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.     

Turbo codes with rate-m/(m+1) constituent convolutional codes

The original turbo codes (TCs), presented in 1993 by Berrou et al., consist of the parallel concatenation of two rate-1/2 binary recursive systematic convolutional (RSC) codes. This paper explains how replacing rate-1/2 binary component codes by rate-m/(m+1) binary RSC codes can lead to better global performance. The encoding scheme can be designed so that decoding can be achieved closer to the theoretical limit, while showing better performance in the region of low error rates. These results are illustrated with some examples based on double-binary (m=2) 8-state and 16-state TCs, easily adaptable to a large range of data block sizes and coding rates. The double-binary 8-state code has already been adopted in several telecommunication standards.     

Recent Advances in Turbo Code Design and Theory

The discovery of turbo codes and the subsequent rediscovery of low-density parity-check (LDPC) codes represent major milestones in the field of channel coding. Recent advances in the design and theory of turbo codes and their relationship to LDPC codes are discussed. Several new interleaver designs for turbo codes are presented which illustrate the important role that the interleaver plays in these codes. The relationship between turbo codes and LDPC codes is explored via an explicit formulation of the parity-check matrix of a turbo code, and simulation results are given for sum product decoding of a turbo code.     

Space-time codes and concatenated channel codes for wirelesscommunications

Following a brief historical perspective on channel coding, an introduction to space-time block codes is given. The various space-time codes considered are then concatenated with a range of channel codecs, such as convolutional and block-based turbo codes as well as conventional and turbo trellis codes. The associated estimated complexity issues and memory requirements are also considered. These discussions are followed by a performance study of various space-time and channel-coded transceivers. Our aim is first to identify a space-time code/channel code combination constituting a good engineering tradeoff in terms of its effective throughput, bit-error-rate performance, and estimated complexity. Specifically, the issue of bit-to-symbol mapping is addressed in the context of convolutional codes (CCs) and convolutional coding as well as Bose-Chaudhuri-Hocquenghem coding-based turbo codes in conjunction with an attractive unity-rate space-time code and multilevel modulation is detailed. It is concluded that over the nondispersive or narrow-band fading channels, the best performance versus complexity tradeoff is constituted by Alamouti's twin-antenna block space-time code concatenated with turbo convolutional codes. Further comparisons with space-time trellis codes result in similar conclusions     

Turbo equalization of serially concatenated turbo codes using a predictive DFE-based receiver

This paper investigates the performance of various “turbo” receivers for serially concatenated turbo codes transmitted through intersymbol interference (ISI) channels. Both the inner and outer codes are assumed to be recursive systematic convolutional (RSC) codes. The optimum turbo receiver consists of an (inner) channel maximum a posteriori (MAP) decoder and a MAP decoder for the outer code. The channel MAP decoder operates on a “supertrellis” which incorporates the channel trellis and the trellis for the inner error-correcting code. This is referred to as the MAP receiver employing a SuperTrellis (STMAP). Since the complexity of the supertrellis in the STMAP receiver increases exponentially with the channel length, we propose a simpler but suboptimal receiver that employs the predictive decision feedback equalizer (PDFE). The key idea in this paper is to have the feedforward part of the PDFE outside the iterative loop and incorporate only the feedback part inside the loop. We refer to this receiver as the PDFE-STMAP. The complexity of the supertrellis in the PDFE-STMAP receiver depends on the inner code and the length of the feedback part. Investigations with Proakis B, Proakis C (both channels have spectral nulls with all zeros on the unit circle and hence cannot be converted to a minimum phase channel) and a minimum phase channel reveal that at most two feedback taps are sufficient to get the best performance. A reduced-state STMAP (RS-STMAP) receiver is also derived which employs a smaller supertrellis at the cost of performance.     

删余型Turbo码分量编码器盲识别算法

摘 要:无法获得完整的递归系统卷积码(Recursive System Code,RSC)码字,传统的盲识别方法就不适用于删余型Turbo码的识别。于是该算法在识别序列的构造上进行了改进,针对Turbo码在删余位上的码字与对应的RSC码有所区别的情况,将该位上的码字视为“0”和“1”等概率出现的误码,从而对删余位进行归零处理并选取合适的截取序列进行匹配度计算,根据最后匹配度的总分布情况对删余型Turbo码分量编码器的参数进行识别。仿真结果表明针对码长为256,码率为1/2的删余型Turbo码,在最大误比特率不超过0.033的情况下正确识别率能保持在80%以上。      

Combined turbo codes and interleaver design

The impact of the distance spectrum and interleaver structure on the bit error probability of turbo codes is considered. A new turbo code design method for Gaussian channels is presented. The proposed method combines a search for good component codes with interleaver design. The optimal distance spectrum is used as the design criterion to construct good turbo component codes at low signal-to-noise ratios (SNRs). In addition, an interleaver design method is proposed. This design improves the code performance at high SNR. Search for good component codes at low SNR is combined with a code matched interleaver design. This results in new turbo codes with a superior error performance relative to the best known codes at both low and high SNR. The performance is verified by both analysis and simulation     

Comparative study of turbo equalization schemes usingconvolutional, convolutional turbo, and block-turbo codes

Turbo equalizers have been shown to be successful in mitigating the effects of inter-symbol interference introduced by partial response modems and by dispersive channels for code rates of R⩽ 1/2. We comparatively studied the performance of a range of binary phase-shift keying turbo equalizers employing block-turbo codes, namely Bose-Chaudhuri-Hocquenghen (1960, 1959) turbo codes, convolutional codes, and convolutional turbo codes having high code rates, such as R=3/4 and R=5/6, over a dispersive five-path Gaussian channel and an equally weighted symbol-spaced five-path Rayleigh fading channel. These turbo equalization schemes were combined with an iterative channel estimation scheme in order to characterize a realistic scenario. The simulation results demonstrated that the turbo-equalized system using convolutional turbo codes was the most robust system for all code rates investigated     

Improving the distance properties of turbo codes using a third component code: 3D turbo codes - [transactions letters]

Thanks to the probabilistic message passing performed between its component decoders, a turbo decoder is able to provide strong error correction close to the theoretical limit. However, the minimum Hamming distance (d_min) of a turbo code may not be sufficiently large to ensure large asymptotic gains at very low error rates (the so-called flattening effect). Increasing the d_min of a turbo code may involve using component encoders with a large number of states, devising more sophisticated internal permutations, or increasing the number of component encoders. This paper addresses the latter option and proposes a modified turbo code in which a fraction of the parity bits are encoded by a rate-1, third encoder. The result is a noticeably increased d_min, which improves turbo decoder performance at low error rates. Performance comparisons with turbo codes and serially concatenated convolutional codes are given.     

The Augmented State Diagram and its Application to Convolutional and Turbo Codes

Convolutional block codes, which are commonly used as constituent codes in turbo code configurations, accept a block of information bits as input rather than a continuous stream of bits. In this paper, we propose a technique for the calculation of the transfer function of convolutional block codes, both punctured and nonpunctured. The novelty of our approach lies in the augmentation of the conventional state diagram, which allows the enumeration of all codeword sequences of a convolutional block code. In the case of a turbo code, we can readily calculate an upper bound to its bit error rate performance if the transfer function of each constituent convolutional block code has been obtained. The bound gives an accurate estimate of the error floor of the turbo code and, consequently, our method provides a useful analytical tool for determining constituent codes or identifying puncturing patterns that improve the bit error rate performance of a turbo code, at high signal-to-noise ratios.     

Turbo processing in transmit antenna diversity systems

We consider turbo-trellis-coded transmission over fading multiple-input-multiple-output (M1M0) channels with transmit diversity using space-time block codes. We give a new view on space-time block codes as a transformation of the fading MIMO channel towards a Gaussian single-input-single-output (siso) channel and provide analytical results on the BER of space-time block codes. Furthermore, we describe the concatenation of Turbo-TCM with a space-time block code and show that in addition to the transmit diversity substantial benefits can be obtained by turbo iterations as long as the channel is time-varying during transmission of a coded block or frequency hopping is applied. Finally, a double iterative scheme for turbo equalization and turbo decoding of the concatenation of Turbo-TCM and space-time block code in frequency-selective MIMO channels is described.     

Design and performance of turbo Gallager codes

The most powerful channel-coding schemes, namely, those based on turbo codes and low-density parity-check (LDPC) Gallager codes, have in common the principle of iterative decoding. However, the relative coding structures and decoding algorithms are substantially different. This paper shows that recently proposed novel coding structures bridge the gap between these two schemes. In fact, with properly chosen component convolutional codes, a turbo code can be successfully decoded by means of the decoding algorithm used for LDPC codes, i.e., the belief-propagation algorithm working on the code Tanner graph. These new turbo codes are here nicknamed "turbo Gallager codes." Besides being interesting from a conceptual viewpoint, these schemes are important on the practical side because they can be decoded in a fully parallel manner. In addition to the encoding complexity advantage of turbo codes, the low decoding complexity allows the design of very efficient channel-coding schemes.     

Multiple serial and parallel concatenated single parity-check codes

Single parity-check (SPC) codes are applied in both parallel and serial concatenated structures to produce high-performance coding schemes. The number of concatenations or stages, M, is increased to improve system performance at moderate-to-low bit-error rates without changing the overall code parameters (namely, code rate and code block length). Analytical bounds are presented to estimate the performance at high signal-to-noise ratios. The SPC concatenated codes are considered with binary phase-shift keying and with 16-quadrature amplitude modulation bit-interleaved coded modulation on the additive white Gaussian noise channel and the independent Rayleigh fading channel. Simulations show that the four-stage serial or parallel concatenated SPC codes can, respectively, outperform or perform as well as 16-state turbo codes. Furthermore, decoding complexity is approximately 9-10 times less complex than that of 16-state turbo codes. The convergence behavior of both serial and parallel concatenated SPC codes is also discussed.     

Fast Chase algorithm with an application in turbo decoding

Turbo product codes (TPCs) provide an attractive alternative to recursive systematic convolutional (RSC)-based turbo systems. Rather than employ trellis-based decoders, an algebraic decoder may be repeatedly employed in a low-complexity, soft-input/soft-output errors-and-erasures decoder such as the Chase algorithm. Taking motivation from efficient forced erasure decoders, this implementation re-orders the Chase algorithm's repeated decodings such that the inherent computational redundancy is greatly reduced without degrading performance. The result is a highly efficient fast Chase implementation. The algorithm presented here is principally applicable to single error-correcting codes although consideration is also given to the more general case. The new decoder's value in practical turbo schemes is demonstrated via application to decoding of the (64,57,4) extended Hamming TPC     

Iterative decoding for partial response (PR), equalized,magneto-optical (MO) data storage channels

Turbo codes and low-density parity check (LDPC) codes with iterative decoding have received significant research attention because of their remarkable near-capacity performance for additive white Gaussian noise (AWGN) channels. Previously, turbo code and LDPC code variants are being investigated as potential candidates for high-density magnetic recording channels suffering from low signal-to-noise ratios (SNR). We address the application of turbo codes and LDPC codes to magneto-optical (MO) recording channels. Our results focus on a variety of practical MO storage channel aspects, including storage density, partial response targets, the type of precoder used, and mark edge jitter. Instead of focusing just on bit error rates (BER), we also study the block error statistics. Our results for MO storage channels indicate that turbo codes of rate 16/17 can achieve coding gains of 3-5 dB over partial response maximum likelihood (PRML) methods for a 10^-4 target BER. Simulations also show that the performance of LDPC codes for MO channels is comparable to that of turbo codes, while requiring less computational complexity. Both LDPC codes and turbo codes with iterative decoding are seen to be robust to mark edge jitter     

Evaluating the performance of turbo codes and turbo-codedmodulation in a DS-CDMA environment

Turbo codes have received great attention due to their outstanding performance. Unfortunately, a high performance is associated with large transmission delays, prohibiting an application for speech transmission. Hence, the aim of this paper is the comparison of turbo codes employing short interleavers with convolutional codes in terms of bit error rate performance and computational effort. Additionally, a pragmatic approach of bandwidth-efficient turbo-coded modulation is considered. Analyzing the structure of the transmitter and receiver, interesting results are presented concerning the design of the mapper. Furthermore, a new very simple soft-output demodulation algorithm is derived. In order to compare turbo codes with convolutional codes under realistic conditions, both are embedded in a direct sequence (DS) code division multiple access system. Besides this comparison, a compromise between a high coding gain (low code rate) and high direct-sequence spreading is worked out, including the consideration of the turbo-coded modulation scheme. Simulation results indicate that turbo codes with small block interleavers do not outperform conventional convolutional codes. Furthermore, it is shown that for coherent demodulation, low code rates and low DS spreading is superior to high code rates and high DS spreading     

On the performance of hybrid FEC/ARQ systems using rate compatiblepunctured turbo (RCPT) codes

This paper introduces a hybrid forward-error correction/automatic repeat-request (ARQ) system that employs rate compatible punctured turbo (RCPT) codes to achieve enhanced throughput performance over a nonstationary Gaussian channel. The proposed RCPT-ARQ system combines the performance of turbo codes with the frugal use of incremental redundancy inherent in the rate compatible punctured convolutional codes of Hagenauer (1988). Moreover, this paper introduces the notion of puncturing the systematic code symbols of a turbo code to maximize throughput at signal-to-noise ratios (SNRs) of interest. The resulting system provides both an efficient family of achievable code rates at middle to high SNR and powerful low-rate error correction capability at low SNR     

The Development of Turbo and LDPC Codes for Deep-Space Applications

The development of error-correcting codes has been closely coupled with deep-space exploration since the early days of both. Since the discovery of turbo codes in 1993, the research community has invested a great deal of work on modern iteratively decoded codes, and naturally NASA's Jet Propulsion Laboratory (JPL) has been very much involved. This paper describes the research, design, implementation, and standardization work that has taken place at JPL for both turbo and low-density parity-check (LDPC) codes. Turbo code development proceeded from theoretical analyses of polynomial selection, weight distributions imposed by interleaver designs, decoder error floors, and iterative decoding thresholds. A family of turbo codes was standardized and implemented and is currently in use by several spacecraft. JPL's LDPC codes are built from protographs and circulants, selected by analyses of decoding thresholds and methods to avoid loops in the code graph. LDPC encoders and decoders have been implemented in hardware for planned spacecraft, and standardization is under way.     

KBturbo码中交织器的设计

已知比特(KB,Known Bits)turbo码是采用将KB周期性的插入到信息比特中,不显著增加系统复杂程度而较大提高turbo码译码性能。但因为一般采用了随机交织器,所以在引入KB方法时,产生了不等差错保护的问题。这个问题可以通过采用KB交织器,将KB在交织器中均匀分配来解决。仿真证实,以伪随机交织器和S随机交织器为基础的KB交织器可以显著提高系统性能。而且采用KB turbo码可以方便地实现速率适配,满足第三代移动通信系统的要求。     

Turbo and Turbo-Like Codes: Principles and Applications in Telecommunications

For decades, the de facto standard for forward error correction was a convolutional code decoded with the Viterbi algorithm, often concatenated with another code (e.g., a Reed-Solomon code). But since the introduction of turbo codes in 1993, much more powerful codes referred to collectively as turbo and turbo-like codes have eclipsed classical methods. These powerful error-correcting techniques achieve excellent error-rate performance that can closely approach Shannon's channel capacity limit. The lure of these large coding gains has resulted in their incorporation into a widening array of telecommunications standards and systems. This paper will briefly characterize turbo and turbo-like codes, examine their implications for physical layer system design, and discuss standards and systems where they are being used. The emphasis will be on telecommunications applications, particularly wireless, though others are mentioned. Some thoughts on the use of turbo and turbo-like codes in the future will also be given.