Cryptography Reference
In-Depth Information
The evolution of phase
φ
(
t
)
can be represented by a trellis whose states are
defined by
(
a
i−L
+1
,a
i−L
+2
,
···
,a
i−
1
;
θ
i−L
)
,thatis:
(
M
L−
1
p
)
states
if
m
even
(2.95)
(
M
L−
1
2
p
)
states
if
m
odd
Note that the complexity of the trellis increases very rapidly with the parameters
M
and
L
. For example, for a modulation with quaternary symbols
(
M
=4)
with a partial response with modulation index
h
=1
/
2
and with parameter
L
=4
, the trellis has 256 states. For MSK and GMSK, the symbols
a
i
are
Figure 2.20 - Trellis associated with phase
φ
(
t
)
for MSK modulation.
binary
(
M
=2)
and the modulation index
h
is
1
/
2
,thatis,
m
=1
and
p
=2
.
Phase
θ
i−L
therefore takes its values in the set
{
0
,π/
2
,π,
3
π/
2
}
and the trellis
associated with phase
φ
(
t
)
has
2
L−
1
4
states. Figure 2.20 shows the trellis
associated with phase
φ
(
t
)
for MSK modulation.
To decode symbols
a
i
we use the Viterbi algorithm whose principle is recalled
below. For each time interval
[
iT,
(
i
+1)
T
[
, proceed in the following way:
×
•
for each branch
l
leaving a state of the trellis at instant
iT
calculate metric
z
i
as defined later, that is, for MSK and GMSK,
2
L
×
4
metrics have to
be calculated;
•
for each path converging to instant
(
i
+1)
T
towards a state of the trel-
lis, calculate the cumulated metric, then select the path with the largest
cumulated metric, called the survivor path;
•
among the survivor paths, trace back along
s
branches of the path having
the largest cumulated metric and decode symbol
a
i−s
;
•
continue the algorithm on the following time interval.
Branch metric
z
i
has the expression:
(
i
+1)
T
z
i
=
r
(
t
)cos(2
πf
0
t
+
φ
i
(
t
)+
ϕ
0
)
dt
iT
