Cryptography Reference
In-Depth Information
where
r
(
t
)=
s
(
t
)+
b
(
t
)
is the signal received by the receiver and
b
(
t
)
is a
AWGN, with zero mean and power spectral density equal to
N
0
/
2
.Quantity
φ
i
(
t
)
represents a realization of phase
φ
(
t
)
associated with branch
l
of the trellis
on time interval
[
iT,
(
i
+1)
T
[
.
Taking into account the fact that the noise can be put in the form
b
(
t
)=
b
c
(
t
)cos(2
πf
0
t
+
ϕ
0
)
b
s
(
t
)sin(2
πf
0
t
+
ϕ
0
)
and that
f
0
>>
1
/T
, the branch
metric can again be written:
−
(
i
+1)
T
(
i
+1)
T
z
i
=
r
c
(
t
)cos
φ
i
(
t
)
dt
+
r
s
(
t
)sin
φ
i
(
t
)
dt
(2.96)
1
T
iT
where the signals
r
c
(
t
)
and
r
s
(
t
)
are obtained after transposition into baseband
of
r
(
t
)
(multiplying
r
(
t
)
by
cos(2
πf
0
t
+
ϕ
0
)
and
−
sin(2
πf
0
t
+
ϕ
0
)
respectively,
then lowpass filtering).
cos
φ
i
(
t
)=cos
2
πh
nT
)+
θ
i−L
i
a
l
n
q
(
t
−
n
=
i
−
L
+1
(2.97)
sin
φ
i
(
t
)=sin
2
πh
nT
)+
θ
i−L
i
a
l
n
q
(
t
−
n
=
i−L
+1
Putting:
i
ψ
i
(
t
)=2
πh
a
l
n
q
(
t
−
nT
)
n
=
i
−
L
+1
branch metric
z
i
can again be written in the form:
z
i
=cos
θ
i−L
A
l
+sin
θ
i−L
B
l
(2.98)
with:
(
i
+1)
T
A
i
=
(
r
c
(
t
)cos
ψ
i
(
t
)+
r
s
(
t
)sin
ψ
i
(
t
))
dt
iT
(
i
+1)
T
B
i
=
(
r
s
(
t
)cos
ψ
i
(
t
)
r
c
(
t
)sin
ψ
i
(
t
))
dt
−
iT
For MSK modulation, it is possible to decode symbols
a
i
by using a receiver
similar to that of 4-PSK modulation. Indeed, the
MSK
signal can be written in
the following form:
S
(
t
)=
A
i
2
iT
)cos
πt
c
2
i−
1
h
(
t
−
2
T
cos(2
πf
0
t
+
ϕ
0
)
2
T
sin(2
πf
0
t
+
ϕ
0
)
(2.99)
−
i
(2
i
+1)
T
)sin
πt
c
2
i
h
(
t
−
