The second phase of the TP-SMAP decoder is basically identical to the optimal code construction algorithm, except that the sequential search algorithm is applied on the trellis TL,N instead of the search tree. Therefore, proving optimality here is similar to the proof provided in Appendix A, except that we need to prove that the path metric is nondecreasing along any path on TL,N.
Lemma 2. In the second phase, the decoding metric is nondecreasing along any path on trellis TL,N, i.e.,
Proof. Based on the backward VA of the first phase, h(Sj) is the minimal metric among all paths from level j to the final node; i.e.,
h(Sj) = min
Therefore,
m x(i,j)(0,0)
= g x(i,j)(0,0)
+ h(Sj) (B.3)
≤ g x(i,j)(0,0)
+ g
x(i+1,j+l)(i,j)
+ h (Sj+l) (B.4)
= g
x(i+1,j+l)(0,0)
+ h(Sj+l) (B.5)
= m
x(i+1,j+l)(0,0)
. (B.6)
The second phase of the TP-SMAP repeatedly pops out the node with smallest m(·) from the Decoding Stack. Suppose that the algorithm encounters the first top node which reaches the final state SL,N; then by the above Lemma, no matter how the algorithm continues, extending any node in the Decoding Stack will generate a node with decoding metric m(·) no smaller then the top node. Hence, according to the MAP decision in (2.4), the TP-SMAP certainly outputs the MAP decision.
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