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 ```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 runt...