The resulting two code words, x and y, are XOR-ed together and weighed by their
corresponding signal strength. We wish to choose a code word candidate that minimizes the changes
in the bit positions where the signal strength is large. We have made use of a relatively safe and reliable
assumption that errors don??™t occur often in the high signal areas.
An alternative way to view error correction is to recall the transmit encoder mapping, where only
2k n-tuple code words were used. We can place rings of radius equal to the error correction capability
of the code word, say t, around valid code words. The significance of this ring is to be able to make
the following statement: Any received code word that falls within the circle, can be corrected to the
intended code word (see Fig. 5.21).
gi
min
j a7
i1
gi(yi, j xi, j)
Decode Encode
yj xj
0 0 y
0
0 1
1 0
1 1
y1
y2
y
3
X X
Let us now discuss the probability of making an erroneous decision. The probability of m errors
in a block of n bits is denoted as, P(m, n) and given below where p the decoder input bit error
probability.
(5.46)
The probability of a code word being in error is upper-bounded as
(5.47)
(5.48)
The corresponding bit error probability is given as
(5.
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