Two decoding algorithms for tailbiting codes

The paper presents two efficient Viterbi decoding-based suboptimal algorithms for tailbiting codes. The first algorithm, the wrap-around Viterbi algorithm (WAVA), falls into the circular decoding category. It processes the tailbiting trellis iteratively, explores the initial state of the transmitted sequence through continuous Viterbi decoding, and improves the decoding decision with iterations. A sufficient condition for the decision to be optimal is derived. For long tailbiting codes, the WAVA gives essentially optimal performance with about one round of Viterbi trial. For short- and medium-length tailbiting codes, simulations show that the WAVA achieves closer-to-optimum performance with fewer decoding stages compared with the other suboptimal circular decoding algorithms. The second algorithm, the bidirectional Viterbi algorithm (BVA), employs two wrap-around Viterbi decoders to process the tailbiting trellis from both ends in opposite directions. The surviving paths from the two decoders are combined to form composite paths once the decoders meet in the middle of the trellis. The composite paths at each stage thereafter serve as candidates for decision update. The bidirectional process improves the error performance and shortens the decoding latency of unidirectional decoding with additional storage and computation requirements. Simulation results show that both proposed algorithms effectively achieve practically optimum performance for tailbiting codes of any length.     

Two step SOVA-based decoding algorithm for tailbiting codes

In this work we propose a novel decoding algorithm for tailbiting convolutional codes and evaluate its performance over different channels. The proposed method consists on a fixed two-step Viterbi decoding of the received data. In the first step, an estimation of the most likely state is performed based on a SOVA decoding. The second step consists of a conventional Viterbi decoding that employs the state estimated in the previous step as the initial and final states of the trellis. Simulations results show a performance close to that of maximum-likelihood decoding.     

VLSI architectures for turbo codes

A great interest has been gained in recent years by a new error-correcting code technique, known as “turbo coding”, which has been proven to offer performance closer to the Shannon's limit than traditional concatenated codes. In this paper, several very large scale integration (VLSI) architectures suitable for turbo decoder implementation are proposed and compared in terms of complexity and performance; the impact on the VLSI complexity of system parameters like the state number, number of iterations, and code rate are evaluated for the different solutions. The results of this architectural study have then been exploited for the design of a specific decoder, implementing a serial concatenation scheme with 2/3 and 3/4 codes; the designed circuit occupies 35 mm², supports a 2 Mb/s data rate, and for a bit error probability of 10^-6, yields a coding gain larger than 7 dB, with ten iterations     

Interleaver design for turbo codes

The performance of a turbo code with short block length depends critically on the interleaver design. There are two major criteria in the design of an interleaver: the distance spectrum of the code and the correlation between the information input data and the soft output of each decoder corresponding to its parity bits. This paper describes a new interleaver design for turbo codes with short block length based on these two criteria. A deterministic interleaver suitable for turbo codes is also described. Simulation results compare the new interleaver design to different existing interleavers     

Bolt interleavers for turbo codes

A procedure is described to produce efficient turbo-code interleavers. It is well suited to the selection of interleavers that produce a large minimum distance for the resulting turbo codes. A feature of these interleavers is that the final state of the two elementary encoders is simultaneously zero. Examples are given and the corresponding codes are simulated. They seem to be among the best ones for rate 1/3 turbo codes.     

Symbol-based turbo codes

We present a new turbo-coding method which parses the input block into n-bit symbols and interleaves on a symbol-by-symbol basis. This is used in conjunction with different modulation techniques to take advantage of tradeoffs between bit error rate performance, code-rate, spectral efficiency, and decoder complexity. The structure of the encoder and decoder of these codes, which We call symbol-based turbo codes, are outlined. The bit error rate performance of a few specific codes are examined. A discussion on decoder complexity is also included. Symbol-based turbo codes are good candidates for low delay transmission of speech and data in spread spectrum communication systems     

Time-varying turbo codes

Turbo codes typically utilize time-invariant component codes whose total encoder memory is no greater than 4. In this letter, turbo codes with time-varying component codes are studied via EXIT analysis, distance spectrum analysis and simulation. Turbo codes with time-varying component codes and memory 6 are shown to have similar performance in the waterfall region and superior performance in the error floor region when compared to both the original memory 4 Berrou turbo code and the memory 8 big numerator-little denominator turbo codes.     

Nonsystematic turbo codes

In this paper, we introduce the concept of nonsystematic turbo codes and compare them with classical systematic turbo codes. Nonsystematic turbo codes can achieve lower error floors than systematic turbo codes because of their superior effective free distance properties. Moreover, they can achieve comparable performance in the waterfall region if the nonsystematic constituent encoder has a low-weight feedforward inverse. A uniform interleaver analysis is used to show that rate R=1/3 turbo codes using nonsystematic constituent encoders have larger effective free distances than when systematic constituent encoders are used. Also, mutual information-based transfer characteristics and extrinsic information transfer charts are used to show that rate R=1/3 turbo codes with nonsystematic constituent encoders having low-weight feedforward inverses achieve convergence thresholds comparable to those achieved with systematic constituent encoders. Catastrophic encoders, which do not possess a feedforward inverse, are shown to be capable of achieving low convergence thresholds by doping the code with a small fraction of systematic bits. Finally, we give tables of good nonsystematic turbo codes and present simulation results comparing the performance of systematic and nonsystematic turbo codes.     

Stream-oriented turbo codes

This work considers the design and performance of a stream-oriented approach to turbo codes which avoids the need for data framing. The stream paradigm applies to both serial and parallel turbo codes using continuous, free-running constituent encoders along with continuous, periodic interleavers. A stream-oriented turbo code based on parallel concatenated convolutional codes (PCCC) is considered and interleaver design criteria are developed for both block and nonblock periodic interleavers. Specifically, several nonblock interleavers, including convolutional interleavers, are considered. Interleaver design rules are verified using simulations where it is shown that nonblock interleavers with small-to-moderate delay and small synchronization ambiguity can outperform block interleavers of comparable delay. For large-delay designs, nonblock interleavers are found which perform within 0.8 dB of the capacity limit with a synchronization ambiguity of N=11     

The rate-allocation problem for turbo codes

In this letter, we view the implicit unequal error protection observed in the asymptotic performance of most turbo codes as an impetus to the formulation of a rate-allocation problem, associated with the distribution of a fixed quota of coded bits in the trellis sections of the upper and lower recursive systematic convolutional codes of the turbo-code structure. We then present an effective greedy approach for solving this rate-allocation problem via a two-phase process of puncturing and repetition coding, resulting in the improvement of the asymptotic performance of the code and lowering of its error floor. Sample simulation results are presented, confirming the potential gains of the approach.     

确定性交织器的研究

本文分析了交织器在turbo码中的重要作用，以及随机交织器存在的缺陷，并提出了一种确定性交织器的设计方法。该交纪念品器具有简明的解析表达式，易于实现。计算机仿真结果表明，该确定交织器可以获得的性能优于随机交织器的平均性能。     

New performance bounds for turbo codes

We derive a new upper bound on the word- and bit-error probabilities of turbo codes with maximum-likelihood decoding by using the Gallager bound. Since the derivation of the bound for a given interleaver is intractable, we assume uniform interleaving as in the derivation of the standard union bound for turbo codes. The result is a generalization of the transfer function bound and remains useful for a wider range of signal-to-noise ratios, particularly for some range below the channel cutoff rate. The new bound is also applicable to other linear codes     

Extended tail-biting schemes for turbo codes

In this letter, we investigate the problem of using tailbiting recursive systematic convolutional (RSC) codes in turbo codes. Tailbiting is not always possible for a given RSC code with fixed length. We propose an extended tailbiting method for RSC codes and compare it with another extension method proposed by Van Stralen, et al. (IEE Electron. Lett., vol. 35, pp. 1461-1462, 1999). Several schemes using these extended tailbiting RSC codes in turbo code systems are developed and compared.     

An efficiently implementable maximum likelihood decoding algorithm for tailbiting codes

Convolutional tailbiting codes are widely used in mobile systems to perform error-correcting strategies of data and control information. Unlike zero tail codes, tailbiting codes do not reset the encoder memory at the end of each data block, improving the code efficiency for short block lengths. The objective of this work is to propose a low-complexity maximum likelihood decoding algorithm for convolutional tailbiting codes based on the Viterbi algorithm. The performance of the proposed solution is compared to that of another maximum likelihood decoding strategy which is based on the A* algorithm. The computational load and the memory requirements of both algorithms are also analysed in order to perform a fair comparison between them. Numerical results considering realistic transmission conditions show the lower memory requirements of the proposed solution, which makes its implementation more suitable for devices with limited resources.     

A note on tailbiting codes and their feedback encoders

Tailbiting codes encoded by feedback convolutional encoders are studied. A condition for when tailbiting will work is given and it is described how the encoder starting state can be obtained for feedback encoders in both controller and observer canonical forms. Finally, results from a search for systematic feedback encoders that encode tailbiting codes with good decoding bit error probabilities are presented     

Multilayer turbo space-time codes

This letter describes a multilayer turbo space-time coding scheme. Based on a carefully designed power allocation strategy, performance reasonably close to the theoretical limits can be achieved at a rate of two bits per channel use with very low complexity.     

Differential space-time turbo codes

Serial concatenation of simple error control codes and differential space-time modulation is considered. Decoding is performed iteratively by passing symbol-wise a posteriori probability values between the decoders of the inner space-time code and the outer code. An extrinsic information transfer analysis is used to predict thresholds for outer convolutional codes of various memory orders and a simple outer parity-check code. This parity-check code is well matched to the inner differential space-time code and achieves a bit-error rate (BER) of 10/sup -6/ less than 2 dB from the Shannon capacity of the fast fading multiple antenna channel. The differential space-time code can also be used to generate a priori information in the absence of channel knowledge. This information can be exploited by a channel estimator inserted into the decoding iteration. It is demonstrated that the inner space-time code provides soft training symbols from periodically inserted training symbols. The reliability of these soft training symbols does not depend on the speed of the channel variations, but on the structure of the inner code and the signal-to-noise ratio (SNR). Simulation studies confirm these findings and show that the proposed system with no initial channel knowledge achieves a performance very close to that of the system with perfect channel knowledge.     

Test-pattern-reduced decoding for turbo product codes with multi-error-correcting eBCH codes

We present a method to reduce the number of test patterns (TPs) decoded in the Chase-II algorithm for turbo product codes (TPCs) constructed with multi-error-correcting extended Bose-Chaudhuri-Hocquengem (eBCH) codes. We classify TPs into different conditions based on the relationship between syndromes and the number of errors so that TPs with the same codeword are not decoded except the one with the least number of errors. For eBCH with code length of 64, simulation results show that over 50% of TPs need not to be decoded without any performance degradation.     

Joint source-channel decoding of variable-length codes for convolutional codes and turbo codes

Several recent publications have shown that joint source-channel decoding could be a powerful technique to take advantage of residual source redundancy for fixed- and variable-length source codes. This letter gives an in-depth analysis of a low-complexity method recently proposed by Guivarch et al., where the redundancy left by a Huffman encoder is used at a bit level in the channel decoder to improve its performance. Several simulation results are presented, showing for two first-order Markov sources of different sizes that using a priori knowledge of the source statistics yields a significant improvement, either with a Viterbi channel decoder or with a turbo decoder.     

Rate-1/2 component codes for nonbinary turbo codes

The iterative decoding structure and component maximum a posteriori decoders used for decoding binary concatenated codes can be extended to the nonbinary domain. This paper considers turbo codes over nonbinary rings, specifically ternary, quaternary, penternary, hexernary, and octernary codes. The best rate-1/2 component codes are determined using a practical search algorithm. The performance of the resulting rate-1/3 turbo codes on an additive white Gaussian noise channel using q-ary phase-shift keying modulation is given.