Cryptography Reference
In-Depth Information
2.3
Transmission on a band-limited channel
2.3.1 Introduction
In this chapter, so far we have assumed that the bandwidth allocated to the
transmission is infinite. We will now envisage a more realistic situation where
a bandwidth W is available to transmit the modulated signal. In this band W
the channel is assumed to be frequency non-selective. We shall restrict ourselves
to the case of linear modulations of the M-ASK, M-PSK and M-QAM types
that have a power spectral density made up of a main lobe of width 2 /T ,where
1 /T is the modulation speed, and sidelobes with zero crossing at n/T .The
bandwidth of a linearly modulated signal is therefore, strictly speaking, infinite.
The modulated signal must consequently be filtered by an emission filter before
being placed at the input of the transmission channel. We are now going to
determine what the minimum band W is necessary to transmit the modulated
signal without degradation of the performance compared to a transmission with
an infinite bandwidth. The frequency response of the emission and reception
filters will also be established.
In what follows, we will consider the complex envelope of the modulated
signal and the equivalent baseband response of the emission filter. Without
compromising the generality of our remarks, this avoids introducing the carrier
frequency, which complicates the notations.
The complex envelope of an M-ASK, M-PSK and M-QAM signal has the
expression:
s e ( t )= A
i
c i h ( t
iT )
(2.103)
where h ( t ) is a unit amplitude rectangular pulse shape of width T ,and c i =
a i + jb i is a modulation symbol with:
M-ASK
c i = a i
b i =0
M-PSK
a i =cos φ i
b i =sin φ i
symbols M
a i and b i
M-QAM
ary
Let g ( t ) be the impulse response of the emission passband filter centred on
the carrier frequency. This waveform can be written:
g ( t )= g c ( t )cos(2 πf 0 t + θ 0 )
g s ( t )sin(2 πf 0 t + θ 0 )
(2.104)
or equivalently:
g ( t )=
e {
g e ( t )exp[ j (2 πf 0 t + θ 0 )]
}
(2.105)
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