mohatta2015.pdf

128 MAXIMUM LIKELIHOOD SEQUENCE DETECTION
rewrite the branch metric in (6.51) as
B(m)=Re 5*(m)
min(m,L., — 1) 1
2z(m) - S(0)q{m) - 2(1 - ¿(m)) ^ S(i)qi(m - C) \
(6.53)
Now we can use the Viterbi algorithm. After an initial start-up phase, we end up
with the Viterbi algorithm with memory L s — 1. When processing z(m), we have
current states defined by different values of q m = \q(m), q(m — 1),..., q(m — (L s —
2)] and previous states defined by different values of q m _i = [q{m — l),q{m —
2),...,q(m-(L s -l)\.
The branch metric in (6.53) is sometimes referred to as the Ungerboeck metric.
Observe that the front-end signal processing consists of a matched filter. In
practice, this can be implemented using the partial-MF samples and matching to
the medium response, as shown in (2.60).
Because we need only work with the set of N s matched filter decision variables,
this set sometimes referred to as a set of sufficient statistics. Here the term statistic
means decision variable. Thus, another way to motivate the matched filter is that
it provides a set of sufficient statistics for MLSD.
6.3.2.1 Direct and Forney forms In the beginning of the chapter, we used a Euclidean
distance metric with a symbol-spaced medium response. We will call this
form the "direct form." We can derive this result from the MLSD formulation. A
symbol-spaced medium response implies
L-l
h{t) = Σ gtM* - itT), (6.54)
where we have used l\ instead of Î to avoid confusion with index Í for the s-
parameters. Substituting (6.54) into (6.46) and (6.47) and assuming p(i) is root-
Nyquist,
L-\
z{m)=Y j g tl v{{m + e l )T) (6.55)
£,=()
L-l
S(£) = Σ ».+/. ( 6·56 )
where v(t) is the partial matched filter signal defined in (2.60) and we define gi 1 to
be zero for t\ less than zero or greater than L — 1.