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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 61: 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
ﬁftyﬁfty chance of guessing correctly when it receives a codeword with an equal number
of 0’s and 1’s.
Figure 61 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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 Fall '13
 HariBalakrishnan

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