Design and decoding of optimal high-rate convolutional codes

This correspondence deals with the design and decoding of high-rate convolutional codes. After proving that every (n,n-1) convolutional code can be reduced to a structure that concatenates a block encoder associated to the parallel edges with a convolutional encoder defining the trellis section, the results of an exhaustive search for the optimal (n,n-1) convolutional codes is presented through various tables of best high-rate codes. The search is also extended to find the "best" recursive systematic convolutional encoders to be used as component encoders of parallel concatenated "turbo" codes. A decoding algorithm working on the dual code is introduced (in both multiplicative and additive form), by showing that changing in a proper way the representation of the soft information passed between constituent decoders in the iterative decoding process, the soft-input soft-output (SISO) modules of the decoder based on the dual code become equal to those used for the original code. A new technique to terminate the code trellis that significantly reduces the rate loss induced by the addition of terminating bits is described. Finally, an inverse puncturing technique applied to the highest rate "mother" code to yield a sequence of almost optimal codes with decreasing rates is proposed. Simulation results applied to the case of parallel concatenated codes show the significant advantages of the newly found codes in terms of performance and decoding complexity.     

一种8VSB Turbo格状编码调制方案

本文阐述这了一种适用于带限信道的８ＶＳＢ Ｔｕｒｂｏ格状编码制方案,它把纠错极强的Ｔｕｒｔｏ码和频带效率极高的格状编码调制结合,具有极强的信道传输能力。同时给出了相应的迭代译码算法及其误码率曲线。     

Coded modulation in the block-fading channel: coding theorems and code construction

We consider coded modulation schemes for the block-fading channel. In the setting where a codeword spans a finite number N of fading degrees of freedom, we show that coded modulations of rate R bit per complex dimension, over a finite signal set /spl chi//spl sube//spl Copf/ of size 2/sup M/, achieve the optimal rate-diversity tradeoff given by the Singleton bound /spl delta/(N,M,R)=1+/spl lfloor/N(1-R/M)/spl rfloor/, for R/spl isin/(0,M/spl rfloor/. Furthermore, we show also that the popular bit-interleaved coded modulation achieves the same optimal rate-diversity tradeoff. We present a novel coded modulation construction based on blockwise concatenation that systematically yields Singleton-bound achieving turbo-like codes defined over an arbitrary signal set /spl chi//spl sub//spl Copf/. The proposed blockwise concatenation significantly outperforms conventional serial and parallel turbo codes in the block-fading channel. We analyze the ensemble average performance under maximum-likelihood (ML) decoding of the proposed codes by means of upper bounds and tight approximations. We show that, differently from the additive white Gaussian noise (AWGN) and fully interleaved fading cases, belief-propagation iterative decoding performs very close to ML on the block-fading channel for any signal-to-noise ratio (SNR) and even for relatively short block lengths. We also show that, at constant decoding complexity per information bit, the proposed codes perform close to the information outage probability for any block length, while standard block codes (e.g., obtained by trellis termination of convolutional codes) have a gap from outage that increases with the block length: this is a different and more subtle manifestation of the so-called "interleaving gain" of turbo codes.     

Trellis‐Based Decoding of High‐Dimensional Block Turbo Codes

This paper introduces an efficient iterative decoding method for high‐dimensional block turbo codes. To improve the decoding performance, we modified the soft decision Viterbi decoding algorithm, which is a trellis‐based method. The iteration number can be significantly reduced in the soft output decoding process by applying multiple usage of extrinsic reliability information from all available axes and appropriately normalizing them. Our simulation results reveal that the proposed decoding process needs only about 30% of the iterations required to obtain the same performance with the conventional method at a bit error rate range of 10^?5 to 10^?6.     

Soft-decision and erasure decoding in fast frequency-hoppingsystems with convolutional, turbo, and Reed-Solomon codes

In this contribution we present an exhaustive treatment of various coding and decoding techniques for use in fast frequency-hopping/multiple frequency shift keying multiple-access systems. One of the main goals is to show how reliability information on each received bit can be derived to enable soft-decision decoding. Convolutional codes as well as turbo codes are considered applying soft-decision, erasure, and hard-decision decoding. Their performance is compared to that of previously proposed Reed-Solomon with either errors-only or errors-and-erasures decoding. A mobile radio environment yielding a frequency-selective fading channel is assumed. It is shown that the application of turbo codes and convolutional codes with soft decision decoding can allow for a comparable number of simultaneously transmitting users to Reed-Solomon codes with errors-and-erasures decoding. Furthermore, the advantage of soft decisions is shown, which can be applied to a widely and growing range of channel codes. The pertinent technique of calculating soft decisions is described in the paper     

On the theory and performance of trellis termination methods forturbo codes

The performance of a turbo code can be severely degraded if no trellis termination is employed. This paper investigates the implications of the choice of trellis termination method for turbo codes, and explains the origin of the performance degradation often experienced without trellis termination. An efficient method to derive the distance spectrum of turbo codes for different trellis termination methods is presented. Further, we present interleaver design rules that are tailored to each termination method. Using interleavers designed with these restrictions, we demonstrate that the performance difference between various termination methods is very small, including no trellis termination at all. For example, we demonstrate a turbo code with a 500-bit interleaver that exhibits no sign of an error floor for frame error rates as low as 10^-8, even though no trellis termination is employed     

Space-Time Trellis Code Design Based on Super Quasi-Orthogonal Block Codes With Minimum Decoding Complexity

In this letter, we propose a new family of space-time trellis codes, which are constructed by combining a super set of quasi-orthogonal space-time block codes with minimum decoding complexity with an outer multiple trellis coded modulation encoder. A systematic set-partitioning method for quadratic amplitude modulation constellations is given. The proposed scheme can be used for systems with four or more than four transmit antennas. Furthermore, its decoding complexity is low because its branch metric calculation can be implemented in a symbolwise way. Simulation results demonstrate that the proposed scheme has a comparable performance as super quasi-orthogonal space-time trellis codes proposed by Jafarkhani and Hassanpour while providing a lower decoding complexity.     

Syndrome trellis codes based on minimal span generator matrix

To improve the embedding efficiency of steganography, syndrome coding based on the coding theory has attracted many researchers’ attentions. In this paper, we make use of the relationship between syndrome coding for minimizing additive distortion and maximum likelihood decoding for linear codes to analyze the main parameters of convolutional codes which influence the embedding efficiency. And, the new syndrome trellis codes based on minimal span generator matrix is proposed. It can be considered an alternative construction of the state-of-the-art syndrome trellis codes (STCs) proposed by Filler and Fridrich recently. Experimental results show that the proposed scheme owns the same embedding performance to STCs and achieve the reduced time complexity and storage requirement meanwhile.     

Assembled space-time turbo trellis codes

A novel full rate space-time turbo trellis code, referred to as an assembled space-time turbo trellis code (ASTTTC), is presented in this paper. For this scheme, input information binary sequences are first encoded using two parallel concatenated convolutional encoders. The encoder outputs are split into four parallel streams and each of them is modulated by a QPSK modulator. The modulated symbols are assembled by a predefined linear function rather than punctured as in the standard schemes. This results in a lower code rate and a higher coding gain over time-varying fading channels. An extended two-dimensional (2-D) log-MAP (maximum a posteriori probability) decoding algorithm, which simultaneously calculates two a posteriori probabilities (APP), is developed to decode the proposed scheme. Simulation results show that, under the same conditions, the proposed code considerably outperforms the conventional space-time turbo codes over time-varying fading channels.     

Low-Density Parity-Check Codes with 2-State Trellis Decoding

A class of low-density parity-check (LDPC) codes with a simple 2-state trellis structure is presented. For LDPC decoding, the conventional belief propagation (BP) algorithm consists of numerous sub-decoders of single-parity check codes and exchanges information between sub-decoders in an iterative manner. If the single-parity check codes can be constructed and grouped in a proper way, the decoder can be decomposed into few identical 2-state trellis decoders. Therefore, instead of numerous sub-decoders of single-parity check codes, an iterative decoding algorithm based on few sub-decoders over 2-state trellis is proposed. The proposed decoding algorithm improves the efficiency of message passing between sub-decoders and hence provides a fast convergent rate as compared to the standard BP algorithm. Simulation results show that the proposed scheme provides a better performance and a fast convergent rate as compared to those of standard BP algorithm. The result also shows that the proposed algorithm has a similar performance as that of asynchronous replica shuffled BP algorithm and has a slightly inferior performance than that of synchronous replica shuffled BP algorithm. However, complexity analysis shows that our proposed algorithm has complexity that is lower than that of the replica shuffled BP algorithm.     

Turbo码概率-代数联合解码算法

Turbo码采用修正的BAHL et al.算法实现解码.这是一种基于软值的概率迭代解码算法.本文在保持Turbo码迭代软解码算法优点的基础上,充分利用Turbo码编码器结构这一确知条件,结合代数解码原理,提出了一种Turbo码概率-代数联合解码算法.该算法结合了概率解码和代数解码的优点,又有效避免了误差传播的发生,使Turbo码的纠错性能在原经典算法的基础上得到进一步的提高.该算法不仅为降低Turbo码的比特误码率和误差地板值提供了一种新的研究途径,而且因其更好的纠错性能而具有十分明显的实用价值.仿真实验结果显示,在比特误码率(BER)为10^-3~10^-4时,与经典Turbo码解码算法相比,采用该算法能获得0.1dB左右的编码增益.     

Iterative Equalization and TCM Decoding with Refined Channel Value

In this correspondence, we present a new method for the iterative equalization and decoding of multilevel trellis coded modulation (TCM) signals over frequency selective channels. Results show that the proposed algorithm achieves better performance compared to the previous work on the MMSE filter- based turbo equalization for a non-binary coded modulation scheme. The performance gain is accomplished by utilizing the combined modulation and coding nature of TCM and passing the refined signal obtained from different paths to the TCM decoder as the channel value in addition to the a priori probabilities.     

Turbo decoding as an instance of Pearl's “beliefpropagation” algorithm

We describe the close connection between the now celebrated iterative turbo decoding algorithm of Berrou et al. (1993) and an algorithm that has been well known in the artificial intelligence community for a decade, but which is relatively unknown to information theorists: Pearl's (1982) belief propagation algorithm. We see that if Pearl's algorithm is applied to the “belief network” of a parallel concatenation of two or more codes, the turbo decoding algorithm immediately results. Unfortunately, however, this belief diagram has loops, and Pearl only proved that his algorithm works when there are no loops, so an explanation of the experimental performance of turbo decoding is still lacking. However, we also show that Pearl's algorithm can be used to routinely derive previously known iterative, but suboptimal, decoding algorithms for a number of other error-control systems, including Gallager's (1962) low-density parity-check codes, serially concatenated codes, and product codes. Thus, belief propagation provides a very attractive general methodology for devising low-complexity iterative decoding algorithms for hybrid coded systems     

On low-complexity space-time coding for quasi-static channels

We propose a new space-time coding scheme for the quasi-static multiple-antenna channel with perfect channel state information at the receiver and no channel state information at the transmitter. In our scheme, codewords produced by a trellis encoder are formatted into space-time codeword arrays such that decoding can be implemented efficiently by minimum mean-square error (MMSE) decision-feedback interference mitigation coupled with Viterbi decoding, through the use of per-survivor processing. We discuss the code design for the new scheme, and show that finding codes with optimal diversity is much easier than for conventional trellis space-time codes (STCs). We provide an upper bound on the word-error rate (WER) of our scheme which is both accurate and easy to evaluate. Then, we find upper and lower bounds on the information outage probability with discrete independent and identically distributed (i.i.d). inputs (as opposed to Gaussian inputs, as in most previous works) and we show that the MMSE front-end yields a large advantage over the whitened matched filter (i.e., zero-forcing) front-end. Finally, we provide a comprehensive performance/complexity comparison of our scheme with coded vertical Bell Labs layered space-time (V-BLAST) architecture and with the recently proposed threaded space-time codes. We also discuss the concatenation of our scheme with block space-time precoders, such as the linear dispersion codes.     

Low-complexity iterative joint source-channel decoding for variable-length encoded Markov sources

In this paper, we present a novel packetized bit-level decoding algorithm for variable-length encoded Markov sources, which calculates reliability information for the decoded bits in the form of a posteriori probabilities (APPs). An interesting feature of the proposed approach is that symbol-based source statistics in the form of the transition probabilities of the Markov source are exploited as a priori information on a bit-level trellis. This method is especially well-suited for long input blocks, since in contrast to other symbol-based APP decoding approaches, the number of trellis states does not depend on the packet length. When additionally the variable-length encoded source data is protected by channel codes, an iterative source-channel decoding scheme can be obtained in the same way as for serially concatenated codes. Furthermore, based on an analysis of the iterative decoder via extrinsic information transfer charts, it can be shown that by using reversible variable-length codes with a free distance of two, in combination with rate-1 channel codes and residual source redundancy, a reliable transmission is possible even for highly corrupted channels. This justifies a new source-channel encoding technique where explicit redundancy for error protection is only added in the source encoder.     

Turbo decoding of simple product codes in a two user binary adder channel employing the Bahl–Cocke–Jelinek–Raviv algorithm

The main goal in this paper is an investigation of the Bahl–Cocke–Jelinek–Raviv (BCJR) algorithm applied in a turbo decoding scheme. Binary product codes are employed in a turbo coding scheme and the channel model considered is the two user binary adder channel (2-BAC) with additive white Gaussian noise. A trellis for two users is constructed for a pair of product codes tailored for use in the 2-BAC in order to employ the BCJR decoding algorithm. Computer simulation is employed to show that product codes on the 2-BAC, employing low-complexity component codes, produces considerable gain with few iterations under iterative BCJR decoding.     

Space-time turbo trellis codes for two, three, and four transmit antennas

New space-time turbo trellis codes (ST turbo TCs) with 4-phase-shift keying (PSK) and 8-PSK for two, three, and four transmit antennas in slow and fast fading channels are proposed in this paper. The component codes of the space-time turbo schemes are constructed by choosing the feedforward coefficients to maximize the minimum squared Euclidean distance and the feedback coefficients to minimize the iterative decoding threshold. The performance of the proposed ST turbo TCs with various memory orders, transmit antennas, and interleaver structures is evaluated by simulation. It is shown that the new codes achieve better performance than previously designed codes. The impact of antenna correlation and imperfect channel estimation on the code performance is also discussed.     

Turbo Decoding on the Binary Erasure Channel: Finite-Length Analysis and Turbo Stopping Sets

This paper is devoted to the finite-length analysis of turbo decoding over the binary erasure channel (BEC). The performance of iterative belief-propagation decoding of low-density parity-check (LDPC) codes over the BEC can be characterized in terms of stopping sets. We describe turbo decoding on the BEC which is simpler than turbo decoding on other channels. We then adapt the concept of stopping sets to turbo decoding and state an exact condition for decoding failure. Apply turbo decoding until the transmitted codeword has been recovered, or the decoder fails to progress further. Then the set of erased positions that will remain when the decoder stops is equal to the unique maximum-size turbo stopping set which is also a subset of the set of erased positions. Furthermore, we present some improvements of the basic turbo decoding algorithm on the BEC. The proposed improved turbo decoding algorithm has substantially better error performance as illustrated by the given simulation results. Finally, we give an expression for the turbo stopping set size enumerating function under the uniform interleaver assumption, and an efficient enumeration algorithm of small-size turbo stopping sets for a particular interleaver. The solution is based on the algorithm proposed by Garello et al. in 2001 to compute an exhaustive list of all low-weight codewords in a turbo code.     

A new high rate code scheme with highly parallel and low complexity decoding algorithm

A new high rate code scheme is proposed in this paper. It consists of serial concatenated recursive systematic ordinary (nonpunctured) convolutional codes with only 8 states in the trellis of the corresponding reciprocal dual codes. With a low complexity and highly parallel decoding algorithm, over additive white Gaussian noise channels, the proposed codes can achieve good bit error rate (BER) performance comparable to that of turbo codes and low density parity check (LDPC) codes. At code rate R=16/17, the overall decoding complexity of the proposed code scheme is almost half that of the LDPC codes.     

Multilevel turbo coding with short interleavers

The impact of the interleaver, embedded in the encoder for a parallel concatenated code, called the turbo code, is studied. The known turbo codes consist of long random interleavers, whose purpose is to reduce the value of the error coefficients. It is shown that an increased minimum Hamming distance can be obtained by using a structured interleaver. For low bit-error rates (BERs), we show that the performance of turbo codes with a structured interleaver is better than that obtained with a random interleaver. Another important advantage of the structured interleaver is the short length required, which yields a short decoding delay and reduced decoding complexity (in terms of memory). We also consider the use of turbo codes as component codes in multilevel codes. Powerful coding structures that consist of two component codes are suggested. Computer simulations are performed in order to evaluate the reduction in coding gain due to suboptimal iterative decoding. From the results of these simulations we deduce that the degradation in the performance (due to suboptimal decoding) is very small