Cryptography Reference
In-Depth Information
is then written in the following way in the form of a z -transform:
L
1
h k ( i ) z −k
h ( z )=
(11.2)
k =0
The impulse response of the channel is assumed to have finite duration ( L coef-
ficients), which is a realistic hypothesis in practice in most scenarios.
Equation (11.1) shows that generally, received symbol y i is a function of the
symbols transmitted before, or after (if the channel introduces a propagation de-
lay) information symbol x i considered at instant i . In accordance with what was
introduced in Chapter 2, we then say that the received signal is spoiled by inter-
symbol interference (ISI). If we now assume that the transmission channel does
not vary (or very little) on the duration of a transmitted block of information,
model(11.1) can be simplified as follows:
L− 1
y i =
h k x i−k + w i
(11.3)
k =0
where we have suppressed the time dependency from the coecients of the equiv-
alent discrete channel. The representation of the equivalent discrete channel in
the form of a digital filter with finite impulse response presented in Figure 11.1
comes directly from (11.3). The coe cients of the filter are precisely those of
the impulse response of the channel.
Figure 11.1 - Representation of the equivalent discrete channel in the form of a digital
filter.
ISI can be a major obstacle for establishing a good quality digital trans-
mission, even in the presence of very low noise. As an illustration, we have
shown in Figure 11.2 the constellation of the symbols received at the output of
a channel highly perturbed by ISI, for a signal to noise ratio of 20 dB 1 ,given
that we have transmitted a sequence of discrete symbols with four phase states
(QPSK modulation). We thus observe that when the ISI is not processed by an
1 We recall that a signal to noise ratio of 20 dB corresponds to a power of the transmitted
signal 100 times higher than the power of the additive noise on the link.
 
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