Cryptography Reference
In-Depth Information
Taking into account the fact that the variables
b
n
,whatever
n
is, are zero mean
and non-correlated, we obtain:
∞
r
p
b
(
t
)
E
{
}
=
E
{
b
p
b
n
}
ν
n
(
t
)=0
∀
p
=1
,
2
,
···
,N
(2.57)
n
=
N
+1
The quantities
r
p
and the noise
b
(
t
)
are therefore independent since Gaussian.
In conclusion, the optimal receiver can base its decision only on the quantities
r
p
,
p
=1
,
2
,
···
,N
with:
T
r
p
=
r
(
t
)
ν
p
(
t
)
dt
(2.58)
0
Passing the signal
r
(
t
)
provided by the transmission channel to the
N
quantities
r
p
is called demodulation.
Example
Let us consider an M-PSK modulation for which the
s
j
(
t
)
signals are of the
form:
s
j
(
t
)=
Ah
(
t
)cos(2
πf
0
t
+
ϕ
0
+
φ
j
)
The signals
s
j
(
t
)
define a space with
N
=2
dimensions if
M>
2
. The normed
and orthogonal functions
ν
p
(
t
)
,
p
=1
,
2
can be expressed respectively as:
ν
1
(
t
)=
T
cos(2
πf
0
t
+
ϕ
0
)
ν
2
(
t
)=
T
sin(2
πf
0
t
+
ϕ
0
)
and the signals
s
j
(
t
)
can be written:
s
j
(
t
)=
A
T
A
T
2
2
cos
φ
j
h
(
t
)
ν
1
(
t
)
−
sin
φ
j
h
(
t
)
ν
2
(
t
)
After demodulation, the observation
R
=(
r
1
,r
2
)
is equal to:
r
1
=
A
T
2
r
2
=
A
T
2
cos
φ
j
+
b
1
sin
φ
j
+
b
2
The observation
R
=(
r
1
,r
2
)
depends only on the states of phase
φ
j
andonthe
noise. We say that observation
R
=(
r
1
,r
2
)
is in
baseband
since independent of
the carrier frequency
f
0
.
The demodulation operation requires knowledge of the frequency
f
0
and
the phase
ϕ
0
of the carrier, the signals
ν
p
(
t
)
having to be synchronous with
the carrier generated by the modulator. That is the reason why we speak of
synchronous demodulation or coherent demodulation.
