mohatta2015.pdf
signal processing from power amplifier operation control point of view
signal processing from power amplifier operation control point of view
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128 MAXIMUM LIKELIHOOD SEQUENCE DETECTION<br />
rewrite the branch metric in (6.51) as<br />
B(m)=Re 5*(m)<br />
min(m,L., — 1) 1<br />
2z(m) - S(0)q{m) - 2(1 - ¿(m)) ^ S(i)qi(m - C) \<br />
(6.53)<br />
Now we can use the Viterbi algorithm. After an initial start-up phase, we end up<br />
with the Viterbi algorithm with memory L s — 1. When processing z(m), we have<br />
current states defined by different values of q m = \q(m), q(m — 1),..., q(m — (L s —<br />
2)] and previous states defined by different values of q m _i = [q{m — l),q{m —<br />
2),...,q(m-(L s -l)\.<br />
The branch metric in (6.53) is sometimes referred to as the Ungerboeck metric.<br />
Observe that the front-end signal processing consists of a matched filter. In<br />
practice, this can be implemented using the partial-MF samples and matching to<br />
the medium response, as shown in (2.60).<br />
Because we need only work with the set of N s matched filter decision variables,<br />
this set sometimes referred to as a set of sufficient statistics. Here the term statistic<br />
means decision variable. Thus, another way to motivate the matched filter is that<br />
it provides a set of sufficient statistics for MLSD.<br />
6.3.2.1 Direct and Forney forms In the beginning of the chapter, we used a Euclidean<br />
distance metric with a symbol-spaced medium response. We will call this<br />
form the "direct form." We can derive this result from the MLSD formulation. A<br />
symbol-spaced medium response implies<br />
L-l<br />
h{t) = Σ gtM* - itT), (6.54)<br />
where we have used l\ instead of Î to avoid confusion with index Í for the s-<br />
parameters. Substituting (6.54) into (6.46) and (6.47) and assuming p(i) is root-<br />
Nyquist,<br />
L-\<br />
z{m)=Y j g tl v{{m + e l )T) (6.55)<br />
£,=()<br />
L-l<br />
S(£) = Σ ».+/. ( 6·56 )<br />
where v(t) is the partial matched filter signal defined in (2.60) and we define gi 1 to<br />
be zero for t\ less than zero or greater than L — 1.