A Tightened Upper Bound on the Error Probability of Binary Convolutional Codes with Viterbi Decoding

A new algorithm for erasurefree sequential decoding of convolutional codes is introduced which achieves low error probabilities at substantially higher decoding speeds than the Viterbi decoding algorithm. The algorithmic properties of the Multiple Stack Algorithm (MSA) are investigated and it is demonstrated that the MSA reaches a decision with an exponentially rather than Pareto distributed computational effort. The MSA's error probability on the binary symmetric channel is studied as a function of its parameters and its performance and complexity compared to that of the Viterbi algorithm. The MSA is seen to achieve equal and lower error probabilities with a significantly lower average decoding effort. The new algorithm can thus be considered an attractive alternative to the Viterbi algorithm where low error probabilities and high decoding speeds are required simultaneously.

The performance of maximum-likelihood (ML) decoded binary linear block codes is addressed via the derivation of tightened upper bounds on their decoding error probability. The upper bounds on the block and bit error probabilities are valid for any memoryless, binary-input and output-symmetric communication channel, and their effectiveness is exemplified for various ensembles of turbo-like codes over the additive white Gaussian noise (AWGN) channel. An expurgation of the distance spectrum of binary linear block codes further tightens the resulting upper bounds

Performance evaluation of maximum-likelihood (ML) soft-decision-decoded binary block codes is usually carried out using bounding techniques. Many tight upper bounds on the error probability of binary codes are based on the so-called Gallager's first bounding technique (GFBT). The tangential sphere bound (TSB) of Poltyrev which has been believed for many years to offer the tightest bound developed for binary block codes is an example. Within the framework of the TSB and GFBT, we apply a new method referred to as the "added-hyper-plane" (AHP) technique, to the decomposition of the error probability. This results in a bound developed upon the application of two stages of the GFBT with two different Gallager regions culminating in a tightened upper bound beyond the TSB. The proposed bound is simple and only requires the spectrum of the binary code.

In 1995, Best etal. published a formula for the exact bit error probability for Viterbi decoding of the rate R =1/2, memory m = 1 (two-state) convolutional encoder with generator matrix G(D) = (1 1 + D) when used to communicate over the binary symmetric channel. Their formula was later extended to the rate R = 1/2, memory m = 2 (four-state) convolutional encoder with generator matrix G(D) = (1 + D2 1 + D + D2) by Lentmaier et al. In this paper, a different approach to derive the exact bit error probability is described. A general recurrent matrix equation, connecting the average information weight at the current and previous states of a trellis section of the Viterbi decoder, is derived and solved. The general solution of this matrix equation yields a closed-form expression for the exact bit error probability. As special cases, the expressions obtained by Best et al. for the two-state encoder and by Lentmaier et al. for a four-state encoder are used. The closed-form expression derived in this paper is evaluated for various realizations of encoders, including rate R = 1/2 and R = 2/3 encoders, of as many as 16 states. Moreover, it is shown that it is straightforward to extend the approach to communication over the quantized additive white Gaussian noise channel.

New lower bounds on the error probability of block codes with maximum-likelihood decoding are proposed. The bounds are obtained by applying a new lower bound on the probability of a union of events, derived by improving on de Caen's lower bound. The new bound includes an arbitrary function to be optimized in order to achieve the tightest results. Since the optimal choice of this function is known, but leads to a trivial and useless identity, we find several useful approximations for it, each resulting in a new lower bound. For the additive white Gaussian noise (AWGN) channel and the binary-symmetric channel (BSC), the optimal choice of the optimization function is stated and several approximations are proposed. When the bounds are further specialized to linear codes, the only knowledge on the code used is its weight enumeration. The results are shown to be tighter than the latest bounds in the current literature, such as those by Seguin (1998) and by Keren and Litsyn (2001). Moreover, for the BSC, the new bounds widen the range of rates for which the union bound analysis applies, thus improving on the bound to the error exponent compared with the de Caen-based bounds.

Maximum likelihood decoding of convolutional codes by Viterbi decoding method is an effective error correcting technique. However, little study on Viterbi decoding of high rate codes has been done since the number of paths entering each trellis state increases as the code rate increases. This paper presents a new maximum likelihood decoding of convolutional codes of rate (n ‐ 1)/n using a trellis. the trellis is drawn according to the state transitions of the syndrome former of the code, and has only one or two paths entering each state rather than 2n‐1 paths as in the case of conventional trellises. Thus, we can replace the 2n‐1‐ary comparisons at each state needed in the conventional Viterbi decoding for the code of rate (n ‐ 1)/n by binary comparisons. the implementation of maximum likelihood decoders for high rate convolutional codes is simplified greatly by this approach. Moreover, this decoding method shows the same performance as that of conventional Viterbi decoding method from the viewpoint of selecting the maximum likelihood code sequence.

When a scarce state transition (SST) Viterbi decoder is used to decode general nonsystematic convolutional codes (not included among quick look‐in (QLI) codes), metrics based on the maximum a posteriori probability (MAP) estimation method must be computed in order for the main decoder to operate under the criterion of minimizing the decoding error probability. In this case, however, it becomes necessary to extend the original code trellis structure by introducing augmented trellis states in order to incorporate the newly induced correlation among the apparent information sequence.This paper shows that the extended trellis degenerates into the original one as the channel error probability q approaches one‐half. By performing an equivalent transformation of metrics, it is shown also that the branch metrics derived based on the MAP estimation method converge to those computed using the conventional one as q approaches one‐half.

This paper considers the optimal generator matrices of a binary cyclic code over a binary symmetric channel with crossover probability p rarr 0 when the goal is to minimize the probability of an information bit error. A given code has many encoder realizations and the information bit error probability is a function of this realization. We derive some sufficient conditions for a binary cyclic code to have systematic optimal generator matrices under bounded distance decoding and determine many binary cyclic codes satisfying these properties.

Complete decoding of systematic linear block codes by standard arrays with coset leaders not necessarily of minimal weight gives, in several interesting cases on the binary symmetric channel with error probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> , a smaller symbol error probability than decoding using the coset leaders of minimum weight. The proposed choice of the coset leaders allows for the classes of uniformly packed and nearly perfect binary codes to achieve a lower information bit error probability, at least for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> sufficiently near zero.

This paper considers the optimal generator matrices of a given binary cyclic code over a binary symmetric channel with crossover probability p → 0 when the goal is to minimize the probability of an information bit error. A given code has many encoder realizations and the information bit error probability is a function of this realization. Our goal here is to seek the optimal realization of encoding functions by taking advantage of the structure of the codes, and to derive the probability of information bit error when possible. We derive some sufficient conditions for a binary cyclic code to have systematic optimal generator matrices under bounded distance decoding and determine many cyclic codes with such properties. We also present some binary cyclic codes whose optimal generator matrices are non-systematic under complete decoding.

A note on the computation of the bit error rate for binary block codes

The block error probability under maximum likelihood decoding of general linear block codes and code ensembles over the q-ary symmetric channel is analyzed. The analysis makes use of two types of upper bounds and it is based on the codes (code ensembles) distance spectra. The first bound generalizes Poltyrev's union bound for the binary symmetric channel. The second bound relies on typical pairs decoding. The two bounds are developed in an asymptotic setting to analyze the behavior of linear block code ensembles, in terms of error exponent and maximum-likelihood decoding threshold. A sphere-packing lower bound is also developed, to provide a reference for the finite-length performance.

We consider communication over the binary erasure channel (BEC) using\nlow-density parity-check (LDPC) codes and belief propagation (BP) decoding. For\nfixed numbers of BP iterations, the bit error probability approaches a limit as\nblocklength tends to infinity, and the limit is obtained via density evolution.\nOn the other hand, the difference between the bit error probability of codes\nwith blocklength $n$ and that in the large blocklength limit is asymptotically\n$\\alpha(\\epsilon,t)/n + \\Theta(n^{-2})$ where $\\alpha(\\epsilon,t)$ denotes a\nspecific constant determined by the code ensemble considered, the number $t$ of\niterations, and the erasure probability $\\epsilon$ of the BEC. In this paper,\nwe derive a set of recursive formulas which allows evaluation of the constant\n$\\alpha(\\epsilon,t)$ for standard irregular ensembles. The dominant difference\n$\\alpha(\\epsilon,t)/n$ can be considered as effects of cycle-free and\nsingle-cycle structures of local graphs. Furthermore, it is confirmed via\nnumerical simulations that estimation of the bit error probability using\n$\\alpha(\\epsilon,t)$ is accurate even for small blocklengths.\n

Convolutional coding and Viterbi decoding, along with binary phase-shift keyed modulation, is presented as an efficient system for reliable communication on power limited satellite and space channels. Performance results, obtained theoretically and through computer simulation, are given for optimum short constraint length codes for a range of code constraint lengths and code rates. System efficiency is compared for hard receiver quantization and 4 and 8 level soft quantization. The effects on performance of varying of certain parameters relevant to decoder complexity and cost is examined. Quantitative performance degradation due to imperfect carrier phase coherence is evaluated and compared to that of an uncoded system. As an example of decoder performance versus complexity, a recently implemented 2-Mbit/s constraint length 7 Viterbi decoder is discussed. Finally a comparison is made between Viterbi and sequential decoding in terms of suitability to various system requirements.

We introduce a very efficient (linear-time) method for finding the best metrics from among a number of contenders during breadth-first decoding of block or convolutional codes. This method can provide a sorted set of survivor metrics during M algorithm decoding of binary linear block codes using only /spl Lambda/1 comparisons in the worst case. This is a significant improvement over comparison-based sorting, which requires O(M log/sub 2/ M) comparisons. A similar improvement is attained over Hadian and Sohel's comparison-based selection method (where the best metrics are found without regard to order). The method is also readily applicable to the decoding of convolutional codes. >