On the complexity of exact maximum likelihood decoding on tail-biting trellises

Abstract

An algorithm for exact maximum likelihood(ML) decoding on tail-biting trellises is presented, which exhibits very good average case behavior. An approximate variant is proposed, whose simulated performance is observed to be virtually indistinguishable from the exact one at all values of signal to noise ratio, and which effectively performs computations equivalent to at most two rounds on the tail-biting trellis. The approximate algorithm is analyzed, and the conditions under which its output is different from the ML output are deduced. The results of simulations on an AWGN channel for the exact and approximate algorithms on the 16 state tail-biting trellis for the (24,12) Extended Golay Code, and tail-biting trellises for two rate 1/2 convolutional codes with memories of 4 and 6 respectively, are reported. An advantage of our algorithms is that they do not suffer from the effects of limit cycles or the presence of pseudocodewords. I. INTRODUCTIONTail-biting trellises are perhaps the simplest instances of decoding graphs with cycles. A tail-biting trellis has a Tanner graph [31] with a single cycle and usually approximate algorithms are used for decoding, as exact algorithms are believed to be too expensive. These approximate algorithms iterate around the trellis until either convergence is reached, or for a preset number of cycles. To the best of our knowledge, no exact decoding algorithms other than the brute force algorithm have been proposed so far for the general case, though there are several approximate algorithms for maximum-likelihood decoding [28], [22], [34], [33], [7], [20] and exact algorithms for bounded distance decoding [4]. The problem of Maximum A-Posteriori Probability(MAP)decoding is not addressed here. We propose an exact recursive algorithm, which exhibits very good average case behavior. The algorithm exploits the fact that a linear tail-biting trellis can be viewed as a coset decomposition The results in this paper appear in part in ISIT 2001[25] Priti Shankar acknowledges support fron the Scientific Analysis Group, DRDO, Delhi October 30, 2018 DRAFT

Approximate Linear Time ML Decoding on Tail-Biting Trellises in Two Rounds

“…However, the time complexity increases proportional to the number of stored paths. Practice has shown that memorizing the best two paths corresponding to the minimum value of M etric at each node gives performance almost indistinguishable from the ideal maximum likelihood decoder [11] An interesting failure condition of the algorithm is the following: The algorithm may fail to assign a value to the M etric field for a node if in the second phase a node fail to belong to any of the trellises assigned to the T rellis field of its predecessors by the algorithm. If this happens along all paths to all final states, the algorithm may fail to output a codeword in the second phase.…”

“…Some of these algorithms may fail to converge on certain inputs. Algorithms with guaranteed convergence were studied in [11], but they fail to achieve linear complexity. In particular, although the approximate algorithm proposed in [11], achieves performance close to an ideal ML decoder, it has a worst case time complexity of O(m log m), where m is the number of nodes in the TBT.…”

Design and implementation of an ML decoder for tail-biting convolutional codes

“…Since this definition does not overestimate the actual metric from s i k to s L j , A* algorithm will find a ML path. Interested readers are referred to [7] for details.…”

“…As opposed to the suboptimum techniques discussed above, an ML decoder based on the A* search algorithm was proposed in [7]. The algorithm consists of two phases: In phase-1, the Viterbi algorithm [8] is applied to the TBCC in order to obtain the trellis information; in phase-2, A* search algorithm uses the heuristic information obtained in phase-1 to yield the ML decoded output.…”

A low-complexity maximum likelihood decoder for tail-biting trellis

“…For linear block codes, conventional trellis and tail-biting trellis representations have gained a great deal of attention in the past decades [1][2][3]. Trellis representations not only reveal the code structure, but also lead to efficient trellisbased decoding algorithms [4][5][6][7][8]. For the same linear block codes, the number of states in its tail-biting trellis can be as low as the square root of the number of states in its minimal conventional trellis [1,8], e.g., for the (24, 12) extended Golay code, the maximum number of states in its conventional trellis is 512 [8], while the maximum number of states in its time-varying tail-biting trellis is only 16 [1].…”