Cryptography Reference
In-Depth Information
be either analogue or digital, with memory or memoryless. In all that follows
below, we restrict ourselves to digital modulations that differ from each other
according to their form or number of points in their constellation, and perhaps
by a memory effect. When the latter is obtained by combining the symbol ap-
plied with later symbols, this can be interpreted as a preliminary transformation
of the digital message. Moreover, it is often necessary to ensure the continuity
of the phase to improve the shape of the spectrum of the modulated signals,
which implies a memory effect.
The choice of a modulation system depends on many factors. Modulated
signals are emitted on an imperfect channel, perturbed by the addition of par-
asitic signals collectively called
noise
and often, in radio-electricity, affected by
variations in the amplitude of the received signal, due for example to a rapid
change in the propagation conditions, a phenomenon called fading. In spite of
these channel faults, we want to receive modulating messages with a small er-
ror probability, which implies that the signals associated with them should be
as different as possible. On the other hand, whether it be radio-electricity or
multiplexing, the radio-electric spectrum is common to several users, each of
which perturbs the others. Therefore we wish to concentrate the power emitted
in a frequency interval that is limited as far as possible. This implies choosing
the modulation parameters in order to give the spectrum the most appropriate
shape. The signal spectrum of the general (2.3) shape is made up of a
main
lobe
centred on
f
0
that concentrates most of the power emitted, and
tails
or
sidelobes
where the spectral density decreases more or less rapidly in relation
to the central frequency
f
0
. Whatever the modulation system, the width of the
main lobe is proportional to the
symbol rate
R
=
D/m
=1
/mT
b
, expressed in
symb/s. The decrease in the spectral density far enough away from the cen-
tral frequency depends only on the discontinuities of the modulating signal and
its derivations. It varies in
1
/
(
f
f
0
)
2(
d
+1)
where
d
is the smallest order of a
derivation of the discontinuous signal (
d
=0
if the modulated signal itself is
discontinuous). We note that it is only possible to increase the value of
d
by
introducing an increasing delay between the instant when a modulating symbol
is applied to the modulator and the characteristic instant corresponding to it.
The main parameters associated with modulation are therefore the size and
the shape of the constellation used (on which the error probability depends, on
a given channel), the width of the main lobe of the spectrum of the modulated
spectrum and the decrease in its spectral density away from the central fre-
quency. They are largely in conflict: for example, we can only reduce the width
of the central lobe by increasing the size of the constellation, to the detriment
of the error probability for the same power. The choice of a modulation system
can therefore only result from a compromise adapted to a particular application.
Apart from the parameters indicated, the complexity of implementation must
be taken into account. For example, shaping that improves the decrease in the
secondary lobes of the spectrum by the increase of order
d
in the first discontin-
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