Erasure-free sequential decoding of trellis codes

This paper describes the use of the sequential slack algorithm to decode cyclic(or extended cyclic) block codes. Once a block code is endowed with a trellis structure decoding with any of the convolutional decoding algorithms is viable. Since trellises for block codes are very wide a sequential algorithm, working at moderate signal-to-noise ratios, is an ineffective decoding alternative to the Viterbi algorithm. Using Wolf's trellis, Chang and Yao's sequential stack algorithm and the Fano metric the (24,12) Golay code can be efficiently decoded. Computer simulations show that by 6 dB the sequential algorithm is the most efficient (using Be'ery and Snyders' definition of complexity) soft decoding algorithm for the (24,12) Golay code.

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.

Fast Maximum Likelihood Decodable Space-Time Block Codes (STBCs) have recently gained a lot of interest for use in wireless communication systems. Several strategies have been introduced in the literature to build such codes, among them we mention the conditional detection strategy, where the code matrix can be expressed as the sum of two Orthogonal STBCs by means of a unitary matrix. Thus, the decoder benefits from the orthogonality by estimating an orthogonal set of symbols assuming knowledge of the other set of symbols. Even if this strategy permits high rates, numerical optimization of the coding gain becomes unrealizable for high-order constellations. Another strategy has been proposed in the literature, the g-group decodable codes, where the minimization of the Maximum Likelihood (ML) metric can be made over several independent subsets of symbols. Moreover, the g-group decodability strategy enables us to optimize the coding gain for each subset of symbols separately which permits the optimization of the coding gain for high-order constellations. On the other hand, the rate of g-group decodable codes is limited. In this paper, we reformulate the problem of finding Unitary Weight (UW)-2-group decodable codes in a simpler way based on necessary and sufficient conditions. Then, a simple routine is used to search for the desired code. We determine the maximum achievable rate for a specified type of weight matrices.

We introduce the notion of innovations for Viterbi decoding of convolutional codes. First, we define a kind of innovation corresponding to the received data, i.e., the input to a Viterbi decoder. Then, the structure of a scarce-state-transition (SST) Viterbi decoder is derived in a natural manner. It is shown that the newly defined innovation is just the input to the main decoder in an SST Viterbi decoder and generates the same syndrome as the original received data does. A similar result holds for quick-look-in codes as well. In this case, however, the precise innovation is not defined. We see that this innovation-like quantity is related to the linear smoothed estimate of the information. The essence of innovations approach to a linear filtering problem is first to whiten the observed data, and then to treat the resulting simpler white-noise observations problem. In our case, this corresponds to the reduction of decoding complexity in the main decoder in an SST Viterbi decoder. We show that the distributions related to the main decoder (i.e., the input distribution and the state distribution in the code trellis for the main decoder) are much biased under moderately noisy conditions. We see that these biased distributions actually lead to the complexity reduction in the main decoder. Furthermore, it is shown that the proposed innovations approach can be extended to maximum-likelihood decoding of block codes as well.

This paper presents new computationally efficient and accurate techniques for estimating the performance of specific high-rate punctured convolutional codes and uses these techniques to evaluate the performance of sequential and Viterbi decoders for the best known codes. In particular, it demonstrates that the disparity between sequential and Viterbi decoding increases dramatically for long memory codes with high rates and for such codes, the union bound cannot be used as a criterion for selecting good codes for sequential decoders. In contrast, it shows that the proposed methods can be used as efficient tools for performance evaluation and/or identification of good high-rate punctured convolutional codes for use with sequential decoders.

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. >

Sequential decoding of large memory codes has been investigated for 8-PSK, 16-QAM, and larger constellations. The results show that high reliability can be achieved at channel signal-to-noise ratios (SNRs) where the code rate is nearly equal to R/sub 0/, yielding 1-1.5 dB coding gain over Viterbi decoding of small memory codes. Since bootstrap hybrid decoding (BHD) is known to improve the cut-off rate of a sequential decoder for binary modulation, further improvements can be expected using this method applied to TCM. We present a lower bound to the computational cut-off rate for the extension of the BHD scheme to TCM. Our analysis is based on the original derivation for the case of binary modulation proposed by Jelinek and Cocke (1971). Numerical evaluations of the expressions obtained for the cases of hard-decision 8-PSK and 8-level quantized 4AM modulation show significant improvements over the usual cut-off rate, similar to the results obtained by Jelinek et al. for BPSK modulation. These results suggest that performance close to capacity can be achieved with sufficiently powerful TCM systems and bootstrap sequential decoding.

Viterbi decoders are either implemented as a Trace Back model or Register Exchange model. The trace back model is more popular, despite having large latency, because it requires less power and has less hardware complexity. In this paper a low complexity FPGA implementation of Register Exchange method is presented which gave the same error performance as that of trace back method at the same time having less hardware requirement. This is achieved using continuous decoding that normalises the state metric for every symbol processed, which reduces the size of registers used for storing the state metric. Split search that finds the best path every clock cycle was used for the normalisation and choosing the data output. This makes it possible to have output rate of one (1) symbol which reduces the number of registers required to save the path history bits to be outputted and reduces delay. Result of implementation of rate half codes of constraint length 7 and generator vector [1111001, 1011011] shows that for the same performance, the Register Exchange implementation presented, used no RAM as in the case of normal trace back and one-pointer algorithm. It requires less silicon area (5.5% less FPGA slices). It can run as fast as or even faster than the latter implementations and has less latency (1-decoding depth as compared to 4-decoding depth of trace back method) than the rest.

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.

Majority-logic decoding of convolutional codes, proposed by Massey [308], is a sub-optimum but simple decoding scheme that allows a high-speed implementation. Its performance is sub-optimum since each of its decoding decision is based only on one output constraint length of received bits. This “limited vision” is not there in Viterbi or sequential decoding which, in principle, consider the entire received sequence before making decoding decisions.

The stop-and-wait ARQ protocol with convolutional code and sequential decoding is considered. The channel throughput and optimum time-out before retransmission are obtained and the effects of the link parameters discussed. The results are extended to a twohop radio link, where a repeater receives the source's data and forwards them on the same radio channel to the destination. The ARQ protocols for the source and the repeater are discussed and a method presented for choosing the optimum time-out intervals for both.

It is shown that use of a two-stage decoding procedure consisting simply of an inner stage of block decoding and an outer stage employing a single sequential decoder does not result in an improvement in the computational overflow problem for the sequential decoder. Improvement can, however, result from use of multiple sequential decoders or use of a single sequential decoder with appropriate scrambling. Although the performance improvement resulting from application of these techniques to the additive white Gaussian noise channel is not significant, implementation and rate advantages make iterative sequential decoding techniques worth pursuing. Ia particular, with these techniques, a sequential decoder for a binary symmetric channel can be used regardless of the physical channel characteristics. Such a universal decoder is expected to be both simple and capable of high rate operation.

Sequential decoding of the channel code in a trellis-coded modulation system with trellis shaping can be used to reduce the system complexity and to achieve high coding gain with large constraint-length codes. It is shown that almost all the shaping gain that can be achieved when Viterbi decoding is used for the channel code can also be achieved when sequential decoding is used for the channel code. It is also shown that the real shaping gain is a function of both the SNR and the spectral efficiency.

Recently, Arikan has proposed a new scheme to transfer the error correction part to external codes for encoding, namely polarization-adjusted convolutional (PAC) coding. However, all existing decoding methods of current research for PAC codes focused on the classic sequential decoding-Fano decoding. The stack decoding which is another well-known sequential decoding, has not been reported. In this paper, we try to do a thorough study of the stack decoding for PAC codes. First, we find that the traditional stack decoding (SD) is not very effective for the PAC codes, especially in terms of complexity. Therefore, we propose a soft output stack decoding (SOSD) for PAC codes, which can reduce the computation very much. In the SOSD algorithm, the discarded paths are recorded and used in the final decision of decoding codeword. The SOSD algorithm can reduce the stack size (amount of computation) very much with almost no decoding performance loss. By carefully choosing the stack size and soft output threshold, in the case of a PAC code with code length 128 and rate 1/2, the SOSD algorithm can reduce 82% of the computation by comparing the conventional SD algorithm.

The parallel Viterbi decoding is discussed for two different cases: uncontrollable sources and controllable sources. For general, uncontrollable Markov processes, a previously known parallel method is extended to a hierarchical parallel decoding approach, which achieves a lower latency. For controllable Markov sources in telecommunications applications, new parallel decoding methods are obtained by controlling the source processes in appropriate ways. The focus is on the parallel decoding methods for controllable sources because these methods have zero processing overhead. Because the methods modify the coding process, they bring positive changes to framing and negative changes to latency and code performance. However, one can adjust the parameters of the methods to make the degradation negligible. Because of their low overhead, the methods are most attractive for high-speed decoders for convolutional and trellis codes, and they are also applicable to other sequential algorithms for suboptimal decoding and estimation of complex Markov sources. >