mohatta2015.pdf

120 MAXIMUM LIKELIHOOD SEQUENCE DETECTION
6.3 THE MATH
Mathematically, MLSD tries to find the sequence of symbol values that maximizes
the likelihood of the received signal. Let s denote the sequence of N s transmitted
symbols. Let S N " denote the set of all possible sequences. For M-ary modulation,
there would be M N " possible sequences. Then, MLSD detects the sequence using
s = arg max Pr{r(í) Ví|s = q}. (6.29)
Our notation is a little sloppy in that we use Pr{} to denote likelihood, which
can be either a discrete probability (sum to one) or a PDF value (integrates to
one). The key is that we are trying to find the hypothetical symbol sequence q that
maximizes the conditional likelihood of the received signal.
Before considering the SISO TDM scenario, we develop the Viterbi algorithm,
an approach to reduce complexity of the tree search without sacrificing performance
(at least when the channel coefficients are known). Then the SISO TDM scenario
is examined. Certain approximate forms are introduced. Performance results are
also provided.
6.3.1 The Viterbi algorithm
The Viterbi algorithm is a form of dynamic programming for efficiently performing
the tree search without explicitly forming all possible paths through the tree. To
understand how the Viterbi algorithm works, it helps to consider an analogous
problem of the traveling salesperson.
Suppose a traveling salesperson needs to get from point A to point F via airplane.
As shown in Fig. 6.4,'there are flights from point A to point B or point C. From
points B and C, there are flights to points D and E. Finally, from points D and E,
there are flights to point F.
Figure 6.4
Traveling salesperson problem.
The salesperson wants to minimize the cost of travel. Ticket costs for each
possible leg of the journey are shown in Fig. 6.4. We can use the tree search
method to consider all possible routes. The tree search considering the first two