite TV, and 5G data channels; turbo codes in 3G and 4G LTE; and polar codes (2009), the first codes proven to reach capacity, in 5G control channels. Shannon told us the wall was there in 1948; it took the industry until the 2000s to build codes that press right up against it.

Key Takeaways

C = B log₂(1 + SNR) is the maximum error-free bit rate of a channel — a ceiling no modulation or coding can exceed.
SNR is the linear power ratio; convert from dB with 10^(dB/10) before using the formula.
Bandwidth is linear, SNR is logarithmic: widening the channel pays far more than adding power, where 3 dB buys only ~1 more bit per symbol.
Systems are either bandwidth-limited (use higher-order QAM) or power-limited (spread over more bandwidth).
Shannon’s achievability + converse make C a real wall; he proved good codes exist without building them.
The ultimate floor is E_b/N₀ ≥ ln 2 = −1.59 dB; LDPC, turbo, and polar codes now operate within a fraction of a dB of capacity.

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Shannon’s Channel Capacity Theorem