Gaussian Channel Capacity

The capacity C = 12 _2(1 + P/N) from a Gaussian input.

The idea

The Gaussian channel is the standard model of a noisy physical link. It accepts a real-valued input $X$ and returns

$Y = X + Z, \qquad Z \sim \mathcal{N}(0, N),$

with the noise $Z$ independent of $X$. Without further restriction the model sets no limit on communication, since a large enough input overwhelms any fixed noise; the realistic ingredient is an average power constraint $\mathbb{E}[X^{2}] \le P$ on the input. Under that constraint the capacity — the largest rate at which information passes reliably — has a closed form.

Theorem (Capacity of the Gaussian channel).

The Gaussian channel $Y = X + Z$, with noise $Z \sim \mathcal{N}(0, N)$ independent of the input and the power constraint $\mathbb{E}[X^{2}] \le P$, has capacity $C = \tfrac{1}{2}\log_2\!\Bigl(1 + \frac{P}{N}\Bigr) \ \text{bits per channel use},$ attained by a Gaussian input of variance exactly $P$.

The ratio $P/N$ is the signal-to-noise ratio. It sits inside the logarithm, so doubling the capacity requires squaring the signal-to-noise ratio, not doubling it.

Ways to work on it

• Walkthrough. Derive C = 12 _2(1+P/N) from h(Y) - h(Z).
• Practice. Evaluate the capacity for a given signal-to-noise ratio.
• Hardest. The bandlimited Shannon–Hartley capacity in bits per second.

Not sure where to start? Take the ten-question placement test.