Digital Signal Processing Reference
In-Depth Information
for the outer code
(
1
=
2
K
,
if
c ¼
(
c
1
,
...
,
c
K
,
c
Kþ
1
,
...
,
c
N
) is a code word;
Pr(
c
)
¼
0,
otherwise.
Since the scale factor 1
/
2
K
will contribute a common factor to all terms in the
calculations to follow, we may remove it, thus obtaining the indicator function
f
(
c
)
for the code
1,
if
c
is a code word;
f
(
c
)
¼
0,
otherwise
:
The likelihood function Pr(
djc
) is in principle a deterministic function that factors are
Pr(
djc
)
¼
Y
N
Pr(
d
i
jc
j
),
with
j ¼P
(
i
)
i¼
1
since
d
i
is determined from
c
j
alone, with
j ¼ P
(
i
) the index map from the interleaver.
The values in
d
are not known with certainty; however, only estimates of them are
available from the inner decoder. Many techniques may be used to inject these esti-
mates; the most common is to inject the extrinsic probabilities introduced above.
Specifically, first de-interleave the values
fT
i
(
d
i
)
g
and associate them with the bits
(
c
j
) according to
T
j
(
c
j
¼
0)
¼ T
i
(
d
i
¼þ
1)
T
j
(
c
j
¼
1)
¼ T
i
(
d
i
¼
1)
with
j ¼P
(
i
)
:
Observe that since each bit
d
i
is determined from a sole bit
c
j
, we may equally regard
T
i
(
d
i
) as a function of
c
j
rather than
d
i
, and reindex the values according to the inter-
leaver
j ¼ P
(
i
). (The use of
T
j
and
T
i
can, admittedly, be confusing at first sight,
although we shall always include the argument
c
j
or
d
i
to distinguish interleaved
from de-interleaved values.) We may then usurp the channel likelihood values as
Pr(
d
i
jc
j
)
T
j
(
c
j
)
:
This then gives, for the marginal calculations
Pr(
c
j
jd
)
X
c
i
,
i
=
j
f
(
c
)
Y
N
T
l
(
c
l
)
l¼
1
¼ T
j
(
c
j
)
X
c
i
,
i
=
j
f
(
c
)
Y
l
T
l
(
c
l
)
|
{z
}
extrinsic probability
i
=
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