Digital Signal Processing Reference
In-Depth Information
received sequence
r[k-4]
r[k-3]
r[k-2]
r[k-1]
r[k]
S[k]={d[k-1],d[k-2]}
0
0
0
0
0 0 0
1 0 1
2 1 0
3 1 1
1
1
1
1
0
0
0
0
0
0
0
1
1
1
1
0
1
1
1
1
0
0
0
0
1
1
1
1
1= arg min {
M
(
s
[
k
])}
most likely sequence
{0,1,2,3}
S
(
s
[
k
] = 1)
=
{0,2,3,3,1} →
d
= {1,1,1,0}
s
[
k
]
∈
Trellis
ˆ
ˆ
Fig. 10
Maximum likelihood sequence estimation (MLSE): A trellis for a 4-state (
L
c
=
2) MLSE
algorithm with
L
r
5 is shown. The trellis diagram illustrates valid state sequences with solid lines
as paths through the trellis. Above each line connecting two states is the corresponding symbol that
would have been transmitted for that transition
=
found that setting
D
equal to a small multiple of the channel memory
L
c
works well
in practice, and setting
D
3
L
c
does not improve performance significantly. For
D
in this range and when more than one survivor path exists at
k
>
D
, selecting the one
with the smallest path metric at
k
appears to work well in practice.
current received sample
r
−
to compute the branch metrics, and update the four
path metrics that correspond to the likelihood of the best path (survivor path) ending
at each of the four states in one symbol period. The survivor paths are stored in
memory. The path metric update step is followed by the trace-back step where one
of the survivor paths is traced back to a predetermined depth,
D
, and a decision is
made. The survivor path metric update is performed using an add-compare-select
(ACS) operation, which is recursive and therefore also difficult to implement at
optical line rates.
A probabilistic model for the observations
r
[
n
]
[
k
]
given the state transitions
b
[
k
]
,
i.e., given the transmitted bits
d
, is required for branch metric
computations. In a practical applications, this model must be determined adaptively
from the channel observations
r
[
k
]
,...,
d
[
k
−
L
c
]
without the aid of any training, i.e. without
any knowledge of the true bit sequence
d
[
n
]
. One such model for a baud-sampled
receiver uses an adaptive system identification algorithm based on a Volterra series
[
n
]
L
c
k
=
0
a
k
d
[
n
−
k
]+
L
c
k
=
0
L
c
∑
=
[
]=
+
[
−
]
[
−
]+
···,
r
n
c
0
b
k
,
d
n
k
d
n
(22)
0
,
=
k
where,
r
are assumed
transmitted channel symbols, and whose parameters
c
0
,
a
k
and
b
k
,
can be adaptively
estimated using the LMS algorithm.
[
n
]=
r
(
nT
)
, are baud-sampled receiver outputs,
d
[
n
]
∈{
0
,
1
}
