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

12.6 Decision Feedback Equalizer 69 1
then we have
N -u
v[k] = d [k] - L b/sk-u-i
i=l
N
N-u
= Sk-u + L CjSk-i - L biSk-u-i + w[n]
i= u +l i=l
N
N-u
= Sk-u + L CiSk-i - L biSk-u-i + w[n]
i=u+l i=l
N -u
= Sk-u + L (c u+i - bj)Sk-u-i + w[n]
i=l
(12.59)
To eliminate the residual ISi, the feedback filter should have coefficients
bi = C u+i i = 1, 2, ... , N - U - 1
With these matching DFE parameters, the residual ISi is completely canceled. Hence, the input
to the decision device
v[k] = Sk-u + w[n]
contains zero ISL The only nuisance that remains in v[k] is the noise. Because the noise term
in d[k] is not affected or amplified by the feedback filter, the decision output for the next time
instant would be much more accurate after all residual ISi has been canceled.
Our DFE analysis so far has focused on the ideal operation of DFE when the decision
results are correct. Traditionally, the design and analysis of DFE has often been based on such
an idealized operating scenario. The design of DFE filters must include both the feedforward
filters and the feedback filter. Although historically there have been a few earlier attempts to
fully decouple the design of the feedforward filter and the feedback filter, the more recent work
by Al Dhahir and Cioffi 9 provides a comprehensive and rigorous discussion.
In the analysis of a DFE, the assumption of con-ect decision output leads to the removal
of ISi in v[k], and hence, a better likelihood that the decision output is accurate. One cannot
help but notice this circular "chicken or egg" argument. The truth of the matter is that the
DFE is inherently a nonlinear system. More importantly the hard decision device is not even
differentiable. As a result, most traditional analytical tools developed for linear and nonlinear
systems no longer apply. For this reason, the somewhat ironic chicken-egg analysis becomes
the last resort. Fortunately, for high-SNR systems, this circular argument does yield analytical
results that can be closely matched by experiments.
Error Propagation in DFE
Because of its feedback structure, the D FE does suffer from the particular phenomenon known
as en-or propagation. For example, when the decision device makes an error, the en-oneous
symbol will be sent to the feedback filter and used for ISi cancellation in Eq. (12.59). However,
because the symbol is incon-ect, instead of canceling the ISi caused by this symbol, the
canceling subtraction may instead strengthen the ISi in v[k]. As a result, the decision device is
more likely to make another subsequent error, and so on. This is known as error propagation.