Cryptography Reference
In-Depth Information
If all the
s
j
(
t
)
signals are transmitted with the same probability (
π
j
=1
/M
)
,
the term
C
j
reduces to
E
j
/
2
. In addition, if all the
s
j
(
t
)
signals have in
addition the same energy
E
j
=
E
(the case of phase or frequency shift keying
modulation), then the MAP criterion is simplified and becomes:
−
T
T
s
j
(
t
)
if
r
(
t
)
s
j
(
t
)
dt >
r
(
t
)
s
n
(
t
)
dt
∀
n
=
jn
=1
,
2
,
···
,M
(2.61)
0
0
2.2.2 Performance of the coherent receiver
Amplitude shift keying with
M
states
For an M-ASK modulation, the
s
j
(
t
)
signals are of the form:
s
j
(
t
)=
A
j
h
(
t
)cos(2
πf
0
t
+
ϕ
0
)
with:
A
j
=(2
j
−
1
−
M
)
Aj
=1
,
2
,
···
,M
They define a space of dimension
N
=1
and thus observation
R
at the output
of the demodulator reduces to a component
r
.
T
r
=
r
(
t
)
ν
(
t
)
dt
0
with
ν
(
t
)=
T
cos(2
πf
0
t
+
ϕ
0
)
.
For a continuous data stream, the estimation of the symbols
a
i
is done by
integrating the product
r
(
t
)
ν
(
t
)
on each time interval
[
iT,
(
i
+1)
T
[
.Ifamatched
filter is used rather than an integrator, the sampling at the output of the filter
is realized at time
(
i
+1)
T
.
On the interval
[0
,T
[
, assuming
iid
information data
d
i
, all the amplitude
states have the same probability and decision rule (2.59) leads to:
1
2
s
j
>rs
n
−
1
2
s
n
A
j
if
rs
j
−
∀
n
=
j
(2.62)
where
A
j
represents the amplitude estimated with:
A
j
cos(2
πf
0
t
+
ϕ
0
)
ν
(
t
)
dt
=
A
j
T
2
T
1
T
s
j
=
if
f
0
>>
(2.63)
0