Mathematical Principles & Theorems
Claude Shannon's Mathematical Theory of Communication. Entropy \(H(X) = -\sum_{i=1}^n p(x_i) \log_2 p(x_i)\) quantifies average uncertainty in bits. Binary Entropy function \(H_b(p) = -p\log_2 p - (1-p)\log_2(1-p)\). Huffman coding produces optimal prefix-free binary trees with average length \(H(X) \le \bar{L} < H(X) + 1\). Channel Capacity of a Binary Symmetric Channel (BSC) with error probability \(p\) is \(C = 1 - H_b(p)\).
Operating Instructions
- Select module: Shannon Entropy Calculator, Huffman Optimal Encoder, or Binary Symmetric Channel.
- Enter text or custom symbol probability distributions; adjust channel crossover error \(p\) via numeric input.
- Click Compute to generate optimal prefix tree codes, bit efficiency, and redundancy metrics.
- Observe simulated noisy bit transmission and bit error rate (BER) statistics.