In this semi-tutorial paper, we will investigate the computational complexity of an abstract version of the Viterbi algorithm on a trellis, and show that if the trellis has e edges, the complexity of the Viterbi algorithm is /spl Theta/(e). This result suggests that the "best" trellis representation for a given linear block code is the one with the fewest edges. We will then show that, among all trellises that represent a given code, the original trellis introduced by Bahl, Cocke, Jelinek, and Raviv in 1974, and later rediscovered by Wolf (1978), Massey (1978), and Forney (1988), uniquely minimizes the edge count, as well as several other figures of merit. Following Forney and Kschischang and Sorokine (1995), we will also discuss "trellis-oriented" or "minimal-span" generator matrices, which facilitate the calculation of the size of the BCJR trellis, as well as the actual construction of it.

/pdf/on-the-bcjr-trellis-for-linear-block-codes-3o3r6iopiv.pdf

On the BCJR trellis for linear block codes

The recent development of turbo codes has stirred new interest in trying to un-derstand how closely practical codes can approach the theoretical limits on codeperformance. This article reformulates Shannon’s sphere-packing bound in a nor-malized form conducive to analyzing the relative performance of families of turbocodes. Even though this bound is achievable only by a (generally mythical)

/pdf/code-performance-as-a-function-of-block-size-2qm6l3qjlv.pdf

Code Performance as a Function of Block Size

From the Publisher:
Trellises and Trellis-Based Decoding Algorithms for Linear Block Codes combines trellises and trellis-based decoding algorithms for linear codes together in a simple and unified form. The approach is to explain the material in an easily understood manner with minimal mathematical rigor. Trellises and Trellis-Based Decoding Algorithms for Linear Block Codes is intended for practicing communication engineers who want to have a fast grasp and understanding of the subject. This book can also be used as a text for advanced courses on the subject.

/pdf/trellises-and-trellis-based-decoding-algorithms-for-linear-3zti1b5gn8.pdf

Trellises and Trellis-Based Decoding Algorithms for Linear Block Codes

While the complexity of trellis decoding for a given block code is essentially a function of the number of states and branches in its trellis, the decoding complexity may be often reduced by means of an appropriate sectionalization of the trellis. Notwithstanding the many examples of such sectionalizations for particular codes that appeared in the literature, no systematic method for finding the best sectionalization of a given trellis is presently known. We present a polynomial-time algorithm which produces the optimal sectionalization of a given trellis T in time O(n/sup 2/), where n is the length of the code generated by T. The algorithm is developed in a general setting of certain operations and functions defined on the set of trellises; it therefore applies to both linear and nonlinear codes, and easily accommodates a broad range of optimality criteria. The particular optimality criterion based on minimizing the total number of additions and comparisons required for maximum-likelihood trellis decoding is investigated in detail: several different methods for decoding a given trellis are discussed and compared in a number of examples. Finally, analysis of the dynamical properties of certain optimal sectionalizations is presented.

Optimal sectionalization of a trellis

In this paper, the basic theory of interleavers is revisited in a semi-tutorial manner, and extended to encompass noncausal interleavers. The parameters that characterize the interleaver behavior (like delay, latency, and period) are clearly defined. The input-output interleaver code is introduced and its complexity studied. Connections among various interleaver parameters are explored. The classes of convolutional and block interleavers are considered, and their practical implementation discussed. The trellis complexity of turbo codes is tied to the complexity of the constituent interleaver. A procedure of complexity reduction by coordinate permutation is also presented, together with some examples of its application.

Interleaver properties and their applications to the trellis complexity analysis of turbo codes

It has long been known that convolutional codes have a natural, regular trellis structure that facilitates the implementation of Viterbi's algorithm. It has gradually become apparent that linear block codes also have a natural, though not in general a regular, 'minimal' trellis structure, which allows them to be decoded with a Viterbi-like algorithm. In both cases, the complexity of the Viterbi decoding algorithm can be accurately estimated by the number of trellis edges per encoded bit. It would, therefore, appear that we are in a good position to make a fair comparison of the Viterbi decoding complexity of block and convolutional codes. Unfortunately, however, this comparison is somewhat muddled by the fact that some convolutional codes, the punctured convolutional codes, are known to have trellis representations that are significantly less complex than the conventional trellis. In other words, the conventional trellis representation for a convolutional code may not be the minimal trellis representation. Thus, ironically, at present we seem to know more about the minimal trellis representation for block than for convolutional codes. In this article, we provide a remedy, by developing a theory of minimal trellises for convolutional codes. (A similar theory has recently been given by Sidorenko and Zyablov). This allows us to make a direct performance-complexity comparison for block and convolutional codes. A by-product of our work is an algorithm for choosing, from among all generator matrices for a given convolutional code, what we call a trellis-minimal generator matrix, from which the minimal trellis for the code can be directly constructed. Another by-product is that, in the new theory, punctured convolutional codes no longer appear as a special class, but simply as high-rate convolutional codes whose trellis complexity is unexpectedly small.

/pdf/the-trellis-complexity-of-convolutional-codes-s1s70a53b5.pdf

The trellis complexity of convolutional codes

In this partially tutorial paper, we examine minimal trellis representations of linear block codes and analyze several measures of trellis complexity: maximum state and edge dimensions, total span length, and total vertices, edges and mergers. We obtain bounds on these complexities as extensions of well-known dimension/length profile (DLP) bounds. Codes meeting these bounds minimize all the complexity measures simultaneously; conversely, a code attaining the bound for total span length, vertices, or edges, must likewise attain it for all the others. We define a notion of "uniform" optimality that embraces different domains of optimization, such as different permutations of a code or different codes with the same parameters, and we give examples of uniformly optimal codes and permutations. We also give some conditions that identify certain cases when no code or permutation can meet the bounds. In addition to DLP-based bounds, we derive new inequalities relating one complexity measure to another, which can be used in conjunction with known bounds on one measure to imply bounds on the others. As an application, we infer new bounds on maximum state and edge complexity and on total vertices and edges from bounds on span lengths.

/pdf/trellis-decoding-complexity-of-linear-block-codes-w2gqkdwu4r.pdf

Trellis decoding complexity of linear block codes

We consider the problem of finding a trellis for a linear block code that minimizes one or more measures of trellis complexity. The domain of optimization may be different permutations of the same code or different codes with the same parameters. Constraints on trellises, including relationships between the minimal trellis of a code and that of the dual code, are used to derive bounds on complexity. We define a partial ordering on trellises: If a trellis is optimum with respect to this partial ordering, if has the desirable property that it simultaneously minimizes all of the complexity measures examined. We examine properties of such optimal trellises and give examples of optimal permutations of codes, most notably the (48,24,12) quadratic residue code.

https://ipnpr.jpl.nasa.gov/progress_report/42-121/121K.pdf

Trellis Complexity Bounds for Decoding Linear Block Codes

We consider the problem of finding a trellis for a linear block code that minimizes one or more measures of trellis complexity for a fixed permutation of the code. We examine constraints on trellises, including relationships between the minimal trellis of a code and that of the dual code. We identify the primitive structures that can appear in a minimal trellis and relate this to those for the minimal trellis of the dual code.

/pdf/minimal-trellises-for-linear-block-codes-and-their-duals-4mmvn6xpe7.pdf

Minimal Trellises for Linear Block Codes and Their Duals

http://ieeexplore.ieee.org/iel3/4005/11520/00531335.pdf

The Trellis Complexity of Convolutio €?S1

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