Cryptography Reference
In-Depth Information
the modulating signal if the latter is analogue. In the case of a digital signal, the
modulating signal is a series of elements of a finite set, or symbols, applied to the
modulator at discrete instants that are called
significant instants
.Thisseries
is called the
digital message
and we assume that the symbols are binary data
applied periodically at the input of the modulator, every
T
b
seconds, therefore
with a binary rate
D
=1
/T
b
bits per second. The binary data of this series are
assumed to be independent and identically distributed (iid). A given digital, for
example binary, message can be replaced by its "
m
th extension" obtained by
grouping the initial symbols into packets of
m
. Then the symbols are numbers
with
m
binary digits (or
m
-tuples), the total number of which is
M
=2
m
,
applied to the modulator at significant instants with period
mT
b
.Ifthe
m
th
extension is completely equivalent to the message (of which it is only a different
description), then the signal modulated by the original message and the signal
modulated by its
m
th extension do not have the same properties, in particular
concerning their bandwidth, since the larger
m
is, the narrower the bandwidth.
The choice of integer
m
thus allows the characteristics of the modulated message
to be varied.
Consider the complex signal
σ
(
t
)=
a
exp[
j
(2
πf
0
t
+
ϕ
)]
(2.2)
whose
s
(
t
)
is the real part, where
j
is the solution to the equation
x
2
+1=0
.
We can represent
σ
(
t
)
as the product
σ
(
t
)=
α
exp(2
πjf
0
t
)
,
(2.3)
where only the first factor
α
=
a
exp(
jϕ
)
(2.4)
depends on the parameters
a
and
ϕ
that represent the data to be transmitted.
The values taken by this first factor for all possible values of the parameters
can be represented by points in the complex plane. The set of these points is
then called a
constellation
and the complex plane is called a
Fresnel plane
.The
modulated signal (2.1) is the real part of the complex signal
σ
(
t
)
defined by
(2.3).
If the correspondence established between the modulating signal and the
variable parameters is instantaneous, the modulation is said to be
memoryless
.
It can be useful for this correspondence to be established between the variable
parameters and a function of the values taken later by the modulating signal. For
example, if the latter is analogue, a conventional modulation process (called fre-
quency) involves making
ϕ
vary proportionally to the integral of the modulating
signal in relation to time. In the same way, in the case of a digital modulation,
the constellation point can be chosen to represent the symbol present at the in-
stant considered, and the modulation is then said to be
memoryless
,orasymbol
obtained by combining it with other later symbols. A modulation can therefore
