mohatta2015.pdf

116 MAXIMUM LIKELIHOOD SEQUENCE DETECTION
Mathematically, we form a metric
M, = (r 1 -f 1 ) 2 + (r 2 -f 2 ) 2 + ..., (6.1)
where the subscript q indicates a particular symbol sequence hypothesis. This
metric is referred to as the Euclidean distance metric, because it is related to the
distance between two points in geometry.
Let's try it for the Alice and Bob example. We form predicted values using
h = -lOtt+Qfl, (6.2)
f 2 = -1092+9(7!. (6.3)
For example, consider the hypothesis g () = +1, q\ = — 1, and q 2 = +1 (an incorrect
hypothesis). Then
h = -10(-1) +9(+l) = 19 (6.4)
f 2 = -l0(+l)+9(-l) = -19. (6.5)
Recall that r\ = 1 and r 2 — —7, so that the metric would be
M q = (1 - 19) 2 + (-7 - (-19)) 2 = 468. (6.6)
We would then need to consider all other hypothetical symbol combinations. All
combinations and their associated metrics are shown in Table 6.1. Observe that
the metric is minimized for q n — +1, q x = +1 and qi = +1. Thus, the detected
symbol values would be s\ = +1 and s 2 = +1·
Table 6.1
Example of sequence metrics
Index Hypothesis Metric
9o 9i 92
~1 +1 +1 +1 40
2 +1 +1 -1 680
3 +1 -1 +1 468
4 +1 -1 -1 388
5 -1 +1 +1 436
6 -1 +1 -1 1076
7 -1 -1 +1 144
8 -1 -1 -1 64
This approach is referred to as Maximum Likelihood Sequence Detection (MLSD).
The "sequence detection" part of the name comes from the fact that we detect the
entire symbol sequence together, rather than detecting symbols one at a time. The
"maximum likelihood" part comes from the fact that the metric is related to the
likelihood of noise values taking on certain values. Specifically, the log of the likelihood
that a noise value n\ equals the number 7 is related to —7 2 or —49. The
closer the number is to zero, the more likely the noise value. Thus, the highest
log-likelihood value is —0 2 or 0. A block diagram for forming an MLSD metric is
given in Fig. 6.1. Generation of the predicted received values is shown in Fig. 6.2.
In our approach above, instead of maximizing the likelihood, we minimize the
negative of the log-likelihood. For example, instead of maximizing — {r\ — f\) 2 , we
minimize the Euclidean distance (r\ - r\) 2 .