Channel Capacity

When to Use

Use this skill when working on channel-capacity problems in information theory.

Decision Tree

1. Mutual Information

• I(X;Y) = H(X) + H(Y) - H(X,Y)
• I(X;Y) = H(X) - H(X|Y) = H(Y) - H(Y|X)
• Symmetric: I(X;Y) = I(Y;X)
• scipy.stats.entropy(p) + scipy.stats.entropy(q) - joint_entropy

1. Channel Model

• Input X, output Y, channel P(Y|X)
• Channel matrix: rows = inputs, columns = outputs
• Element (i,j) = P(Y=j | X=i)

1. Channel Capacity

• C = max_{p(x)} I(X;Y)
• Maximize over input distribution
• Achieved by capacity-achieving distribution

1. Common Channels

| Channel | Capacity | |---------|----------| | Binary Symmetric (BSC) | 1 - H(p) where p = crossover prob | | Binary Erasure (BEC) | 1 - epsilon where epsilon = erasure prob | | AWGN | 0.5 log2(1 + SNR) |

1. Blahut-Arimoto Algorithm

• Iterative algorithm to compute capacity
• Alternates between optimizing p(x) and p(y|x)
• Converges to capacity
• z3_solve.py prove "capacity_upper_bound"

Tool Commands

Scipy_Mutual_Info

bash
uv run python -c "from scipy.stats import entropy; p = [0.5, 0.5]; q = [0.6, 0.4]; H_X = entropy(p, base=2); H_Y = entropy(q, base=2); print('H(X)=', H_X, 'H(Y)=', H_Y)"

Sympy_Bsc_Capacity

bash
uv run python -m runtime.harness scripts/sympy_compute.py simplify "1 + p*log(p, 2) + (1-p)*log(1-p, 2)"

Z3_Capacity_Bound

bash
uv run python -m runtime.harness scripts/z3_solve.py prove "I(X;Y) <= H(X)"

Key Techniques

From indexed textbooks:

• Elements of Information Theory] Elements of Information Theory -- Thomas M_ Cover & Joy A_ Thomas -- 2_, Auflage, New York, NY, 2012 -- Wiley-Interscience -- 9780470303153 -- 2fcfe3e8a16b3aeefeaf9429fcf9a513 -- Anna’s Archive. Using a randomly generated code, Shannon showed that one can send information at any rate below the capacity C of the channel with an arbitrarily low probability of error. The idea of a randomly generated code is very unusual.

Cognitive Tools Reference

See .claude/skills/math-mode/SKILL.md for full tool documentation.