Cryptography Reference
In-Depth Information
For a continuous data stream, the signal can be written in the form:
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
)
(2.27)
b
i
h
(
t
−
√
M
)
,for
where the modulation symbols
a
i
and
b
i
take the values
(2
j
−
1
−
,
√
M
and for
M
=2
m
j
=1
,
2
,
···
with even
m
.
The signal
S
(
t
)
can be
expressed by the relations (2.11) and (2.21):
S
(
t
)=
e
{
s
e
(
t
)exp
j
(2
πf
0
t
+
ϕ
0
)
}
with
s
e
(
t
)=
A
i
iT
)
,
i
=
a
i
+
jb
i
The binary data
d
i
provided by the information source being
iid
, the modulation
symbols
c
i
are independent, with zero mean and variance equal to
2(
M
c
i
h
(
t
−
−
1)
/
3
.
The psd of the signal
S
(
t
)
is again given by (2.13) with:
A
2
T
sin
πfT
πfT
2
γ
s
e
(
f
)=
2(
M
−
1)
(2.28)
3
The spectral width of a modulated M-QAM signal is therefore, to within an
amplitude, the same as that of M-ASK and M-PSK signals.
2.1.3 Memoryless modulation with M states (M-FSK)
For this modulation, also called
Frequency Shift Keying
(M-FSK), it is the fre-
quency that is the modulated value. The modulator generates signals of the
form:
s
j
(
t
)=
Ah
(
t
)cos(2
π
(
f
0
+
f
j
)
t
+
ϕ
j
)
(2.29)
where
f
j
=
j
Δ
f
,
j
=1
,
2
,
,M
and
h
(
t
)
is a rectangular pulse with unit
amplitude and width
T
.The
ϕ
j
are random independent phases with constant
realization on the interval
[0
,T
[
.The
s
j
(
t
)
signals can therefore be generated
by independent oscillators since there is no relation between the phases
ϕ
j
.
Let us compute the correlation coecient between two modulated signals taking
different frequency states.
···
T
A
2
cos(2
π
(
f
0
+
j
Δ
f
)
t
+
ϕ
j
)cos(2
π
(
f
0
+
n
Δ
f
)
t
+
ϕ
n
)
dt
ρ
j,n
=
(2.30)
0
After integration, and assuming
f
0
>>
1
/T
, we obtain:
sin(2
π
(
j
ρ
j,n
=
A
2
T
2
−
n
)Δ
fT
+
ϕ
j
−
ϕ
n
)
sin(
ϕ
j
−
ϕ
n
)
−
(2.31)
2
π
(
j
−
n
)Δ
fT
2
π
(
j
−
n
)Δ
fT
