# Consider a digital communication system that uses a repetition c

# Consider a digital communication system that uses a repetition c

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 Pe = 3p2(l â p) + p3 where p is the transition probability of the channel.

(b) For is 5, show that the average probability of error is given by Pe = 10p3 (1 â p)2 + 5p4(1 â p) + p5

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