Coping with bit errors figure 6 1 probability of a

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Unformatted text preview: arbitrary decision. (We have tacitly assumed that the a priori probability of the sender sending a message 0 is the same 4 LECTURE 6. COPING WITH BIT ERRORS Figure 6-1: Probability of a decoding error with the replication code that replaces each bit b with c copies of b. The code rate is 1/c. as a 1.) We can write the probability of decoding error for the replication code as, being a bit careful with the limits of the summation: ￿ ￿c ￿ c￿ i c− i c if c odd i=￿ 2 ￿ i p (1 − p) ￿ c￿ i ￿￿ ￿c P (decoding error) = (6.2) c−i + 1 c pc/2 (1 − p)c/2 if c even i=￿ c ￿ i p (1 − p) 2 c/2 2 When c is even, we add a term at the end to account for the fact that the decoder has a fifty-fifty chance of guessing correctly when it receives a codeword with an equal number of 0’s and 1’s. Figure 6-1 shows the probability of decoding error from Eq.(6.2) as a function of the code rate for the replication code. The y -axis is on a log scale, and the probability of error is more or less a straight line with negative slope (if you ignore the...
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