Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 593

19.1 shows the digital communication system described earlier in Chap. 2, with
one modification. The detector output s est ( n ) , which is the decision about the
transmitted symbol s ( n ) , is delayed and fed back through a digital filter to
modify the value of the detector input

s ( n ) . As explained in Sec. 5.7, this
receiver is called a decision feedback receiver, or a decision feedback equalizer ,
abbreviated as DFE . Here the filter B ( z ) has the form

B ( z )= b 1 z − 1 + b 2 z − 2 + ... + b N z −N .

(19 . 1)

Since b 0 = 0, there is no delay-free loop. That is, the detector input s ( n )attime

n is affected only by the past detected symbols or past decisions

s est ( n − 1) ,s est ( n − 2) ,...,s est ( n − N ) . (19 . 2)

Figure 19.2 shows the detailed structure of the DFE system for the case of a
third order feedback filter B ( z ) .
The basic idea behind DFE is that if the past symbols have been identified
with low probability of error, then those decisions can be successfully used to
improve the detection performance of the present symbol. This idea works when
there is a su cient amount of statistical dependence between the received sig-
nal samples r ( n ). Even though the transmitted symbol stream s ( n )mayhave
independent samples, such a dependence can arise in r ( n ) owing to the filtering
effect of the cascaded system G ( jω ) H ( jω ) F ( jω ) . Since the DFE system involves
a nonlinear device (the detector) in the feedback loop, the equalizer is nonlinear ,
and we say that we have a nonlinear transceiver. The transceivers considered in
earlier sections, which do not contain DFEs, are called linear transceivers; more
specifically, they have linear equalizers.
The performance of the DFE system depends on the mean square error at
the detector input, namely

| 2 . E = E | s ( n ) − s ( n ) (19 . 3)

For fixed F ( jω ) ,H ( jω ) , and G ( jω ), this error depends on how the feedback filter
B ( z ) is chosen. The mean square error can be estimated theoretically under the
assumption that the past decisions have been perfect, that is, the decisions fed
back via B ( z ) are correct. More specifically, one assumes that

s est ( n − k )= s ( n − k ) , 1 ≤ k ≤ N, (19 . 4)

and proceeds with the theoretical analysis to compute . Under the preceding

assumption of perfect past decisions , it is in fact possible to optimize the prefilter
F ( jω ), equalizer G ( jω ) , and feedback filter B ( z ) such that the mean square error
E

E

is minimized. The details of such analysis can be found in Proakis [1995] and
references therein (also see the end of Sec. 5.7 for historical remarks). When
the error probabilities are small, the “perfect past decisions” assumption is quite
reasonable, but when the receiver operates in the range of larger error proba-
bilities, such assumptions tend to be unsatisfactory. The error in the decision
s est ( n ) tends to “propagate” via the feedback loop, and many future decisions
can be affected this way.