On bit-error probability of a concatenated coding scheme

This paper presents a method for evaluating the bit-error probability of a concatenated coding system for BPSK transmission over the AWGN channel. In the concatenated system, a linear binary block code is used as the inner code and is decoded with the soft-decision maximum likelihood decoding, and a maximum distance separable code (or its interleaved code) is used as the outer code and is decoded with a bounded distance decoding. The method is illustrated through a specific example in which the inner code is a binary (64.40.8) Reed-Muller subcode and the outer code is the NASA standard (255, 223, 33) Reed-Solomon code over GF(2⁸) interleaved to a depth of 5. This specific concatenated system is being considered for NASA's high-speed satellite communications. The bit-error performance is evaluated by a combination of simulation and analysis. The split weight enumerators for the maximum distance separable codes are derived and used for the analysis     

Reliability and Throughput Analysis of a Concatenated Coding Scheme

In this paper, the performance of a concatenated coding scheme for error control in ARQ systems is analyzed for both randomerror and burst-error channels. In particular, the probability of undetected error and the system throughput are calculated. In this scheme, the inner code is used for both error correction and error detection, and the outer code is used for error detection only. Interleaving/deinterleaving of the outer code is assumed. A retransmission is requested if either the inner code or the outer code detects the Presence of errors. Various coding examples are considered. The results show that concatenated coding can provide extremely high system reliability (i.e., low probability of undetected error) and high system throughput.     

A Concatenated Coding Scheme for Error Control

In this paper, a concatenated coding scheme for error control in data communications is presented and analyzed. In this scheme, the inner code is used for both error correction and detection; however, the outer code is used only for error detection. A retransmission is requested if either the inner code decoder fails to make a successful decoding or the outer code decoder detects the presence of errors after the inner code decoding. Probability of undetected error (or decoding error) of the proposed scheme is derived. An efficient method for computing this probability is presented. Throughput efficiency of the proposed error control scheme incorporated with a selective-repeat ARQ retransmission strategy is also analyzed. Three specific examples are presented. One of the examples is proposed for error control in the NASA Telecommand System.     

A cascaded coding scheme for error control and its performanceanalysis

A coding scheme for error control in data communication systems is investigated. The scheme is obtained by cascading two error-correcting codes, called the inner and outer codes. Its error performance is analyzed for a binary symmetric channel with a bit-error rate ϵ<1/2. It is shown that, if the inner and outer codes are chosen properly, high reliability can be attained even for a high-channel bit-error rate. Specific examples with inner codes ranging from high rates to low rates and Reed-Solomon codes as outer codes are considered, and their error probabilities evaluated. They all provide high reliability even for high bit-error rates, say 10^-1-10 ^-2. Several example schemes are being considered for such satellite and spacecraft downlink error control     

Recent Results on the Use of Concatenated Reed-Solomon/Viterbi Channel Coding and Data Compression for Space Communications

The development of sophisticated adaptive source coding algorithms together with inherent error sensitivity problems fostered the need for efficient space communication at very low bit error probilbilities(leq 10^{-6}). This led to the specification and implementation of a concatenated coding system using an interleaved Reed-Solomon code as the outer code and a Viterbi-decoded convolutionai code as the inner code. This paper presents the experimental results of this channel coding system under an emulatedS-band uplink andX-band downlink two-way space communication channel, where both uplink and downlink have strong carrier power. Test results verify that at a bit error probability of 10^-6or less, this concatenated coding system does provide a coding gain of 2.5 dB or more over the viterbi-decoded convolutional-only coding system. These tests also show that a desirable interleaving depth for the Reed-Solomon outer code is 8 or more. The imptict of this "virtually" error-free space communication link on the transmission of images is discussed and examples of Simulation results are given.     

A versatile time-domain Reed-Solomon decoder

A versatile Reed-Solomon (RS) decoder structure based on the time-domain decoding algorithm (transform decoding without transforms) is developed. The algorithm is restructured, and a method is given to decode any RS code generated by any generator polynomial. The main advantage of the decoder structure is its versatility, that is, it can be programmed to decode any Reed-Solomon code defined in Galois field (GF) 2^m with a fixed symbol size m. This decoder can correct errors and erasures for any RS code, including shortened and singly extended codes. It is shown that the decoder has a very simple structure and can be used to design high-speed single-chip VLSI decoders. As an example, a gate-array-based programmable RS decoder is implemented on a single chip. This decoder chip can decode any RS code defined in GF (2⁵) with any code word length and any number of information symbols. The decoder chip is fabricated using low-power 1.5-μ, two-layer-metal, HCMOS technology     

Protocol configuration and verification of an adaptive errorcontrol scheme over analog cellular networks

A high-speed and error-free voiceband data communication method using a hybrid automatic repeat request (ARQ) protocol over an analog cellular system is described. The present method adopts an adaptive error control scheme. This error control scheme automatically selects the optimal error correction code according to circuit bit error rate (BER), so as to match it to the frequently changing mobile radio propagation path conditions. This method adopts multiframe rejection as a retransmission scheme for a high throughput efficiency on the burst error circuit. Actual field evaluation was made by mounting this protocol on a CCITT V.22 bis modem with a data transmission speed of 2400 b/s and a modulation method using 16 carrier states over the Advanced Mobile Phone Service (AMPS) in Atlanta, GA, verifying that data communications can be achieved with an average throughput efficiency of 70% over a radio channel having a BER up to 10^-2     

High rate concatenated coding systems using bandwidth efficienttrellis inner codes

High-rate concatenated coding systems with bandwidth-efficient trellis inner codes and Reed-Solomon (RS) outer codes are investigated for application in high-speed satellite communication systems. Two concatenated coding schemes are proposed. In one the inner code is decoded with soft-decision Viterbi decoding, and the outer RS code performs error-correction-only decoding (decoding without side information). In the other the inner code is decoded with a modified Viterbi algorithm, which produces reliability information along with the decoded output. In this algorithm, path metrics are used to estimate the entire information sequence, whereas branch metrics are used to provide reliability information on the decoded sequence. This information is used to erase unreliable bits in the decoded output. An errors-and-erasures RS decoder is then used for the outer code. The two schemes have been proposed for high-speed data communication on NASA satellite channels. The rates considered are at least double those used in current NASA systems, and the results indicate that high system reliability can still be achieved     

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.     

Forced sequence sequential decoding: a concatenated coding systemwith iterated sequential inner decoding

We describe a new concatenated decoding scheme based on iterations between an inner sequentially decoded convolutional code of rate R=1/4 and memory M=23, and block interleaved outer Reed-Solomon (RS) codes with nonuniform profile. With this scheme decoding with good performance is possible as low as E_b/N₀=0.6 dB, which is about 1.25 dB below the signal-to-noise ratio (SNR) that marks the cutoff rate for the full system. Accounting for about 0.45 dB due to the outer codes, sequential decoding takes place at about 1.7 dB below the SNR cutoff rate for the convolutional code. This is possible since the iteration process provides the sequential decoders with side information that allows a smaller average load and minimizes the probability of computational overflow. Analytical results for the probability that the first RS word is decoded after C computations are presented. These results are supported by simulation results that are also extended to other parameters     

Interleaved concatenated coding for the turbulent atmosphericdirect detection optical communication channel

The performance of two concatenated coding systems using a K =3, R=1/2 convolutional inner code and a Reed-Solomon (RS) (15, 9) or (15, 7) outer code was measured over a 5.76-km-long atmospheric direct detection optical communication channel. Inner code interleaving of 100 μs combined with outer code interleaving of 240 bits (60 RS symbols) was found to be sufficient to obtain a decoded BER of less than 10^-6 under conditions of moderate channel turbulence and an average of 6-10 detected photons per channel bit     

一种基于位反转置换的系统极化码的缩短方法

极化码作为 3GPP 标准制定中的一种信道编码技术方案,具有良好的纠错性能。为了进一步提高删余极化码的误码率性能,将极化码中基于位反转置换的缩短算法应用到系统极化码,提出一种系统极化码的缩短方法。仿真结果表明,在AWGN信道中,在高码率条件下,建议的系统极化码的缩短方法的误码率性能优于系统极化码的准均匀凿孔方法,也优于极化码的缩短方法。当码率为3/4、误码率为10^-4时,系统极化码的缩短方法比极化码的缩短方法约有0.5 dB的增益,比系统极化码的准均匀凿孔方法约有0.25 dB的增益。     

Comments on “Weighted sum codes for error detection and theircomparison with existing codes”

A.J. McAuley (IEEE/ACM Trans. Networking, vol.2, p.16-22, 1994) proposed a new family of error detection codes called weighted sum codes. In the present paper it is noted, that these codes are equivalent to lengthened Reed Solomon codes, resp. shortened version of lengthened Reed Solomon codes over GF(2^h/2). It is also shown, that it is possible to use these codes for error correction of one error in the code word over GF(2^h/2)     

Iterative Viterbi decoding, trellis shaping, and multilevelstructure for high-rate parity-concatenated TCM

We define and apply a new algorithm called the iterative Viterbi decoding algorithm (IVA) to decode a high-rate parity-concatenated TCM system in which a trellis code is used as the inner code and a simple parity-check code is used as the outer code. With trellis shaping, the IVA can achieve a performance 1.25 dB away from the Shannon limit at a BER of 3×10^-5 with low complexity. By augmenting the system with a binary BCH code, the error floor can be reduced to 10^-9 with very little additional cost     

采用光硬限幅器和RS码的多波长OCDMA性能分析

提出了一种采用多个光硬限幅器和RS信道编码的多波长光码多分址（MW OCDMA）系统,理论分析了其误码率和（BER）和归一化吞吐量性能.结果表明,采用光硬限幅器的MW OCDMA系统BER性能将明显改善,进一步采用RS信道编码后,MW OCDMA系统BER性能将大大改善.另一方面,采用光硬限幅器的MW OCDMA系统的归一化吞吐量明显提高,特别是归一化吞吐量峰值提高了约1倍,但进一步采用RS信道编码后,MW OCDMA系统的归一化吞吐量并没有提高,在系统负载量不是很大的情况下还略有下降.     

On the probability of symbol error in Viterbi decoders

An upper bound is derived on the probability that at least one of a sequence of B consecutive bits at the output of a Viterbi (1979) decoder is in error. Such a bound is useful for the analysis of concatenated coding schemes employing an outer block code over GF(2^B) (typically a Reed-Solomon (RS) code), an inner convolutional code, and a symbol (GF(2^B)) interleaver separating the two codes. The bound demonstrates that in such coding schemes a symbol interleaver is preferable to a bit interleaver. It also suggests a new criterion for good inner convolutional codes     

RS（16,12）缩短码编译码原理及性能分析

在各类数字通信系统以及计算机存储和运算系统经常利用差错控制编码降低误码率,提高通信质量,满足对数据传输通道可靠性的要求。RS码是一种性能优良的前向纠错码,具有同时纠正随机错误和突发错误的能力,它的构造特点决定了其非常适合于纠正突发性错误。文中在阐述RS系统码编译码原理的基础上,提出了RS（16,12）缩短码的编译码方法,利用MATLAB对R S（16,12）缩短码在高斯信道和瑞利信道条件下的纠错能力进行仿真,并分析其纠错性能。     

基于级联交织码的水下LED光通信误码性能分析

为了提升系统误比特率,减小基线漂移以及海水信道的吸收散射等特性对光信号产生的影响,采用了基于水下发光二极管(LED)光通信系统的低密度奇偶校验码(LDPC)-里所(RS)级联交织码方案,在模拟水下LED光通信实验系统的情况下,分析码字方案中RS码、LDPC码以及交织参量对系统误比特率性能的影响,得到了级联交织码方案的优化参量,并进行了实验模拟验证。结果表明,优化后的级联交织码系统与未编码系统、RS码系统、LDPC码系统相比分别可获得3.8dB,2dB,1.2dB的增益,可有效提高系统的误比特率性能。该研究为提高水下无线光通信系统的可靠性提供了参考。     

Reed-Solomon error control coding for Rayleigh fading channels withfeedback

The use of nonbinary block error control codes over Rayleigh fading channels with feedback is examined. It is assumed that the fading is slow with respect to the rate of symbol transmission. Expressions are derived for the probabilities of channel symbol error and erasure, which are in turn used to develop expressions for code symbol error and erasure. Two erasure generation mechanisms are considered, one based on the existence of channel amplitude side information, the other not. This analytical framework is used to evaluate the performance of the Reed-Solomon/hybrid-ARQ protocol (RS/HARQ) over fading channels with feedback. The RS/HARQ system uses erasure decoding in a hybrid-ARQ protocol to provide excellent reliability performance at the expense of a reduction in throughput. The RS/HARQ protocol allows for the variation of the erasure threshold and the effective diameter of the decoding operation     

A New Concatenated Coding SystemUsing Multilevel Inner Codes

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