Cryptography Reference
In-Depth Information
Figure 7.26 - Algorithm for determining the transfer function
IE
=
T
(
IA,Eb/N
0
)
b. Definition of the
apriori
mutual information
Hypotheses:
•
Hyp. 1: when the interleaving is large enough, the distribution of the input
extrinsic information can be approximated by a Gaussian distribution after
a few iterations.
•
Hyp.
2: probability density
f
(
z
|
x
)
satisfies the exponential symmetry
condition, that is,
f
(
z
|
x
)=
f
(
−
z
|
x
)
exp
(
−
z
)
.
The first hypothesis allows the
a priori
LLR
Z
A
ofaSISOdecodertobemod-
elled by a variable having independent Gaussian noise
n
z
, with variance
σ
z
and
expectation
μ
z
, applied to the transmitted information symbol
x
according to
the expression
Z
A
=
μ
z
x
+
n
z
The second hypothesis imposes
σ
z
=2
μ
z
. The amplitude of the extrinsic infor-
mation is therefore modelled by the following distribution:
√
4
πμ
z
exp
μ
z
x
)
2
4
μ
z
1
(
λ
−
f
(
λ
|
x
)=
−
(7.60)
From (7.59) and (7.60), observing that
f
(
z
|
1
)=
f
(
−
z
|−
1)
, we deduce the
a
priori
mutual information:
√
4
πμ
z
exp
log
2
2
1+exp(
dλ
+
∞
μ
z
)
2
4
μ
z
1
(
λ
−
IA
=
−
×
−
λ
)
−∞
or again
exp
λ
σ
z
2
2
2
σ
z
+
∞
1
√
2
πσ
z
−
IA
=1
−
−
×
log
2
[1 + exp (
−
λ
)]
dλ
(7.61)
−∞
