Consider a digital communication system that uses a repetition code for the channel encoding/decoding. In particular, each transmission is repeated times, where n = 2m + 1 is an odd integer. The decoder operates as follows, if in a block of n received hits, the number of 0s exceeds the number of is, the decoder decides in favor of a 0. Otherwise, it decides in favor of a 1. An error occurs when m + 1 or more transmissions out of n = 2m + 1 are incorrect. Assume a binary symmetric channel.

(a) For n = 3, show that the average probability of error is given by P_e = 3p²(l ? p) + p^3?where p is the transition probability of the channel.

(b) For is 5, show that the average probability of error is given by P_e = 10p^3?(1 ? p)² + 5p⁴(1 ? p) + p⁵

(c) Hence, for the general case, deduce that the average probability of error is given by

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P. = Σ p'(1-p"-i