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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124 MAXIMUM LIKELIHOOD SEQUENCE DETECTION<br />
Often, if the decision depth is large enough, there is no difference between this<br />
decision and the decision without a decision depth (i.e., infinite decision depth). If<br />
this is true, then finite decision depth will keep selecting symbols from the same<br />
path. Sometimes a consistency check is made. If the algorithm switches which path<br />
it is taking symbols from, then an error is declared. This is not always necessary,<br />
as error correction coding (see Chapter 8) can be used to correct errors made this<br />
way.<br />
Another practical issue is metric growth. As branch metrics are accumulated,<br />
the path metric may become too large to be represented by the computing device<br />
(overflow). One solution is metric renormalization, in which the same value is<br />
subtracted from each path metric at a certain iteration.<br />
We have assumed that the first symbol is known. If this is not the case, then we<br />
need to hypothesize different values for this symbol as well.<br />
6.3.1.1 Flow diagram A flow diagram of the Viterbi algorithm (assuming infinite<br />
decision depth) is shown in Fig. 6.9. Before we walk through the diagram, let's set<br />
up some assumptions and notation.<br />
We assume that the first and last symbols sent are «o and s^„_i. We have access<br />
to received samples that can be modeled as<br />
L-l<br />
r m \= 5Z c t s m-e + n m . (6.34)<br />
e=u<br />
We also assume we have access to received samples at symbol periods m\ through<br />
rri2, which are not necessarily 0 and N s — 1, respectively.<br />
To simplify the notation, we will denote the symbols that state i represents<br />
at current processing time m as q m (i) = \q m {i) ... g m _(z,_2)(¿)]· T ne number of<br />
current states at processing time m is denoted Ns(m). At the beginning, for M-ary<br />
modulation, there will be between M and M L ~ l states, as m\ should be between<br />
0 and L — \ (ideally 0). In the "middle" of the processing, there will M L ~ l states,<br />
regardless of m. At the end of the processing, there will be between M L ~ l and M<br />
states, as m-2 should be between N s — 1 and N s + L — 2 (ideally N s + L — 2).<br />
Continuing with notation, the path metric for current state i at processing time<br />
m is denoted P m (i). The corresponding path history is denoted Q m (i), which keeps<br />
a list of hypothetical symbol values qk for k = m through k = 0.<br />
We start by initializing the "previous" path metrics at time m — m-y — 1 to zero,<br />
giving<br />
P mi _ 1 (t)=0, i = 0 ... JV s (mi-l)-l. (6.35)<br />
Then we increment m by one so we can process the first received sample available.<br />
For each possible "current" state i, we would compute candidate metrics by adding<br />
a path metric for "previous" state j at time m — 1 to a branch metric. State j<br />
would be one of the states in set A(i), corresponding to valid state combinations.<br />
Thus, for a particular current state ¿n, we would compute the candidate metrics<br />
where<br />
C m (io, j) = Pm-i(j) + B(r m ,i n ,j), (6.36)<br />
B(r m ,i t) ,j)<br />
r m - C,)