mohatta2015.pdf

MORE MATH 143
6.4.3 More approximate forms
Complexity is often dominated by the number of path metrics maintained (number
of states if the Viterbi algorithm is used) and the number of branch metrics
computed. Complexity can be reduced by reducing these numbers (at the expense
of performance). In explaining these approaches, we will assume the direct form
(Euclidean distance) and the Viterbi algorithm. It should be noted that some of
the approaches have issues with regards to the Ungerboeck form.
6.4.3.1 Channel shortening One way to reduce the number of states in the Viterbi
algorithm is to reduce the memory of the channel or at least the memory of the
significant channel coefficients. This can be done by prefiltering the received signal
using a filter that concentrates signal energy into a few taps. Such an approach
is approximate because either noise coloration due to prefiltering is not accounted
for properly (which would lead to the same state space size) or smaller channel
coefficients at large lags are ignored. There has been more recent work on channel
shortening with all-pass filters, avoiding the noise coloration issue.
6.4.3.2 RSSE and DFSE Reduced state sequence estimation (RSSE) provides two
approaches to reducing the number of Viterbi states: decision feedback sequence
estimation (DFSE) and set partitioning (SP). Recall that with the Viterbi algorithm,
there are M L ~ l states. With DFSE, the state space is reduced by reducing
the memory assumed by the Viterbi algorithm (reducing L). For example, consider
16-QAM, root-Nyquist pulse shaping, and a channel with three, symbol-spaced
taps. The Viterbi algorithm would normally use 16 2 = 256 states and compute
16 3 = 4096 branch metrics each iteration. If we ignore the channel path with the
largest delay, we would only have 16 states and 256 branch metrics. Instead of
completely ignoring ISI from the largest delay path, we subtract it using the symbol
value stored in the path history (which depends on the previous state being
considered). Thus, unlike DFE, the value subtracted may be different depending
on the previous state being considered.
With SP, the state space is reduced by grouping possible symbol values into
sets, reducing the effective M. Consider the example of a two-tap, symbol-spaced
channel with 16-QAM and root-Nyquist pulse shaping. The Viterbi algorithm
would have 16 states and form 256 branch metrics. We can partition these states
into sets. For example, we can form four sets of four symbol values each (M' = 4).
For now, assume each set corresponds to a particular quadrant in the I/Q plane.
(This is not the best partition, but it simplifies the explanation). With pure MLSD,
there would be four parallel connections between each of these "super-states." With
SP, a decision is made on the previous symbol, so that the four values in the set
are reduced to one. This reduces the number of parallel connections to one. Thus,
we would like symbols in the same set to be as far apart as possible, which is not
the case when forming sets using quadrants in the I/Q plane.
In practice, we can use a combination of these two approaches. How to form good
set partitions and how to handle memory two or larger are described in [Eyu88].
6.4.3.3 Assisted MLD The idea with assisted MLD (AMLD) is to use a separate,
simpler equalizer to "assist" the MLSD process by reducing the number of sequences