Cryptography Reference
In-Depth Information
For an M-ASK signal, the different states of the modulated signal are situ-
ated on a straight line and its constellation is therefore one-dimensional in the
Fresnel plane. There are many ways to make the association between the value of
the amplitude of the modulated signal and the particular realization of a group
of data of m =log 2 ( M ) data. In general, we associate with two adjacent values
taken by the amplitude, two groups of data that differ by only one binary value.
This particular association is called Gray coding . It enables the errors made by
the receiver to be minimized. Indeed, when the receiver selects an amplitude
adjacent to the emitted amplitude because of noise, which corresponds to the
most frequent situation, we make only one error for m =log 2 ( M ) data trans-
mitted. We show in Figure 2.2 two examples of signal constellations modulated
in amplitude by Gray coding.
Figure 2.2 - Example of 4-ASK and 8-ASK signal constellations with Gray coding
The mean energy E s used to transmit an M -ary symbol is equal to:
T
E A j cos 2 (2 πf 0 t + ϕ 0 ) dt
E s =
0
where E A j , the expectation of A j
has the expression A 2 ( M 2
1) / 3 .
1
T , the previous relation gives the mean
Making the hypothesis that f 0 >>
energy E s :
E s = A 2 T
2
( M 2
1)
(2.8)
3
The mean energy E b used to transmit a bit is:
E s
log 2 ( M )
E b =
(2.9)
For a transmission with a continuous data stream, the amplitude modulated
signal can be written in the form:
S ( t )= A
i
a i h ( t
iT )cos(2 πf 0 t + ϕ 0 )
(2.10)
where the
are a sequence of M -ary symbols, called modulation symbols,
which take the values (2 j
{
a i }
1
M ) , j =1 , 2 ,
···
,M . In the expression of the
modulated signal, i is the time index.
The signal S ( t ) can again be written in the form:
S ( t )=
e {
s e ( t )exp( j (2 πf 0 t + ϕ 0 ))
}
(2.11)
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