mohatta2015.pdf

62 ZERO-FORCING DECISION FEEDBACK EQUALIZATION
3.3 THE MATH
We start this section by determining when a zero-forcing solution is possible. The
basic equations for ZF DFE are then given.
A full zero-forcing solution is only possible in a few special cases. This is because
the pulse shape is typically bandlimited, making it nonzero for many symbol periods
in both the future and the past. Even if the pulse shape is root-Nyquist and
partial MF is employed, ISI from future symbol periods occurs if fractionally spaced
equalization is employed or the medium path delays are fractionally spaced.
As we will consider the more general case in Chapter 5, we will focus on a special
case for which full ZF is possible. Specifically, we will assume the following.
1. The channel can be modeled with symbol-spaced paths (τ( = ÍT).
2. Root-Nyquist pulse shaping is used at the transmitter, and partial MF is used
at the receiver (R p (qT) = 6(q)).
3. The sampling phase is aligned with the first tap delay of the channel (i n = TO).
4. Symbol-spaced sampling is used (T s = T).
With these simplifying assumptions, the received samples (r m = v(mT)) can be
modeled as
L-l
r m = \/E~s Σ 9e s m-e + u m , (3.31)
where u m is zero-mean, complex Gaussian r.v. with variance No.
The traditional block diagram for DFE is given in Fig. 3.3. The received signal
is processed by a feedforward filter (FFF). The output of a feedback filter (FBF) is
then subtracted, removing ISI from past symbol periods. The result is a decision
variable, which is used by a decision device or detector (DET) to determine a
detected symbol value.
We have divided the FFF into two filters: a front-end filter matched to the pulse
shape (partial MF or PMF) and a forward filter (FF). Also, because the FF is
linear, the DFE can be formulated with the FBF being applied prior to the FF, as
shown in Fig. 3.4. This formulation is convenient as it decouples the forward and
backward filter designs under the design assumptions to be used.
The FBF subtracts the influence of past symbols on r m , giving
which can be modeled as
L-l
y m = r m -^2 ges(m - (.), (3.32)
1=1
L-l
Vm h V~Ës9oSm + Σ 9e\s{m - i) - s(m - ί)\ + u m . (3.33)
e=i
We will assume that the past symbol decisions are correct, simplifying (3.33) to
Um \= \/E s gns m + u m . (3.34)