Lemma 6.4 Any non-cycle irreducible closed walk contains at least two cycles (which may be partially or completely overlapping, or disjoint).
From these two results, it appears to be “relatively unlikely” to have a negative cost sum along a non-cycle irreducible closed walk when the cost sums along all cycles are positive.
The reasoning behind this is that the cost sum of such a walk can be written as the sum of the costs of some (at least two) positive terms corresponding to cycles in the walk, and a remain-ing term; to get a negative sum, the remainremain-ing term has to outweigh the sum of the other terms.
Unfortunately, we do not have a formal (nontrivial) result regarding the influence on the error probability of these non-cycle irreducible closed walks. To illustrate the complexity of the problem, we give in Figure 6.3 two non-cycle irreducible closed walks, both rather dif-ferent from the one in Figure 6.1.
3
3
1 1
1 1
1
1 -4
Figure 6.2 An assignment of local costs with a positive cost sum along all cycles but a negative cost sum along the indicated irreducible closed walk.
Figure 6.3 Two examples of irreducible closed walks that are not cycles.
6.2 Tailbiting Trellises 57
For very small codes, it is possible to use simulation to compare the performance of the min-sum algorithm with that of maximum-likelihood decoding. The result of such a simula-tion is illustrated in Figure 6.4, indicating that the min-sum algorithm is very close to maxi-mum-likelihood for that code.
6.2 Tailbiting Trellises
The reason why it is possible to characterize the possible error events for cycle codes is that their deviations have a chain-like structure and may thus be partitioned into finite parts cor-responding to a certain structure on the graphs. This structure is shared with the deviations of trellises (i.e., the detours). For ordinary (not tailbiting) trellises, which are cycle-free, this leads to the standard analysis of the distance structure, which can be found, e.g., in [21]. For tailbiting trellises, which are not cycle-free, we can use an analysis similar to that of cycle codes.
Recall that the walks corresponding to the deviations of cycle codes can be partitioned into irreducible closed walks. Similarly, deviations on a tailbiting trellis (i.e., detours) can be partitioned into closed subpaths whose length are multiples of the trellis length, each starting and ending in the same state; the deviation can only have negative cost if at least one such
Figure 6.4 Decoding performance of the (15, 6, 5) cycle code of the Peterson graph [12, pp. 136-138] used on a Gaussian channel. The upper curve corresponds to 30 iterations of the min-sum algorithm, and the lower curve corresponds to maximum-likelihood decoding.
0 1 2 3 4 5 6 7
10−6 10−5 10−4 10−3 10−2 10−1 100
SNR [dB]
Block Error Rate
closed subpath has negative cost. The situation is more complex than for cycle codes, how-ever, since detours may end before the leaves of the computation tree, i.e., they may have finite support even when the number of iterations tends to infinity.
So, consider the computation tree obtained with the min-sum algorithm on a tailbiting trellis. Let n be the number of trellis sections and let l be the number of sections in the com-putation tree. As long as , the computation tree is cycle-free, and the analysis of Chapter 5 applies. We are interested in the case when . The valid configurations on the tree system are paths in a computation trellis which consists of multiple copies of the origi-nal tailbiting trellis, pasted together. See Figure 6.5.
The deviations on this tree system take the form of detours on the computation trellis, as discussed in Example 4.1. This means that if we follow the path from the nonzero root sec-tion towards one of the leaves (in either direcsec-tion) then the path will either stay away from the zero state altogether or enter the zero state exactly once and stay in the all-zero path afterwards. The length of a detour is the number of nonzero branches.
As usual, we restrict ourselves to binary symbols and assume that the local costs for a
“0” is zero ( ) so that the cost of a path equals the cost sum of its nonzero branches.
(In fact, a branch is actually a local configuration on a section; its cost is the sum of the costs of its visible sites.)
We now define a closed path (in the computation trellis) as a subpath that starts and ends in the same state (of the same state spaces). Just as for cycle codes, a path is called reducible if it contains a closed path that may be removed, leaving a shorter path, so that the cost of the original path is the sum of the costs of the remaining shorter path and the closed path. A path is called irreducible if it does not contain a closed path; the lengths of irreducible paths are
l≤n
l>n 0
0
1 1 0
1 0 0
1 1 0
1 0 0
1 1 0
1
1 2 3 4 5 6
0 0
1 1 0
1 0 0
1 1 0
1 0 0
1 1 0
1 0
0
1 1 0
1
0 0
1 1 0
1
1 2 3 4 5 6 1 2
6 5
Figure 6.5 A tailbiting trellis of length 6 (top) and the “computation trellis” corre-sponding to a computation tree of length 10 (bottom). The numbers below the trellis sections indicate the corresponding codeword component index. The marked path is valid on the computation trellis (in fact, it is a “detour”) although it uses different branches in sections corresponding to the same original section.
γs( )0 = 0
6.3 The General Case 59 limited by , where M is the number of states in the largest trellis state space. Note that closed paths of length n correspond to valid paths on the original tailbiting trellis, i.e., to codewords.
With these definitions, it is easy to see that any path S on the computation trellis may be decomposed into a collection of irreducible closed paths and a remaining irreduc-ible path , so that the cost is the sum . In particular, any detour may be decomposed in this way, and the remaining irreducible path will also be a detour (unless its length is zero). But since a decoding error can only occur if some detour has negative cost, we have proved
Theorem 6.5 Using the min-sum algorithm with a tailbiting trellis, a