mohatta2015.pdf

130 MAXIMUM LIKELIHOOD SEQUENCE DETECTION
Substituting (6.57) and (6.59) into (6.45) and adding the term (6.60) gives
N,+L-2
s = arg max YJ —\v{mT)—v q {mT)\ 2
qeSp ;
ra=0
N. + L-2
= arg min V^ \v(mT) - v q (mT)\ 2 . (6.61)
qeS,, '
Thus, for the case of root-Nyquist pulse shaping and symbol-spaced paths, we have
written the MLSD receiver in terms of minimizing the Euclidean distance metric.
The Euclidean distance metric can also be used in the more general case by preprocessing
the received signal with a matched filter followed by a whitening filter
(makes the noise samples uncorrelated). The uncorrelated noise samples allow the
Euclidean distance metric to be used, and the overall metric is referred to as the
Forney metric.
We can interpret the direct form as a special case of the Forney form [Li]. The
Forney form uses the Euclidean distance metric operating after performing whitened
matched filtering. When the channel is assumed to be symbol-spaced, the whitened
matched filter reduces to a filter matched to pulse shape, giving r(m).
6.3.3 Given statistics
We formulated MLSD assuming we had r(i) for all t. Sometimes we are given a set
of statistics to work with that may or may not be sufficient. However, we can still
formulate a conditional MLSD solution, conditioned on the information available.
The procedure is similar to that given for r(t) except that one typically works with
discrete samples rather than a continuous time function.
6.3.4 Fractionally spaced MLSD
When working with r(i), MLSD is considered fractionally spaced if the medium response
is fractionally spaced, so that the front-end MF can be expressed as matching
to the pulse shape followed by a fractionally spaced MF. Thus, if fractionally spaced
samples are used to complete the matched filtering, then MLSD is considered fractionally
spaced.
Whether fractionally spaced MLSD is needed depends on the excess bandwidth,
the sample timing, and whether the channel is dispersive. See Chapter 2 on fractionally
spaced MF for more details.
6.3.5 Approximate forms
Here we consider three standard approximate MLSD forms, sometimes referred to
as near-ML approaches. Unlike the Viterbi algorithm, which effectively performs
an exhaustive search of all possible sequences, these approximate forms perform a
nonexhaustive search tree search. However, they hopefully perform an intelligent
search, so that performance losses are minimal.