B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

676 DIGITAL COMMUNICATIONS UNDER LINEARLY DISTORTIVE CHANNELS
Fi g
ure 12.5
A SISO discrete
linear channel
model for TSE.
Discrete
channel
H(z)
Equalizer
filter
F(z)
On the other hand, the modulation formats adopted in high-speed dial-up modems are
highly complex. For example, the V.32bis (14.4 kbit/s) modem uses a trellis-coded QAM constellation
of size 128 (with 64 distinct symbols) at the symbol rate of 2400 baud (symbols/s).
In such applications, even a relatively short L = 5 FIR channel would require MLSE to have
over 1 billion states. In fact, at higher bit rates, dial-up modems can use size 256 QAM or even
size 960 QAM. As a result, the large number of states in MLSE makes it completely unsuitable
as a receiver in such systems. Consequently, suboptimal equalization approaches with
low complexity are much more attractive. The design of simple and cost effective equalizers
(deployed in applications including voiceband dial-up modems) is discussed next.
12.3 LINEAR T-SPACED EQUALIZATION (TSE)
When the receiver filter is matched to the transmission pulse p(t) only, it is no longer optimum.*
Even if the ideal matched filter q(-t) is known and applied, it is quite possible in practice for
the sampling instant to have an offset to such that the sampling takes place at t = nT + to .
Such a sampling offset is known as a timing error. When there is a timing error, the receiver
is also not optimum. It is in fact commonplace for practical communication systems to have
unknown distortive channels and timing jitters. Nevertheless, T-spaced equalization is simpler
to implement. Here we discuss the fundamental aspects of TSE design.
Because T -spaced sampling leads to a simple discrete time linear system Eq. ( 12.17) as
shown in Fig. 12.5, the basic linear equalizer is simply a linear filter F(z) followed by a direct
QAM decision device. The operational objective of the equalizer (filter) F(z) is to remove as
much ISi as possible from its output d [n]. We begin our discussion on the T-spaced equalizer
(TSE) by denoting the (causal) equalizer transfer function
F(z) = Lf[i]z - i
If the channel noise w[n] is included, the TSE output is
d [n] = F(z)z[n] = F (z)H (z)sn + F(z)w[n]
'-,-' '-,-'
signal term noise term
(12.28)
We denote the joint channel equalizer transfer function as
(X)
C(z) = F(z)H (z) = L c;z - i
The goal of the equalizer F(z) is to clean up the ISi in d [n] to achieve an error-free decision
i=O
Sn = dee (d [n]) = Sn-u (12.29)
* The sufficient statistics shown by G. D. Forney 1 are not necessarily retained.