gen-define-channel-capacity-and-give-one-short-worked-example-63a8

Define channel capacity and give one short worked example.

# Channel Capacity **Definition:** Channel capacity (C) is the maximum rate at which information can be reliably transmitted over a communication channel with arbitrarily small probability of error. It is measured in bits per channel use (or bits per second for continuous channels). For a discrete memoryless channel, it is defined as: **C = max I(X;Y)** where the maximum is taken over all possible input probability distributions p(X), and I(X;Y) is the mutual information between input X and output Y. ## Worked Example: Binary Symmetric Channel (BSC) **Problem:** Find the capacity of a BSC with crossover probability p = 0.1 **Solution:** A BSC flips each transmitted bit with probability p. The capacity formula for a BSC is: **C = 1 - H(p)** where H(p) = -p log₂(p) - (1-p) log₂(1-p) is the binary entropy function. For p = 0.1: - H(0.1) = -0.1 log₂(0.1) - 0.9 log₂(0.9) - H(0.1) = -0.1(-3.322) - 0.9(-0.152) - H(0.1) = 0.332 + 0.137 - H(0.1) ≈ 0.469 bits Therefore: **C = 1 - 0.469 = 0.531 bits per channel use** This means we can reliably transmit about 0.531 bits of information per transmission through this noisy channel.