mohatta2015.pdf

THE MATH 121
legs of the journey is shown in Fig. 6.5. Observe that there are two ways to end up
at point D: via point B or via point C. We don't know yet whether we will travel
through point D, but if we do, we can go ahead and figure out whether it will be
via point B or point C. As travel via point C is 700 whereas travel via point B is
400, we can eliminate the route to point D via point C. This route is marked with
an X in Fig. 6.5. Similarly, of the two ways to get to point E, we can eliminate the
route via point B, as it costs more.
Such pruning of candidate paths is based on Bellman's law of optimality, which
states that any segment of the optimal path will also be optimal. Thus, if the
optimal path passes through D, then segment from A to D will also be optimal.
Based on this principle, we can redraw the partial tree search in Fig. 6.5 as a
trellis, shown in Fig. 6.6. At point D, there are two candidate paths with candidate
metrics 400 (via B) and 700 (via C). We prune the path via C. A similar pruning
occurs at point E. The next step would be to consider point C and two candidate
paths: one via D (400 + 200 = 600) and one via E (600 + 200 = 800). Clearly, the
path via D would be chosen.
Figure 6.5
Traveling salesperson tree search.
We can also build a trellis for MLSD. Consider the SISO TDM scenario in which
the channel consists of two, symbol-spaced paths. BPSK and root-Nyquist pulse
shaping is used at the transmitter, and the receiver performs filtering matched to
the pulse shape. If the subsequent sampling is aligned with the path delays, then
the resulting samples can be modeled as
r m (= cs m + ds m -i + n m . (6.30)
The trellis for this example is shown in Fig. 6.7. Let's suppose we start with a
known symbol s () = +1- Like the tree search, there are two paths corresponding to
the two values for s\. We could compute the path and branch metrics according