Cryptography Reference
In-Depth Information
The
N
integrators of the demodulator can be replaced by
Nh
(
T
−
t
)
impulse
response filters, each followed by a sampler at time
t
=
T
.
+
∞
s
j
(
t
)
ν
j
(
t
)
∗
h
(
T
−
t
)=
s
j
(
τ
)
ν
j
(
τ
)
h
(
T
−
t
+
τ
)
dτ
−∞
where
represents the convolution product.
Sampling at
t
=
T
, we obtain:
∗
T
s
j
(
t
)
ν
j
(
t
)
∗
h
(
T
−
t
)
|
t
=
T
=
s
j
(
τ
)
ν
j
(
τ
)
dτ
0
which is equal to the output of the integrator.
The filter
h
(
T
t
)
is called the
filter matched
to waveform
h
(
t
)
of width
T
.We
can show that this filter maximizes the signal to noise ratio at its output at time
t
=
T
.
For a continuous data stream, the integration is performed on each interval
[
iT,
(
i
+1)
T
[
i
=1
,
2
,
−
···
and, if we use matched filters, the sampling is realized
at time
(
i
+1)
T
.
After demodulation, the receiver must take a decision about the group of
data transmitted on each time interval
[
iT,
(
i
+1)
T
[
. Todothis,itsearches
for the most probable signal
s
j
(
t
)
by using the maximum
a posteriori
(MAP)
probability criterion:
s
j
(
t
)
if
Pr
{
s
j
(
t
)
|
R
}
>
Pr
{
s
p
(
t
)
|
R
}∀
p
=
jp
=1
,
2
,
···
,M
where
s
j
(
t
)
is the signal that was transmitted and
R
=(
r
1
···
r
N
)
the
output of the demodulator. To simplify the notations, the time reference has
been omitted for the components of observation
R
.
Pr
r
p
···
denotes the
probability of
s
j
(
t
)
conditionally to the knowledge of observation
R
.
Using Bayes' rule, the MAP criterion can again be written:
{
s
j
(
t
)
/R
}
s
j
(
t
)
if
π
j
p
(
R
|
s
j
(
t
))
>π
p
p
(
R
|
s
p
(
t
))
∀
p
=
jp
=1
,
2
,
···
,M
where
π
j
=Pr
{
s
j
(
t
)
}
represents the
a priori
probability of transmitting the sig-
nal
s
j
(
t
)
and
p
(
R
s
j
(
t
))
is the probability density of observation
R
conditionally
to the emission of the signal
s
j
(
t
)
by the modulator.
Taking into account the fact that the components
r
p
=
s
jp
+
b
p
of observation
R
conditionally to the emission of the signal
s
j
(
t
)
are non-correlated Gaussian,
with mean
s
jp
and variance
N
0
/
2
,wecanwrite:
|
p
=1
p
(
r
p
|
p
=1
p
(
r
p
|
N
N
s
j
(
t
)
if
π
j
s
j
(
t
))
>π
n
s
n
(
t
))
∀
n
=
jp
=1
,
2
,
···
,M
