mohatta2015.pdf

144 MAXIMUM LIKELIHOOD SEQUENCE DETECTION
that need to be considered. As an example, consider M-QAM, root-Nyquist pulse
shaping, and a channel consisting of two symbol-spaced paths. Suppose we first
perform linear equalization. However, instead of selecting one value for each symbol,
we keep the N best values, where N < M. Next we perform a reduced search
Viterbi algorithm in which we only consider the N values kept for each symbol.
This means we have N L ~ l states instead of M L_1 and N L branch metrics instead
of M L . The parameter TV is a design parameter, allowing us to trade complexity
and performance. For example, if M — 16 and N = 4, we reduce the number of
states from 16 to 4 and the number of branch metrics from 256 to 16. While we
have added the complexity of linear equalization, the overall complexity can still
be reduced due to the reduced search.
Various extensions are possible. The first equalization stage need not be linear
equalization, though it should be relatively simple. Also, the first stage can actually
consist of multiple sub-stages. For example, the first stage could be LE, keeping
N\ < M best symbol values. The second stage could be a hybrid form (see next
subsection) in which pairs of symbols are jointly detected, considering TVf combinations
and keeping only N2 possible pair values, where N2 < Nf. The reduced
search Viterbi algorithm would then consider symbol pairs.
Multiple stages can also be used in conjunction with centrólas. In the first stage,
the 16-QAM constellation is approximated as QPSK using the centroid of each
quadrant. One or more centroide can be kept for further consideration. In the
second stage, the centroide are expanded into the four constellation points that
they represent.
6.4.3.4 Sub-block forms In previous chapters, we introduced the notions of group
LE and group DFE. We revisit these here, but view them from an MLSD point
of view. We can interpret group LE as a form of sub-block MLSD. By modeling
the symbols in the sub-block as M-ary symbols and modeling the other symbols
as being Gaussian (colored noise), we obtain an approximate form of MLSD. This
gives us a hybrid form that is part MLSD (joint detection of symbols within the
sub-block) and part LE (linear filtering prior to joint detection). We can do the
same with DFE.
6.5 AN EXAMPLE
GSM is a 2G cellular system [Rai91, Goo91]. GSM employs Gaussian Minimum
Shift Keying (GMSK) with a certain precoding. Using [Lau86], this form of GMSK
can be approximated as a form of partial response BPSK [Jun94]. Thus, even in
a nondispersive channel, there is ISI due to the transmit pulse shape which spans
roughly 3 symbol periods. The modem bit rate is 270.833 kbaud (symbol period 3.7
/is). To handle a delay spread of up to 7.4 μβ, the equalizer needs to handle ISI from
4 previous symbols. With MLSD, this leads to 16 states in the Viterbi algorithm.
In [Ave89], the Ungerboeck metric is used in a 32-state Viterbi algorithm (handles
more dispersion). Joint detection of cochannel interference is considered in [Che98].
In [Ben94], the M-algorithm is used to reduce the complexity of a 16-state MLSD.
Setting M between 4 and 6 provides performance comparable to full MLSD for the
scenarios considered. The M algorithm is also used in [Jun95a].