Cryptography Reference
In-Depth Information
amplitude of the received symbol has a value immediately lower or higher than
the value of the transmitted amplitude).
Pes
log
2
(
M
)
E
b
N
0
Peb
=
if
>>
1
(2.66)
Phase shift keying with M states
For M-PSK modulation , the
s
j
(
t
)
signals are of the form:
s
j
(
t
)=
A
cos(2
πf
0
t
+
ϕ
0
+
φ
j
)
(2.67)
with:
φ
j
=(2
j
+1)
π
M
+
θ
0
j
=0
,
1
,
···
,
(
M
−
1)
The
s
j
(
t
)
signals, for
M>
2
, define a two-dimensional space. Observation
R
at
the output of the demodulator is therefore made up of two components
(
r
1
,r
2
)
with:
T
T
r
1
=
r
(
t
)
ν
1
(
t
)
dt
r
2
=
r
(
t
)
ν
2
(
t
)
dt
0
0
where
ν
1
(
t
)=
T
cos(2
πf
0
t
+
ϕ
0
)
et
ν
2
(
t
)=
T
sin(2
πf
0
t
+
ϕ
0
)
.
Using decision rule (2.50) and assuming the information data
iid
, all the states
of phase have the same probability and the decision is the following:
−
2
2
φ
j
if
r
p
s
jp
>
r
p
s
np
∀
n
=
j
(2.68)
p
=1
p
=1
with:
s
j
1
=
A
T
2
s
j
2
=
A
T
1
T
cos
φ
j
2
sin
φ
j
and
if
f
0
>>
(2.69)
Taking into account the expressions of
s
j
1
and of
s
j
2
, the decision rule can again
be written:
φ
j
if
r
1
cos
φ
j
+
r
2
sin
φ
j
>r
1
cos
φ
n
+
r
2
sin
φ
n
∀
n
=
j
(2.70)
The coherent receiver for an M-PSK modulation is represented in Figure 2.15.
It is made up of two components called the phase component (projection of the
received signal on
ν
1
(
t
)=
2
/T
cos(2
πf
0
t
+
ϕ
0
)
) and the quadrature component
(projection of the received signal on
ν
2
(
t
)=
2
/T
sin(2
πf
0
t
+
ϕ
0
)
) and a decision
circuit.
The emission by the modulator of a phase state corresponds to the transmis-
sion of a group of
log
2
(
M
)
information bit. The error probability on a group of
