Cryptography Reference
In-Depth Information
Chapter 2
Digital communications
2.1
Digital Modulations
2.1.1 Introduction
The function of modulation has its origin in radio-electric communications. An
emitter can radiate an electromagnetic wave only in a limited, and generally
narrow, portion of the spectrum, which can roughly be described as a frequency
"window" with
Δ
f
centred on a frequency
f
0
,with
Δ
f<<f
0
. The messages to
be transmitted, that can be either analogue (for example speech) or digital (for
example Morse code), are represented by signals that occupy only the bottom of
the frequency spectrum. The spectrum of the signal, coming from a microphone
in the case of speech, does not extend beyond a few kilohertz. The same thing
applies for a signal that represents the two short (Tit) or long (Tat) elements of
Morse "code", since the speed of handling several dozen signs per second is very
small compared to the frequency
f
0
that is measured in hundreds of kilohertz
or in megahertz. Another use for modulation is frequency multiplexing which
enables several simultaneous communications on the same wideband (cable or
optical fibre) support, that are easily separated due to the fact that they each
occupy a specific bandwidth, not connected to that of any other.
A sinusoidal wave
f
0
can be represented by the function
s
(
t
)=
a
cos(2
πf
0
t
+
ϕ
)
(2.1)
where
t
denotes the time and
f
0
is constant. Modulation involves making one or
other of parameters
a
the amplitude, and
ϕ
the phase, depend on the signal to
be transmitted. The modulated signal
s
(
t
)
then has a narrow spectrum centred
on
f
0
,whichiswhatwewant.
The signal to be transmitted will in the sequel be called the
modulating
signal
. Modulation makes one of the parameters
a
and
ϕ
vary as a function of
