A binary symmetric channel (BSC) has two inputs x1 = 0 and x2 = 1 and two outputs y1 = 0 and y2 = 1. The channel is symmetric because the probability of receiving a 1 if a 0 is transmitted is the same as the probability of receiving a 0 if a 1 is sent. This common transition probability is denoted as b. Consider a simple repetition coding scheme over the BSC in which each bit is repeated n times where n = 2m + 1 is an odd integer. For decoding the received bits, a majority rule is employed. In other words, if in a block of n received bits the number of 0s is greater than the number of 1s, the decoder will decide that a 0 was transmitted. Otherwise, the decoder will decide that a 1 was transmitted. Therefore an error occurs whenever m + 1 or more bits out of n bits are received incorrectly.

Question

a. For this coding-decoding scheme, derive an expression for the probability of bit error Pe.
b. Calculate Pe when b = 0.05 and n = 3, 5, 7.

Answer 1

Analysis: