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

690 DIGITAL COMMUNICATIONS UNDER LINEARLY DISTORTIVE CHANNELS
Figure 12.8
A decision
feedback
equalizer with
fractionally
spaced samples.
z1[k] q-----
I :
I I
1
-
Fi (z)
I
I
I
I
Feedforward
filtcr(s)
Feedback
filter B(z)
equalizers are also known as feedforward (FFW) equalizers. The advantages ofFFW equalizers
lie in their simple implementation as FIR filters and in the straightforward design approaches
they accommodate. FFW equalizers require much lower computational complexity than the
nonlinear MLSE receivers.
On the other hand, FFW equalizers do suffer from several major weaknesses. First, the
TSE or FSE in their FFW forms can cause severe noise amplifications depending on the
underlying channel conditions. Second, depending on the roots of the channel polynomials,
the FFW equalizer(s) may need to be very long to be effective, particular when the channels
are nearly singular. To achieve simple and effective channel equalization without risking noise
amplification, a decision feedback equalizer (DFE) proves to be a very useful tool.
Recall that FFW equalizers generally serve as a channel inverse filter (in ZF design) or a
regularized channel inverse filter (in MMSE design). The DFE, however, comprises another
feedback filter in addition to a feedforward filter. The feedforward filter is identical to linear
TSE or FSE, whereas the feedback filter attempts to cancel ISI from previous data samples
using data estimates generated by a memoryless decision device. The feedforward filter may
be operating on fractionally spaced samples. Hence, there may be m parallel filters as shown
in Fig. 12.8.
The basic idea behind the inclusion of a feedback filter B(z) is motivated by awareness that
the feedforward filter output d[k] may contain some residual ISi that can be more effectively
regenerated by the feedback filter output and canceled from v[k]. More specifically, consider
the case in which the feedforward filter output d[k] consists of
N
d[k] = Sk-u + I: C;Sk-i + W[n]
'-.,-'
i=u+l
'-,.-' noise
residual ISi
(12.58)
There is a residual ISi term and a noise term. If the decision output is very accurate such that
Sk-u = Sk-u
then the feedback filter input will equal to the actual data symbol. If we denote the feedback
filter as
N-u
B(z) = L b;z - i
i=l