Asymptotic performance comparison of concatenated (turbo) codes over GF(4)

In this paper, we investigate and compare the asymptotic performance of concatenated convolutional coding schemes over GF(4) over additive white Gaussian noise (AWGN) channels. Both parallel concatenated codes (PCC) and serial concatenated codes (SCC) are considered. We construct such codes using optimal non‐binary convolutional codes where optimality is in the sense of achieving the largest minimum distance for a fixed number of encoder states. Code rates of the form k₀/(k₀ + 1) for k₀=1, 8, and 64 are considered, which suite a wide spectrum of communications applications. For all of these code rates, we find the minimum distance and the corresponding multiplicity for both concatenated code systems. This is accomplished by feeding the encoder with all possible weight‐two and weight‐three input information patterns and monitoring, at the output of the encoder, the weight of the corresponding codewords and their multiplicity. Our analytical results indicate that the SCC codes considerably outperform their counterpart PCC codes at a much lower complexity. Inspired by the superiority of SCC codes, we also discuss a mathematical approach for analysing such codes, leading to a more comprehensive analysis and allowing for further improvement in performance by giving insights on designing a proper interleaver that is capable of eliminating the dominant error patterns. Copyright © 2004 John Wiley & Sons, Ltd.     

On the asymptotic input-output weight distributions of some accumulate-based codes

In this letter, we show how to compute the asymptotic growth rate of input-output weight enumerator (AGR-IOWE) for some accumulate-based codes by using the sharp tools already developed. Numerical results on the AGR-IOWE for irregular repeat-accumulate (IRA) codes, systematic regular RA (SRA) codes, and concatenated zigzag codes are reported. It is observed that the SRA code has the same AGR-IOWE as a comparable concatenated zigzag code. For both SRA and concatenated zigzag codes, if keeping the code rate fixed, the increase of the grouping factor for the component punctured accumulate code may result in better asymptotic performance under maximum-likelihood decoding, but often worse performance under iterative sum-product decoding.     

Improved upper bounds on the ML decoding error probability ofparallel and serial concatenated turbo codes via their ensemble distancespectrum

The ensemble performance of parallel and serial concatenated turbo codes is considered, where the ensemble is generated by a uniform choice of the interleaver and of the component codes taken from the set of time-varying recursive systematic convolutional codes. Following the derivation of the input-output weight enumeration functions of the ensembles of random parallel and serial concatenated turbo codes, the tangential sphere upper bound is employed to provide improved upper bounds on the block and bit error probabilities of these ensembles of codes for the binary-input additive white Gaussian noise (AWGN) channel, based on coherent detection of equi-energy antipodal signals and maximum-likelihood decoding. The influence of the interleaver length and the memory length of the component codes is investigated. The improved bounding technique proposed here is compared to the conventional union bound and to a alternative bounding technique by Duman and Salehi (1998) which incorporates modified Gallager bounds. The advantage of the derived bounds is demonstrated for a variety of parallel and serial concatenated coding schemes with either fixed or random recursive systematic convolutional component codes, and it is especially pronounced in the region exceeding the cutoff rate, where the performance of turbo codes is most appealing. These upper bounds are also compared to simulation results of the iterative decoding algorithm     

迭代译码的级联Reed-Solomon乘积码与卷积码

该文提出用Reed Solomon(RS)乘积码作为外码,卷积码作为内码的级联码方案并且内外码间用Congruential向量生成的交织图案对RS码符号进行重排列。对此级联码采用的迭代译码基于成员码的软译码算法。当迭代次数达到最大后,通过计算RS码的校正子,提出一种纠正残余错误的方法,进一步提高了系统的误比特性能。仿真结果表明,在AWGN信道中与迭代译码的级联RS/卷积码相比,当误比特率为1e-5时,新系统的编码增益大约有0.4 dB。     

Performance of RS coded M-ary modulation with and without symbol overlapping

In this paper, we present analytical bit error probability results for M-ary modulation concatenated with Reed Solomon (RS) codes. The analysis of bit error probability is nontrivial as the number of bits per symbol for the RS codes may not be an integer multiple of the number of bits per symbol for a modulation symbol. We propose a Markov chain technique which allows analytical evaluation of the bit error probability for such cases. The performance of RS coding with coherent biorthogonal, coherent/non-coherent orthogonal modulation over an additive white Gaussian noise (AWGN) channel is evaluated. Simulation of the bit error probability of RS code concatenated with a Nordstrom Robinson (NR) code as an inner code is performed and compared with the case of biorthogonal modulation. From the results, we notice that a stronger inner code gives better bit error probability. In addition, the throughput of the coded system with biorthogonal modulation over an AWGN channel is discussed. For a Rayleigh flat fading and block fading channel, we analyze the bit error probability of RS codes concatenated with biorthogonal modulation. From the result, we notice that a stronger outer code gives a better bit error probability for the case of Rayleigh flat fading channel.     

Iterative Decoding of Concatenated Convolutional Codes: Implementation Issues

This tutorial paper gives an overview of the implementation aspects related to turbo decoders, where the term turbo generally refers to iterative decoders intended for parallel concatenated convolutional codes as well as for serial concatenated convolutional codes. We start by considering the general structure of iterative decoders and the main features of the soft-input soft-output algorithm that forms the heart of iterative decoders. Then, we show that very efficient parallel architectures are available for all types of turbo decoders allowing high-speed implementations. Other implementation aspects like quantization issues and stopping rules used in conjunction with buffering for increasing throughput are considered. Finally, we perform an evaluation of the complexities of the turbo decoders as a function of the main parameters of the code.     

Concatenated codes with large minimum distance (Corresp.)

Some concatenated codes of length 128 and less are constructed. Nineteen of these codes are superior to the best previously known linear codes, as shown by the fact that the wellknown lower bound on the minimum distance of the concatenated code as the product of the minimum distances of its component codes exceeds the minimum distance of the best previously known code.     

Analysis, design, and iterative decoding of double seriallyconcatenated codes with interleavers

A double serially concatenated code with two interleavers consists of the cascade of an outer encoder, an interleaver permuting the outer codeword bits, a middle encoder, another interleaver permuting the middle codeword bits, and an inner encoder whose input words are the permuted middle codewords. The construction can be generalized to h cascaded encoders separated by h-1 interleavers, where h>3. We obtain upper bounds to the average maximum likelihood bit-error probability of double serially concatenated block and convolutional coding schemes. Then, we derive design guidelines for the outer, middle, and inner codes that maximize the interleaver gain and the asymptotic slope of the error probability curves. Finally, we propose a low-complexity iterative decoding algorithm. Comparisons with parallel concatenated convolutional codes, known as “turbo codes”, and with the proposed serially concatenated convolutional codes are also presented, showing that in some cases, the new schemes offer better performance     

Rice衰落下RS码级联空时分组码系统的性能分析

从理论上给出一种Rice衰落条件下Reed-Solomon码级联空时分组码系统的差错性能分析方法,并推导给出级联码误比特率上界的数学表达式。理论分析和仿真结果表明,随着信噪比的增加,级联码系统的性能曲线迅速变好,获得了很高的编码增益。在误比特率为10-4时,与Reed-Solomon码的级联可以使衰落条件下空时分组码的性能提高大约5 dB。     

New results on serial concatenated and accumulated-convolutional turbo code performance

Previous methods for analyzing serial concatenated turbo codes employing union error bounds are extended to determine the complete output weight enumeration function of the code; this provides the opportunity to employ a more refined bound due to Polytrev, with considerably improved results limited, however, to block lengths of about 256 bits by computational constraints. The method is then applied to a new class of “accumulated-convolutional” codes, which is a simple special subclass of serial concatenated codes inspired by the “repeat-accumulate” codes of Divsalar et al. Performance appears to be superior to that of conventional codes and results are obtained for much longer block lengths, with impressive results in regions approaching channel capacity.     

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     

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.     

Joint iterative decoding of serially concatenated error controlcoded CDMA

Joint iterative decoding of multiple forward error control (FEC) encoded data streams is studied for linear multiple access channels, such as code-division multiple access (CDMA). It is shown that such systems can be viewed as serially concatenated coding systems, and that iterative soft-decision decoding can be performed successfully To improve power efficiency, powerful FEC codes are used. These FEC codes are themselves serially concatenated. The overall transmission system can be viewed as the concatenation of two error control codes with the linear multiple access channel, and soft-decision decoders are used at each stage. A variance transfer function approach applied to the analysis of this system captures the role of the component decoders in an overall iterative decoding system. We show that this approach forms a methodology to study the effects of the component codes as well as that of the iteration schedule. Analysis and simulation examples are presented for transmission systems that operate close to the Shannon limit and illustrate the accuracy of the analysis     

Performance of RS-BCH Concatenated Codes and BCH Single-Stage Codes on an Interference Satellite Channel

In this correspondence, a model is analyzed that was designed to study interference on satellite channels. We developed this model to obtain performance results for a coherent phase-shift keyed (CPSK) system in which RS-BCH concatenated codes and BCH singlestage codes are applied to a satellite channel corrupted by cochannel interference. These results make use of earlier work on performance analysis of anm-phase CPSK system operating in the presence of random Gaussian noise and non-Gaussian interference. Earlier work on performance evaluation of concatenated codes on an equierror channel is also used. Our model incorporates features that account for the burst behavior of the interference sources. Results indicate that the use of RS-BCH concatenated coding provides significant performance improvement over no coding as well as single-stage BCH coding.     

Analysis and design of parallel concatenated Gallager codes

The convergence characteristics of parallel concatenated Gallager codes using the Gaussian approximation of extrinsic information are analysed. A design strategy is introduced to select the component low-density parity-check codes effectively     

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     

Concatenated coding performance for FSK modulation ontime-correlated Rician fading channels

We present an analytical method for evaluating the performance of noninterleaved concatenated codes over channels modeled as a nonfrequency selective correlated Rician fading channel with a known power spectral density. The main idea is to model the communication system from the modulator input to the demodulator output as a finite state channel (FSC) model, and apply powerful enumeration techniques to such a discrete channel in order to gain useful information on the system performance. The concatenated scheme makes use of two codes; Reed-Solomon codes are employed for the outer code, and binary block codes are used as the inner code. Next, the method is extended to study the effect on the performance when an interleaving with finite depth is incorporated into the communication system. A comparison between symbol and bit interleaving is made. Finally, we study the potential gain produced when channel information is passed on to the outer decoder in the form of an erasure symbol. In all cases, analytical expressions for the probability of the number of error symbols produced by the FSC model were obtained in terms of a coefficient in a formal power series. This is an interesting alternative approach with respect to computer simulations     

Woven codes with outer warp: variations, design, and distanceproperties

We consider convolutional and block encoding schemes which are variations of woven codes with outer warp. We propose methods to evaluate the distance characteristics of the considered codes on the basis of the active distances of the component codes. With this analytical bounding technique, we derived lower bounds on the minimum (or free) distance of woven convolutional codes, woven block codes, serially concatenated codes, and woven turbo codes. Next, we show that the lower bound on the minimum distance can be improved if we use designed interleaving with unique permutation functions in each row of the warp of the woven encoder. Finally, with the help of simulations, we get upper bounds on the minimum distance for some particular codes and then investigate their performance in the Gaussian channel. Throughout this paper, we compare all considered encoding schemes by means of examples, which illustrate their distance properties     

Concatenated coding for DS/CDMA transmission in wirelesscommunications

We investigate the use of forward-error correction (FEC) as well as concatenated coding for reliable data transmission in asynchronous direct-sequence code-division multiple-access communications over frequency-selective Rayleigh fading channels. The FEC scheme combines antenna diversity with low complexity concatenated codes which consist of a Reed-Solomon outer code and a convolutional inner code. Under this concatenated coding scheme, we analyze the average bit-error rate performance and capacity tradeoffs between various system parameters under a fixed total bandwidth expansion and concatenated codes constraint requirements     

Unveiling turbo codes: some results on parallel concatenated codingschemes

A parallel concatenated coding scheme consists of two simple constituent systematic encoders linked by an interleaver. The input bits to the first encoder are scrambled by the interleaver before entering the second encoder. The codeword of the parallel concatenated code consists of the input bits to the first encoder followed by the parity check bits of both encoders. This construction can be generalized to any number of constituent codes. Parallel concatenated schemes employing two convolutional codes as constituent codes, in connection with an iterative decoding algorithm of complexity comparable to that of the constituent codes, have been previously shown to yield remarkable coding gains close to theoretical limits. They have been named, and are known as, “turbo codes”. We propose a method to evaluate an upper bound to the bit error probability of a parallel concatenated coding scheme averaged over all interleavers of a given length. The analytical bounding technique is then used to shed some light on some crucial questions, which have been floating around in the communications community since the proposal of turbo codes