Channel Coding Theorem(Claude Shannon) Theorem For allr<c and e>o there exists a code of rate r whose error probability <e e can be arbitrarily small Proof uses large block size n as n -oo capacity is achieved In practice codes that achieve capacity are difficult to find The goal is to find a code that comes as close as possible to achieving capacity Converse of Coding Theorem For all codes of rate R>C, EEo>0, such that the probability of error is always greater than Eo For code rates greater than capacity, the probability of error is bounded away from 0Eytan Modiano Slide 7 Channel Coding Theorem (Claude Shannon) Theorem: For all R < C and ε > o; there exists a code of rate R whose error probability < ε – ε can be arbitrarily small – Proof uses large block size n as n → ∞ capacity is achieved • In practice codes that achieve capacity are difficult to find – The goal is to find a code that comes as close as possible to achieving capacity • Converse of Coding Theorem: – For all codes of rate R > C, ∃ ε 0 > 0, such that the probability of error is always greater than ε 0 For code rates greater than capacity, the probability of error is bounded away from 0