Digital Signal Processing Reference
In-Depth Information
Assuming the encoder is initialized to
x
(0)
¼ S
0
, the forward recursion is initialized as
a
0
(0)
¼
1,
a
0
(1)
¼ a
0
(2)
¼ a
0
(3)
¼
0
:
In the same vein, a recursion for the sequence
b
i
(
m
) may be developed as [17]
b
j
1
(
m
)
¼
X
m
0
g
j
(
m
,
m
0
)
b
j
(
m
0
)
where the sum again is among all transitions
g
j
(
m
,
m
0
) which connect state configur-
ation
S
m
0
at time
j
to state configuration
S
m
at time
j
2
1. These define the backward
recursions. Again for the example encoder of Figure 3.4, these calculations appear as
b
j
1
(0)
¼ g
j
(0, 0)
b
j
(0)
þg
j
(0, 2)
b
j
(2)
b
j
1
(1)
¼ g
j
(1, 0)
b
j
(0)
þg
j
(1, 2)
b
j
(2)
b
j
1
(2)
¼ g
j
(2, 1)
b
j
(1)
þg
j
(2, 3)
b
j
(3)
b
j
1
(3)
¼ g
j
(3, 1)
b
j
(1)
þg
j
(3, 3)
b
j
(3)
as sketched in Figure 3.6(
b
). Again, the values
b
j
2
1
(
m
) should be scaled to sum to one
b
j
1
(0)
þb
j
1
(1)
þb
j
1
(2)
þb
j
1
(3)
¼
1
:
If the input sequence to the encoder is chosen to drive the final state
x
K
to
S
0
, then the
backward recursion is initialized as
b
K
(0)
¼
1,
b
K
(1)
¼ b
K
(2)
¼ b
K
(3)
¼
0
:
If instead no such constraint is placed on the trellis, then the backward recursion may
be initialized with a uniform distribution
1
b
K
(0)
¼ b
K
(1)
¼ b
K
(2)
¼ b
K
(3)
¼
4
:
To specify the terms
g
j
(
m
,
m
0
), we repeat the definition here
g
j
(
m
0
,
m
)
¼
Pr(
x
j
¼ S
m
;
V
j
jx
j
1
¼ S
m
0
)
¼
Pr(
V
j
jx
j
¼ S
m
,
x
j
1
¼ S
m
0
)Pr(
x
j
¼ S
m
jx
j
1
¼ S
m
0
)
:
The transition probability Pr(
x
j
¼ S
m
jx
j
2
1
¼ S
m
0
) is the
a priori
probability that the
encoder input at time
j
provoked the transition from configuration
S
m
0
to
S
m
. The like-
lihood term Pr(
V
j
jx
j
¼ S
m
,
x
j
2
1
¼ S
m
0
) reduces to the channel likelihood function for
the measurement
V
j
, given the encoder output induced by the state transition from
S
m
0
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