Channel capacity

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

2. Channel Model

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

3. Channel Capacity

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

4. 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)

5. 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

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

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

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.