Digital Signal Processing Reference
In-Depth Information
in which
v
j
denotes the channel output at this stage rather than
y
j
, to avoid confusion
with Section 3.5.1 treating the case involving intersymbol interference. Optimal
decoding is accomplished with the aid of the
a posteriori
probability mass function
Pr(
c
)Pr(
v
j
c
)
Pr(
v
)
Pr(
cjv
)
¼
which is evaluated for the particular
v ¼
[
v
1
,
...
,
v
2
K
] received, as
c
varies among the
set of code words
C
from the encoder. The bitwise optimal solution is obtained by
thresholding the marginal evaluations from this function
X
Pr(
cjv
)
_
X
c
[
C
c
j
¼
0
Pr(
cjv
),
j ¼
1, 2,
...
,2
K
c
[
C
c
j
¼
1
and choosing
c
j
¼
1 if the left-hand side is greater than the right-hand side, and
c
j
¼
0
otherwise.
Now, at each time step
j
, the encoder produces two bits,
c
j
and
c
jþK
, which we
shall combine into a quaternary symbol
C
j
¼
[
c
j
c
jþK
] taking four configurations.
Similarly, the channel outputs
v
j
and
v
jþK
[cf. (3.3)] may be combined into a composite
V
j
¼
[
v
j
v
jþK
] conveying information on the encoder output at time
j
.
The trellis description shows that at any time instant
j
, the internal state of the
encoder assumes one of four configurations,
viz.
0
0
0
1
1
0
1
1
x
j
¼
or
x
j
¼
or
x
j
¼
or
x
j
¼
|
{z
}
S
0
|
{z
}
S
1
|
{z
}
S
2
|
{z
}
S
3
in which
S
0
,
...
,
S
3
denote the state configurations so indicated. The
a posteriori
prob-
abilities Pr(
c
j
jv
) may then be related to the state transition probabilities Pr(
x
j
2
1
¼ S
m
0
,
x
j
¼ S
m
jv
). As an example
Pr(
c
j
¼
0
jv
)
¼
Pr(
x
j
1
¼ S
0
,
x
j
¼ S
0
jv
)
þ
Pr(
x
j
1
¼ S
1
,
x
j
¼ S
2
jv
)
þ
Pr(
x
j
1
¼ S
2
,
x
j
¼ S
3
jv
)
þ
Pr(
x
j
1
¼ S
3
,
x
j
¼ S
1
jv
)
because the four state transitions in the probabilities on the right-hand side encompass
the event that
c
j
¼
0.
The state transition probabilities can be expressed using three conditional prob-
ability sequences
a
j
(
m
)
¼
Pr(
x
j
¼ S
m
jV
1
,
...
,
V
j
)
b
j
(
m
)
¼
Pr(
V
jþ
1
,
...
,
V
K
jx
j
¼ S
m
)
g
j
(
m
0
,
m
)
¼
Pr(
x
j
¼ S
m
;
V
j
jx
j
1
¼ S
m
0
)
Search WWH ::
Custom Search