A digital communication system uses a repetition code for channel encoding/decoding. During transmission, each bit is repeated three times (0 is transmitted as 000 and 1 is transmitted as 111). It is assumed that the source puts out symbols assumed independently and with equal probability. The decoder operates as follows: in a block of three received bits, if the number of zeros exceeds the number of ones, the decoder decides in favor of a 0; if the number of ones exceeds the number of zeros, the decoder decides in favor of a 1. Assuming a binary symmetric channel with crossover probability p = 0.1, the average probability of error is ________.
3-Repetition Code over a Binary Symmetric Channel
Consider a Binary Symmetric Channel (BSC) with crossover probability p.
Transmission rule:
- 0 → 000
- 1 → 111
The receiver applies majority decoding on the three received bits.
Condition for Error
A decoding error occurs when two or more bits are flipped by the channel.
Probability of Error
Exactly two bit errors:
C(3,2) · p² · (1 − p) = 3p²(1 − p)
Exactly three bit errors:
C(3,3) · p³ = p³
Final Expression
Pe = 3p²(1 − p) + p³ = 3p² − 2p³