Cryptography Reference
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s ) is therefore finally given by:
Conditional probability Pr( s
|
m
s )=exp
Pr( s
|
X l,j L a ( x i,j )
(11.17)
j =1
As for the second term P ( y i |
s ,s ) , it quite simply represents the likelihood
P ( y i |
z i ) of observation y i relative to branch label z i associated with the tran-
sition considered. The latter corresponds to the symbol that we would have
observed at the output of the channel in the absence of noise:
L− 1
z i =
h k x i−k
(11.18)
k =0
The sequence of symbols ( x i ,x i− 1 ,...,x i−L +1 ) occurring in the computation
of z i is deduced from the knowledge of initial state s and of information symbol
X l associated with transition s
s . In the presence of zero-mean circularly-
symmetric complex additive white Gaussian noise with total variance σ w ,we
obtain:
exp
2
1
πσ w
|
y i
z i |
s ,s )= P ( y i |
P ( y i |
z i )=
(11.19)
σ w
Factor 1 /πσ w is common to all the branch metrics and can therefore be
omitted in the computations. On the other hand, we see here that calculating
branch metrics γ i− 1 ( s ', s ) requires both knowledge of the impulse response of the
equivalent discrete channel and knowledge of the noise variance. In other words,
in the context of a practical implementation of the system, the MAP equalizer
will have to be preceded by a channel estimation procedure.
To summarize, after computing branch metrics γ i− 1 ( s ,s ) then performing
the forward and backward recursions, the a posteriori LLR L ( x i,j ) is finally
given by:
α i− 1 ( s ) γ i− 1 ( s ,s ) β i ( s )
( s ,s ) /x i,j =1
L ( x i,j )=ln
(11.20)
α i− 1 ( s ) γ i− 1 ( s ,s ) β i ( s )
( s ,s ) /x i,j =0
In reality and in accordance with the turbo principle, it is not this a posteriori
information that is propagated to the SISO decoder, but rather the extrinsic
information. Here, the latter measures the equalizer's own contribution in the
global decision process, excluding the information relating to the bit considered
coming from the decoder at the previous iteration, that is to say, the a priori
LLR L a ( x i,j ) . If we develop the expression of branch metric γ i− 1 ( s ,s ) in the
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