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Capacity of the binary symmetric channel

Example

The binary symmetric channel has input and output alphabets ${0, 1}$ ; independently at each use, the output is the input bit flipped with probability $p \in [0, 1]$ . Write $h (p) : = - p \ln p - (1 - p) \ln (1 - p), 0 \ln 0 : = 0.$ The channel capacity is $\ln 2 - h (p)$ nats per use, attained by a uniform input.

Facts & Assumptions

Given: The binary symmetric channel with crossover probability $p \in [0, 1]$ .

[F1]

Finite discrete memoryless channels: the row $W (\cdot ∣ x)$ assigns probability $1 - p$ to $x$ and $p$ to $1 - x$ .

[F2]

Operational capacity formula for a finite DMC: capacity is the maximum of finite mutual information over input laws.

[F3]

Finite entropy and mutual-information chain rules: $I (X; Y) = H (Y) - H (Y ∣ X)$ .

[F4]

Finite conditional entropy and Finite distributions, entropy, divergence, and stochastic matrices: conditional entropy is the finite weighted average of conditional pmf entropies, with $0 \ln 0 = 0$ .

[F5]

Finite KL divergence is nonnegative with equality exactly at equal laws: for a binary law $Q = Bern (q)$ , $D (Q ∥ Bern (1 / 2)) = \ln 2 - h (q) \geq 0$ , with equality exactly when $q = 1 / 2$ .

Verification

Given: An input bit $X$ with $P (X = 1) = r \in [0, 1]$ and the specified channel.

1.1F1F3F4givenalgebra

The output law is Bernoulli with parameter $q = p + r (1 - 2 p)$ . Each row has entropy $h (p)$ , so [F3] and [F4] give $I (X; Y) = h (q) - h (p)$ .

2.1F2F5step 1.1algebracases∎

By [F5], $0 \leq D (Bern (q) ∥ Bern (1 / 2)) = \ln 2 - h (q)$ , so $I (X; Y) \leq \ln 2 - h (p)$ . Taking $r = 1 / 2$ gives $q = 1 / 2$ and equality. Therefore [F2] gives capacity $\ln 2 - h (p)$ . If $p = 0$ or $p = 1$ , $h (p) = 0$ and the channel is a deterministic bijection; if $p = 1 / 2$ , every row is the same and the capacity is zero.

Depends on

Used by

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Sources

MIT 6.441 Information Theory, Lecture 8, Channel Capacity and Binary Symmetric Channels (standard reference, not scraped)
Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, Chapter 19 §19.3 (standard reference, not scraped)