Cryptography Reference
In-Depth Information
bandwidth is finite. If, for example, we decide to transmit 99% of the power of
the modulated signal, which results in only a low distortion of the signal
S
(
t
)
,
then the bandwidth is about
8
/T
where
1
/T
is the symbol rate. We shall see in
Section 2.3 that it is possible to greatly reduce this band without degrading the
performance of the modulation. This remark is valid for all linear modulations.
Phase Shift Keying with M states (M-PSK)
For this modulation, also called
Phase Shift Keying
(PSK), it is the phase of
the carrier that is the modulated value. The modulator provides signals of the
form:
s
j
(
t
)=
Ah
(
t
)cos(2
πf
0
t
+
ϕ
0
+
φ
j
)
(2.15)
where
f
0
is the carrier frequency,
ϕ
0
its phase and
h
(
t
)
a rectangular pulse of
unit amplitude and width
T
. The modulated phase
φ
j
takes a value among
M
=2
m
with:
φ
j
=(2
j
+1)
π
M
+
θ
0
0
≤
j
≤
(
M
−
1)
(2.16)
The different states of the phase are equidistributed on a circle of radius
A
.
The phase
θ
0
is fixed at
π/
2
for a 2-PSK modulation and at 0 for an M-PSK
modulation with
M>
2
states.
The modulated signal can again be written in the form:
−
s
j
(
t
)=
Ah
(
t
)[cos
φ
j
cos(2
πf
0
t
+
ϕ
0
)
−
sin
φ
j
sin(2
πf
0
t
+
ϕ
0
)]
(2.17)
In this form, the M-PSK signal can be expressed as the sum of two quadrature
carriers,
cos(2
πf
0
t
+
ϕ
0
)
and
sin(2
πf
0
t
+
ϕ
0
)
, the amplitude modulated by
cos
φ
j
and
sin
φ
j
with
cos
2
φ
j
+sin
2
φ
j
=1
. We can check that when
M
is
a multiple of 4, the possible values of the amplitude of the two carriers are
identical.
In Figure 2.4 we show two constellations of a phase modulated signal with
Gray coding. The constellations have two dimensions and the different states of
the modulated signal are on a circle of radius
A
. We say that the constellation
is circular.
−
Figure 2.4 - Examples of constellations of a phase modulated signal with Gray coding.
