Cryptography Reference
In-Depth Information
where
f
0
is the frequency of the carrier,
ϕ
0
its phase and
h
(
t
)
a rectangular
pulse of unit amplitude and width
T
.
Two situations can arise depending on whether the length
m
of the groups of
data at the input of the modulator is even or not. If
m
is even, then
M
=2
m
is
a perfect square
(4
,
16
,
64
,
256
, ...
)
; in the opposite case,
M
is simply a power
of two
(8
,
32
,
128
, ...
)
.
When
m
is even, the group of data can be separated into two sub-groups of
length
m/
2
, each being associated respectively with amplitudes
A
j
and
A
j
that
,
√
M
. In Figure 2.5
are represented the constellations of the 16-QAM and 64-QAM modulations.
These constellations are said to be square.
√
M
)
A,
take their values in the set
(2
j
−
1
−
j
=1
,
2
,
···
Figure 2.5 - Constellations of two QAM-type modulations.
When
m
is odd, the M-QAM signal can no longer be obtained as a combina-
tion of two quadrature amplitude-modulated carriers. However, we can build the
M-QAM signal from an N-QAM signal modulated classically on two quadrature
carriers, where
N
is a square immediately higher than
M
by preventing
(
N
M
)
states. For example, 32-QAM modulation can be obtained from 36-QAM modu-
lation where
A
j
and
A
j
take the values
(
−
±
A,
±
3
A,
±
5
A
)
by preventing the four
5
A
)
for the pairs (
A
j
and
A
j
). The constellation of
the 32-QAM modulation is shown in Figure 2.6.
The M-QAM signal can again be written in the form:
states of amplitude
(
±
5
A,
±
s
j
(
t
)=
V
j
h
(
t
)cos(2
πf
0
t
+
ϕ
0
+
φ
j
)
(2.24)
with:
V
j
=
(
A
j
)
2
+(
A
j
)
2
φ
j
=tan
−
1
A
j
A
j
In this form, the M-QAM modulation can be considered as a modulation
combining phase and amplitude. Assuming that the phase takes
M
1
=2
m
1
states and the amplitude
M
2
=2
m
2
states, the modulated signal transmits
log
2
(
M
1
M
2
)=
m
1
+
m
2
data every
T
seconds. Figure 2.7 shows the constellation
