Fast turbo codes with space‐time block codes in fast fading channels

In this paper, space‐time block coding has been used in conjunction with Turbo codes to provide good diversity and coding gains. A new method of dividing turbo encoder and decoder into several parallel encoding and decoding blocks is considered. These blocks work simultaneously and yield a faster coding scheme in comparison to classical Turbo codes. The system concatenates fast Turbo coding as an outer code with Alamouti's G2 space‐time block coding scheme as an inner code, achieving benefits associated with both techniques including acceptable diversity and coding gain as well as short coding delay. In this paper, fast fading Rayleigh and Rician channels are considered for discussion. For Rayleigh fading channels, a fixed frame size and channel memory length of 5000 and 10, respectively, the coding gain is 7.5 dB and bit error rate (BER) of 10^?4 is achieved at 7 dB. For the same frame size and channel memory length, Rician fading channel yields the same BER at about 4.5 dB. Copyright © 2013 John Wiley & Sons, Ltd.     

Transmission performance analysis of a new class of line codes for optical fiber systems

A family of mB(m+1)B binary, nonalphabetic, balanced line codes is presented that is suitable for high bit rate (>or=135 Mb/s) optical fiber transmission due to its relatively simple encoding and decoding rules. Here, B represents a block of m bits, where m is an odd number. The coding, decoding, and bit error rate (BER) performance of the codes are discussed. Statistical and spectral analysis for the specific case in which the number of bits, m, equals seven, is presented. This makes possible a detailed comparison of the proposed code with conventional 7B8B codes.<>     

Block error-correcting codes for systems with a very high BER: Theoretical analysis and application to the protection of watermarks

This paper addresses the problem of error-correction for extremely noisy channels (BER from 0.1 to 0.5), such as those obtained for image or video watermarking. Minimum distance arguments are used to identify a region for which no other code is as efficient as repetition codes, whatever the rate, at least when bounded decoding is considered. However, in order to obtain a reasonable and sufficiently low BER, repetition codes are not very efficient. We present a coding scheme concatenating a repetition code with another one, and design rules in order to select these codes for a given watermarking application are developed. The repetition code lowers the huge channel BER, as no other code can do this part of the job. Then, the second more powerful code working at a lower BER achieves a larger BER reduction. In this paper, this role is devoted to BCH codes, as members of a classical family. Thanks to their moderate decoding complexity, they turn out to be an interesting cost versus performance trade-off, while more efficient coding schemes based on soft decoding are far more complex. However, we also provide an idea of the solutions to consider for watermarking applications with fewer complexity limitations, for which more powerful decoding techniques can be implemented.     

Complexity Reduction in SISO Decoding of Block Codes

SISO decoding for block codes can be carried out based on a trellis representation of the code. However, the complexity entailed by such decoding is most often prohibitive and thus prevents practical implementation. This paper examines a new decoding scheme based on the soft-output Viterbi algorithm (SOVA) applied to a sectionalized trellis for linear block codes. The computational complexities of the new SOVA decoder and of the conventional SOVA decoder, based on a bit-level trellis, are theoretically analyzed and derived for different linear block codes. These results are used to obtain optimum sectionalizations of a trellis for SOVA. For comparisons, the optimum sectionalizations for Maximum A Posteriori (MAP) and Maximum Logarithm MAP (Max-Log-MAP) algorithms, and their corresponding computational complexities are included. The results confirm that the new SOVA decoder is the most computationally efficient SISO decoder, in comparisons to MAP and Max-Log-MAP algorithms. The simulation results of the bit error rate (BER) performance, assuming binary phase -- shift keying (BPSK) and additive white Gaussian noise (AWGN) channel, demonstrate that the performance of the new decoding scheme is not degraded. The BER performance of iterative SOVA decoding of serially concatenated block codes shows no difference in the quality of the soft outputs of the new decoding scheme and of the conventional SOVA.     

Computing the word-, symbol-, and bit-error rates for block error-correcting codes

In order to select error-correcting codes for various applications, their performances have to be determined. However, when targeting error-rate computations for block error-correcting codes, many required results are missing in the coding literature. Even in the simple case of binary codes and bounded-distance decoding, classical texts do not provide a bit-error rate (BER) expression taking into account both decoding errors and failures. In the case of nonbinary codes used to protect binary symbols, such as Reed-Solomon codes in many applications, there is no available result making realistic channel assumptions in order to derive BERs. Finally, for the more complex case of complete decoding, only some bounds are available, such as the union one. This paper presents new approximations of error rates for block error-correcting codes as a function of the channel BER (crossover probability). We extend an existing approximation in order to consider not only bounded-distance decoding, but also complete "nearest-neighbor" decoding. We also develop approximations able to deal with nonbinary codes. Combined with state-of-the-art approximations, these new results enable the computation of bit-, symbol-, and word-error rates in various decoding situations. They can consider separately errors related to erroneously decoded words and decoding failures, and they provide accurate estimates of error rates. As they do not require detailed information about the structure of codes, they are general enough to be used in simple comparisons between different codes, avoiding the need for simulations.     

Efficient Space-Time Block Codes Derived from Quasi-Orthogonal Structures

We propose a new class of block codes that outperforms known space-time block codes at low rates. The new codes are designed by using appropriate rotations and set partitioning on two quasi-orthogonal codes, and combining subsets of their codewords. Using these techniques we are able to obtain higher coding gain at a given rate and improve performance. Simulations confirm the advantages of this code compared to other codes operating at the same rate and signal-to-noise ratio (SNR). We also provide an efficient maximum likelihood (ML) decoding algorithm for the new code     

Neural network decoding of the block Markov superposition transmission

A neural network (NN)-based decoding algorithm of block Markov superposition transmission (BMST) was researched.The decoders of the basic code with different network structures and representations of training data were implemented using NN.Integrating the NN-based decoder of the basic code in an iterative manner,a sliding window decoding algorithm was presented.To analyze the bit error rate (BER) performance,the genie-aided (GA) lower bounds were presented.The NN-based decoding algorithm of the BMST provides a possible way to apply NN to decode long codes.That means the part of the conventional decoder could be replaced by the NN.Numerical results show that the NN-based decoder of basic code can achieve the BER performance of the maximum likelihood (ML) decoder.For the BMST codes,BER performance of the NN-based decoding algorithm matches well with the GA lower bound and exhibits an extra coding gain.     

Nonlinear Product Codes and Their Low Complexity Iterative Decoding

This paper proposes encoding and decoding for nonlinear product codes and investigates the performance of nonlinear product codes. The proposed nonlinear product codes are constructed as N‐dimensional product codes where the constituent codes are nonlinear binary codes derived from the linear codes over higher order alphabets, for example, Preparata or Kerdock codes. The performance and the complexity of the proposed construction are evaluated using the well‐known nonlinear Nordstrom‐Robinson code, which is presented in the generalized array code format with a low complexity trellis. The proposed construction shows the additional coding gain, reduced error floor, and lower implementation complexity. The (64, 24, 12) nonlinear binary product code has an effective gain of about 2.5 dB and 1 dB gain at a BER of 10^?6 when compared to the (64, 15, 16) linear product code and the (64, 24, 10) linear product code, respectively. The (256, 64, 36) nonlinear binary product code composed of two Nordstrom‐Robinson codes has an effective gain of about 0.7 dB at a BER of 10^?5 when compared to the (256, 64, 25) linear product code composed of two (16, 8, 5) quasi‐cyclic codes.     

Iterative decoding of binary block and convolutional codes

Iterative decoding of two-dimensional systematic convolutional codes has been termed “turbo” (de)coding. Using log-likelihood algebra, we show that any decoder can be used which accepts soft inputs-including a priori values-and delivers soft outputs that can be split into three terms: the soft channel and a priori inputs, and the extrinsic value. The extrinsic value is used as an a priori value for the next iteration. Decoding algorithms in the log-likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code. The iteration is controlled by a stop criterion derived from cross entropy, which results in a minimal number of iterations. Optimal and suboptimal decoders with reduced complexity are presented. Simulation results show that very simple component codes are sufficient, block codes are appropriate for high rates and convolutional codes for lower rates less than 2/3. Any combination of block and convolutional component codes is possible. Several interleaving techniques are described. At a bit error rate (BER) of 10^-4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block/convolutional component codes, interleaver sizes less than 1000 and for three to six iterations     

Near-optimum decoding of product codes: block turbo codes

This paper describes an iterative decoding algorithm for any product code built using linear block codes. It is based on soft-input/soft-output decoders for decoding the component codes so that near-optimum performance is obtained at each iteration. This soft-input/soft-output decoder is a Chase decoder which delivers soft outputs instead of binary decisions. The soft output of the decoder is an estimation of the log-likelihood ratio (LLR) of the binary decisions given by the Chase decoder. The theoretical justifications of this algorithm are developed and the method used for computing the soft output is fully described. The iterative decoding of product codes is also known as the block turbo code (BTC) because the concept is quite similar to turbo codes based on iterative decoding of concatenated recursive convolutional codes. The performance of different Bose-Chaudhuri-Hocquenghem (BCH)-BTCs are given for the Gaussian and the Rayleigh channel. Performance on the Gaussian channel indicates that data transmission at 0.8 dB of Shannon's limit or more than 98% (R/C>0.98) of channel capacity can be achieved with high-code-rate BTC using only four iterations. For the Rayleigh channel, the slope of the bit-error rate (BER) curve is as steep as for the Gaussian channel without using channel state information     

Performance comparison of proposed 3-d coherent spatial-phase-time coding/decoding with super structured fiber Bragg grating-based optical CDMA systems

In this paper, performance comparison in terms of bit error rate (BER) of proposed WDM compatible optical CDMA system incorporating 3-D spectral-phase-time encoding/decoding to a 7 chip-super-structured fiber Bragg grating (SSFBG)-based optical code division multiple access (OCDMA) system is investigated. Coding and decoding using binary [0, π] phase chips is demonstrated for six users at 5 Gb/s, and a single coded signal is separated with acceptable bit-error rate ≤10^?9. In our proposed optical CDMA system encoding and decoding is done by converting hadamard codes (used for conventional CDMA system) to phase codes. It is then compared with two optical pulse retiming and reshaping systems incorporating super structured fiber Bragg gratings (SSFBGs) as pulse shaping elements. Simulation results show that with all input bands having same sample rate, size data rates our proposed codes with even larger number of channels perform better in terms of eye opening & BER.     

基于MSK和LT码的联合迭代译码算法

传统的简单级联编码调制系统在译码时会造成软信息损失.提出了一种基于MSK和LT码的联合软迭代译码算法,设计了算法的系统模型.利用LT码的软译码和MSK调制的SISO算法,进行联合软迭代译码,提高了编码调制系统的性能.仿真结果表明:在误码率为10-4时,提出的算法比传统的简单级联编码调制算法约有1.5 dB的编码增益.     

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.     

Error correcting codes for fast frequency-hopped MFSK spread-spectrum satellite communications under worst case jamming

Spread-spectrum techniques such as frequency hopping (FH) have been used to reduce the vulnerability of satellite communications to jamming. In such a system, error correcting (EC) codes play an essential role. Thus a knowledge of the performance of various EC codes is necessary in order to choose an EC code and related system parameters in the design of an anti-jam system, this paper examines the performance of various types of error correcting codes under worst case partial band noise jamming and worst case multitone jamming using fast frequency-hopped, non-coherent M-ary frequency-shift keying (NCMFSK) with optimum diversity. A comprehensive study including convolutional codes, binary and non-binary block codes and concatenated codes has been conducted. This paper presents bit error rate (BER) performances of various error correcting codes. New candidate codes with superior anti-jam performance are identified.     

Universal lossless source controlled channel decoding for i.i.d. sequences

Messages coded and transmitted over a channel usually contain some redundancy which is not utilized by channel decoding techniques, especially if its governing statistical parameters are unknown. We propose to integrate universal lossless source coding techniques into channel decoding of redundant sequences with unknown statistics to improve performance of Viterbi and turbo decoding. Simulation results demonstrate that we achieve identical bit error rate (BER) performance to nonuniversal techniques that utilize prior knowledge of the message statistics and that if redundancy exists even in an a-priori unknown form, we can improve the code performance over standard techniques.     

Noncoherent block-coded MPSK

For coherent detection, block-coded modulation is a bandwidth efficient scheme. In this paper, we propose theorems about the error performance of block-coded modulation using M-ary phase-shift keying (MPSK) for noncoherent detection. Based on these theorems, we propose a novel block-coded modulation scheme for noncoherent detection called noncoherent block-coded MPSK. The proposed scheme provides flexible designs of noncoherent block codes with different code rate, block length and error performance. Good noncoherent block codes can be easily obtained by properly choosing binary linear block codes as the component codes. Moreover, noncoherent block codes of this new scheme can be decoded by multistage decoding, which has the advantage of low complexity and satisfactory error performance. In this paper, two algorithms of multistage decoding for noncoherent detection are proposed as well. The error performance of some designed codes and decoding algorithms is verified by computer simulation.     

SNR-Invariant Importance Sampling for Hard-Decision Decoding Performance of Linear Block Codes

We present an importance sampling (IS) technique for evaluating the word-error rate (WER) and bit-error rate (BER) performance of binary linear block codes under hard-decision decoding. This IS technique takes advantage of the invariance of the decoding outcome to the transition probability of the binary symmetric channel given a received error pattern, and is equivalent to the method of stratification for variance reduction. A thorough analysis of the accuracy of the proposed signal-to-noise-ratio-invariant IS (IIS) estimator based on computing its relative bias and standard deviation is provided. Under certain conditions, which may be achieved fairly easily for certain code and decoder combinations, we demonstrate that it is possible to use the proposed IIS technique to accurately evaluate the WER and BER to arbitrarily low values. Further, in all cases, the probability estimates obtained via IIS always serve as a lower bound on the true probability values     

Interlevel‐correlated orthogonal space–time block codes based on the expanded signal constellation

Orthogonal space–time block codes provide full diversity with a very simple decoding scheme. However, they do not provide much coding gain. For a given space–time block code, we combine several component codes in conjunction with set partitioning of the expanded signal constellation according to the coding gain distance (CGD) criterion. By providing proper interlevel coding between adjacent blocks, we can design an orthogonal space–time block code with high rate, large coding gain, and low decoding complexity. The error performance of an example code is compared with some codes in computer simulation. These codes are compared based on the situation of the same transmission rate, space diversity order, and state complexity of decoding trellis. Copyright © 2010 John Wiley & Sons, Ltd.     

Polar码在OFDM系统中应用研究

Polar码是2007年由ErdalArikan基于信道极化理论提出的一种新的信道编码方法。在理论上,它能够达到信道容量,并且有着较低复杂度的编译码算法。由于Polar码出现不久,其相关应用研究报道较少。OFDM（Orthogonal Frequency Division Multiplexing）即正交频分复用技术将信道分成若干正交子信道,将高速数据信号转换成并行的低速子数据流,调制到每个子信道上进行传输。本文研究Pola鸸在OFDM系统中的应用,讨论在OFDM系统中Polar码的译码迭代次数、码长、码率以及系统输入信噪比（SignaltoNoiseRatio,SNR）等因素对信号传输的影响。通过性能仿真说明在一定条件下译码迭代次数增加,和码率减小,都能使Polar—OFDM系统的性能更好。     

Probabilistic construction of large constraint length trellis codesfor sequential decoding

Probabilistic algorithms are given for constructing good large constraint length trellis codes for use with sequential decoding that can achieve the channel cutoff rate bound at a bit error rate (BER) of 10^-5-10^-6. The algorithms are motivated by the random coding principle that an arbitrary selection of code symbols will produce a good code with high probability. One algorithm begins by choosing a relatively small set of codes randomly. The error performance of each of these codes is evaluated using sequential decoding and the code with the best performance among the chosen set is retained. Another algorithm treats the code construction as a combinatorial optimization problem and uses simulated annealing to direct the code search. Trellis codes for 8 PSK and 16 QAM constellations with constraint lengths v up to 20 are obtained. Simulation results with sequential decoding show that these codes reach the channel cutoff rate bound at a BER of 10^-5-10^-6 and achieve 5.0-6.35 dB real coding gains over uncoded systems with the same spectral efficiency and up to 2.0 dB real coding gains over 64 state trellis codes using Viterbi decoding