Cryptography Reference
In-Depth Information
Probability of erroneous decoding of a codeword
For a linear block code C ( n, k ) of error correction capability t , the codeword
transmitted will be wrongly decoded if there are t + j errors, j =1 , 2 ,
t , in the received word r . For a binary symmetric channel of probability p ,
the probability P e, word of performing an erroneous decoding of the transmitted
codeword is upper bounded by:
···
,n
n
j
p j (1
n
p ) n−j
P e, word <
(4.24)
j = t +1
We can also determine the binary error probability P e, bit on the information data
after decoding. In presence of erroneous decoding, the maximum a posteriori
likelihood decoder adds at most t errors by choosing the codeword with the
minimum distance from the received word. The error probability is therefore
bounded by:
( j + t ) n
j
p j (1
n
1
n
p ) n−j
P e, bit <
(4.25)
j = t +1
If the transmission is performed with binary phase modulation (2-PSK, 4-PSK),
probability p is equal to:
2 erfc RE b
1
p =
N 0
where R is the coding rate, E b the energy received per transmitted information
bit and N 0 the unilateral power spectral density of the noise. Figure 4.5 shows
the binary error probability and word error probability after algebraic decoding
for the (15,7) BCH code. The modulation is 4-PSK and the channel is Gaussian.
The higher bounds expressed by (4.24) and (4.25) respectively are also plotted.
Soft decoding
Considering a channel with additive white Gaussian noise and binary phase mod-
ulation transmission (2-PSK or 4-PSK), the components r j ,
j =0 , 1 ,
···
,n
1
of the received word r have the form:
r j = E s c j + b j ,
c j =2 c j
1
where c j =0 , 1 is the symbol in position j of codeword c , c j is the binary
symbol associated with c j , E s is the energy received per transmitted symbol
and b j is white Gaussian noise, with zero mean and variance equal to σ b .
Maximum a posteriori likelihood decoding
Decoding using the maximum a posteriori likelihood criterion means search-
ing for codeword c such that:
c |
= c
c = c
Pr
{
c
|
r
}
> Pr
{
r
}
,
c
C ( n, k )
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