Cryptography Reference
In-Depth Information
The energy
E
s
for transmitting a phase state, that is, a group of
log
2
(
M
)
binary
data, is equal to:
T
A
2
cos
2
(2
πf
0
t
+
ϕ
0
+
φ
j
)
dt
=
A
2
T
2
E
s
=
if
f
0
>>
1
/T
(2.18)
0
Energy
E
s
is always the same whatever the phase state transmitted. The energy
used to transmit a bit is
E
b
=
E
s
/
log
2
(
M
)
.
For the transmission of a continuous data stream, the modulated signal can
bewrittenintheform:
S
(
t
)=
A
i
a
i
h
(
t
−
iT
)cos(2
πf
0
t
+
ϕ
0
)
−
i
iT
)sin(2
πf
0
t
+
ϕ
0
)
(2.19)
b
i
h
(
t
−
where the modulation symbols
a
i
and
b
i
take their values in the following sets:
cos
(2
j
+1)
M
+
θ
0
a
i
∈
0
≤
j
≤
(
M
−
1)
b
i
∈
sin
(2
j
+1)
M
+
θ
0
(2.20)
0
≤
j
≤
(
M
−
1)
The signal
S
(
t
)
can again be written in the form given by (2.11) with:
s
e
(
t
)=
A
i
c
i
h
(
t
−
iT
)
,
i
=
a
i
+
jb
i
(2.21)
Taking into account the fact that the data
d
i
provided by the source of infor-
mation are
iid
, the modulation symbols
c
i
are independent, with zero mean and
unit variance.
The psd of the signal
S
(
t
)
is again equal to:
γ
S
(
f
)=
1
f
0
)+
1
4
γ
s
e
(
f
−
4
γ
s
e
(
f
+
f
0
)
with this time:
γ
s
e
(
f
)=
A
2
T
sin
πfT
πfT
2
(2.22)
the psd looking like that of Figure 2.3.
Quadrature Amplitude Modulation using two quadrature carriers (M-
QAM)
For this modulation, also called
Quadrature Amplitude Modulation
(M-QAM),
it is two quadrature carriers
cos(2
πf
0
t
+
ϕ
0
)
and
sin(2
πf
0
t
+
ϕ
0
)
that are
amplitude modulated. The modulator provides signals of the form:
−
s
j
(
t
)=
A
j
h
(
t
)cos(2
πf
0
t
+
ϕ
0
)
A
j
h
(
t
)sin(2
πf
0
t
+
ϕ
0
)
−
(2.23)