An alternative way to view this is that the block diagram has two components: a DD and a decision
part. The first dashed block is, in fact, a conventional DD while the second dashed block is performing
the operations of a decision. Earlier we presented a block diagram for the DD where the
decision device was drawn to actually perform the nearest neighbor function. Here the same function
is essentially performed.
What this shows us is that the conventional DD is indeed a max a posteriori probability detector.
Up until this point the conventional DD was derived from basic intuition given the differential encoding
rule used at the transmitter. There we had one equation with one unknown and so the detector rules
were trivial to write down.
We must now present the decision rule where we observe 3 symbols and make a joint decision on
2 symbols in the observation window. We can simply write down the decision rule to be
(4.96)
Here we see the metric consists of three parts. The first part corresponds to an ordinary DD using an
observation window of 2 symbols. The second part also corresponds to an ordinary DD except the
inputs have been delayed by 1 symbol. The third part corresponds to a 2-symbol DD using an observation
window of 3 symbols.
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