纠正随机错误与长突发删除的多进制乘积码研究

考虑多进制LDPC码的符号特性，以及对其残留错误和删除的分析，本文采用多进制LDPC码作为内码，相同Galois域下的高码率RS码作为外码来构造多进制乘积码；并提出了一种低复杂度的迭代译码方案，减少信息传输的各类错误。在译码时，只对前一次迭代中译码失败的码字执行译码，并对译码正确码字所对应的比特初始概率信息进行修正，增强下一次迭代多进制LDPC译码符号先验信息的准确性，减少内码译码后的判决错误，从而充分利用外码的纠错能力。仿真结果显示，多进制乘积码相较于二进制LDPC乘积码有较大的编码增益，并通过迭代进一步改善了性能，高效纠正了信道中的随机错误和突发删除。对于包含2%突发删除的高斯信道，在误比特率为10-6时，迭代一次有0.4 dB左右的增益。      

Hardware Implementation for Bipartite Belief Propagation Polar Decoding with Bit Flipping

For the scenarios with high throughput requirements, the belief propagation (BP) decoding is one of the most promising decoding strategies for polar codes. By pruning the redundant variable nodes (VNs) and check nodes (CNs) in the original factor graph, the graph is condensed to a sparse bipartite graph that is similar to the graph for low-density parity-check (LDPC) codes. In this paper, we introduce the bit flipping scheme into the LDPC-like BP (L-BP) decoding and propose two methods to identify the error-prone VNs. By additional decoding attempts, the L-BP flip (L-BPF) decoding improves the error-correction performance with a similar average complexity for high E_b/N₀ values. The simulation results show that the L-BPF decoding achieves 0.25 dB gain compared with the L-BP decoding. Finally, a parallel decoder with the proposed L-BPF algorithm for an (256,128) polar code is implemented using 65nm CMOS technology, and it delivers a throughput of 1877.3 Mbps.

     

A new series of concatenated codes subject to matrix type-Bcodes

Some important aspects of a series of concatenated codes subjected to matrix type-Bcodes are investigated. Concatenated matrix codes and the concatenated quadratic residue codes especially are emphasized. An analysis of the error patterns, which can be corrected with the matrix coding, also is given. These codes are suitable for compound channels with memory (i.e., channels on which burst, cluster, and random errors occur). Explicit formulas are given for the number of bursts, cluster, and random errors that can be corrected with these codes. Decoding schemes and techniques for studying error propagation in the proposed codes are given. In particular a new decoding algorithm for a concatenated matrix code is given. The performance of coding and decoding schemes of the various types of concatenated codes can be tested in practice.     

Turbo codes-based image transmission for channels with multiple types of distortion

Product codes are generally used for progressive image transmission when random errors and packet loss (or burst errors) co-exist. However, the optimal rate allocation considering both component codes gives rise to high-optimization complexity. In addition, the decoding performance may be degraded quickly when the channel varies beyond the design point. In this paper, we propose a new unequal error protection (UEP) scheme for progressive image transmission by using rate-compatible punctured Turbo codes (RCPT) and cyclic redundancy check (CRC) codes only. By sophisticatedly interleaving each coded frame, the packet loss can be converted into randomly punctured bits in a Turbo code. Therefore, error control in noisy channels with different types of errors is equivalent to dealing with random bit errors only, with reduced turbo code rates. A genetic algorithm-based method is presented to further reduce the optimization complexity. This proposed method not only gives a better performance than product codes in given channel conditions but is also more robust to the channel variation. Finally, to break down the error floor of turbo decoding, we further extend the above RCPT/CRC protection to a product code scheme by adding a Reed-Solomon (RS) code across the frames. The associated rate allocation is discussed and further improvement is demonstrated.     

Carrier phase recovery for turbo‐coded systems with pre‐coded GMSK modulation

A carrier phase recovery scheme suited for turbo‐coded systems with pre‐coded Gaussian minimum shift keying (GMSK) modulation is proposed and evaluated in terms of bit‐error‐rate (BER) performance. This scheme involves utilizing the extrinsic information obtained from the turbo‐decoder to aid an iterative carrier phase estimation process, based on a maximum‐likelihood (ML) strategy. The phase estimator works jointly with the turbo‐decoder, using the updated extrinsic information from the turbo‐decoder in every iterative decoding. A pre‐coder is used to remove the inherent differential encoding of the GMSK modulation. Two bandwidths of GMSK signals are considered: BT=0.5 and 0.25, which are recommended by the European Cooperation for Space Standardization (ECSS). It is shown that the performance of this technique is quite close to the perfect synchronized system within a wide range of phase errors. This technique is further developed to recover nearly any phase error in [?π,+π] by increasing the number of phase estimators and joint decoding units. This, however, will increase the complexity of the system. Copyright © 2008 John Wiley & Sons, Ltd.     

Image/video communications: joint source/channel coding

Transmission of image/video messages over communication networks is becoming a standard way of communication due to very efficient compression algorithms that reduce required channel capacity to an acceptable level. However, all compression standard techniques are strongly sensitivitive to channel disturbances and their application is suitable only for practically noiseless channels. In standard noisy channels, the effect of errors on a compressed data bit stream can be divided into two categories: systematic errors defined by the structure of data blocks, and random errors caused by amplitude changes of transmitted components. A systematic error can be detected at the receiver through control of the data stream structure and corrected by error concealment methods or by automatic repeat request (ARQ) procedures. Random errors, noise and burst‐like errors, as well as impulse noise, should be controlled through channel coding. It is reasonable that an integrated source and channel coding methods should be preferred and should give better coding performance. In this paper, a new framework for an image/video coding approach has been presented in which the source and channel coding is integrated in a unique procedure. Image compression is performed in a standard way of the JPEG algorithm based on discrete cosine transform (DCT) and error control coding is based on the real/complex‐number (N,M) BCH code using discrete Fourier transform (DFT) specified with zeros in the time domain, i.e. with roots in the frequency domain. Efficiency of the proposed method is tested on two examples, an one‐dimensional real‐valued time sequence coded by real‐number (20,16) BCH code using DFT, and an example of an image coded by complex (10,8) BCH code using DFT with the correction ability of up to 8 impulses per transmitted 8×8 block. In addition, two decoding methods based on Berlekamp–Massey algorithm (BMA) and the minimum‐norm algorithm (MNA) have also compared. Copyright © 1999 John Wiley & Sons, Ltd.     

An improved AS-SCLF decoding algorithm of polar codes based on the assigned set

An improved successive cancellation list bit-flip based on assigned set (AS-SCLF) decoding algorithm is proposed to solve the problems that the successive decoding of the successive cancellation (SC) decoder has error propagation and the path extension of the successive cancellation list (SCL) decoder has the decision errors in the traditional cyclic redundancy check aided successive cancellation list (CA-SCL) decoding algorithm. The proposed algorithm constructs the AS firstly. The construction criterion is to use the Gaussian approximation principle to estimate the reliabilities of the polar subchannel and the error probabilities of the bits under SC decoding, and the normalized beliefs of the bits in actual decoding are obtained through the path metric under CA-SCL decoding, thus the error bits containing the SC state are identified and sorted in ascending order of the reliability. Then the SCLF decoding is performed. When the CA-SCL decoding fails for the first time, the decision results on the path of the SC state in the AS are exchanged. The simulation results show that compared with the CA-SCL decoding algorithm, the SCLF decoding algorithm based on the critical set and the decision post-processing decoding algorithm, the improved AS-SCLF decoding algorithm can improve the gain of about 0.29 dB, 0.22 dB and 0.1 dB respectively at the block error rate (BLER) of 10-4 and reduce the number of decoding at the low signal-to-noise ratio (SNR), thus the computational complexity is also reduced.     

Algebraic Soft-Decision Decoding of Reed–Solomon Codes Using Bit-Level Soft Information

The performance of algebraic soft-decision decoding of Reed–Solomon codes using bit-level soft information is investigated. Optimal multiplicity assignment strategies for algebraic soft-decision decoding (SDD) with infinite cost are first studied over erasure channels and the binary-symmetric channel. The corresponding decoding radii are calculated in closed forms and tight bounds on the error probability are derived. The multiplicity assignment strategy and the corresponding performance analysis are then generalized to characterize the decoding region of algebraic SDD over a mixed error and bit-level erasure channel. The bit-level decoding region of the proposed multiplicity assignment strategy is shown to be significantly larger than that of conventional Berlekamp–Massey decoding. As an application, a bit-level generalized minimum distance decoding algorithm is proposed. The proposed decoding compares favorably with many other Reed–Solomon SDD algorithms over various channels. Moreover, owing to the simplicity of the proposed bit-level generalized minimum distance decoding, its performance can be tightly bounded using order statistics.      

The Influence of Short Error Clusters on CCS No 7 Link Availability

We observe the Common Channel Signaling (CCS) No 7 link availability under short error bursts or clusters with high bit error rate (BER), and compare it to the availability of CCS No 7 link under equivalent random errors. It is shown that the mean time to changeover of the CCS No 7 link in the case of error clusters is greater than in the case of random errors of same intensity. The reason for this positive impact of short clusters with high BER is the concentration of more than one errored bits in one errored signal unit. In the same time the mean alignment time of CCS No 7 link (that is out of service) is shortest if errors are grouped in rare short error clusters. Hence, the availability of CCS No 7 link increases and frequency of CCS No 7 link oscillations decreases, if errors are grouped.     

Error-propagation properties of uniform codes

The problem of error propagation in uniform codes is investigated using the concept of parity-parallelogram submatrices and the threshold-decoding algorithm. A set of optimum orthogonalization rules is presented and it is shown that if these rules are incorporated into the decoder, then sufficient conditions can be found for the return of the decoder to correct operation following a decoding error. These conditions are considerably less stringent than the requirement that the channel be completely free of errors following a decoding error. However, this is not the case if the prescribed orthogonalization rules are not followed, as is demonstrated with a simple example. It is also shown that the syndrome memory required with Massey's orthogonalization procedure for definite decoding of uniform codes is the lowest possible. The results of simulation of the ratefrac{1}{4}andfrac{1}{8}uniform codes are presented, and these codes are seen to make fewer decoding errors with feedback decoding than with definite decoding. Comparison of the performance of an ordinary feedback decoder with a genie-aided feedback decoder, which never propagates errors, indicates that error propagation with uniform codes is a minor problem if the optimum orthogonalization rules are used, but that the situation is somewhat worse with nonoptimum orthogonalization.     

基于Logistic混沌序列的错误图样对纠错码的影响

为了在“比特级”层面对纠错码进行更深入的研究,提出了一种服从Logistic混沌序列的错误图样分布模型。通过将混沌序列映射成二进制序列,并采用一定的方法使序列中“1”的数目可控,以此来生成错误图样。将卷积码、RS码以及RS+卷积级联码作为研究对象,以随机错误为参照,分别研究了该错误图样下三者的译码性能。仿真结果表明,基于Logistic混沌序列的错误图样对卷积码和级联码的影响要大于随机错误图样,而随机错误图样对RS码的影响更大。     

LP Decoding Corrects a Constant Fraction of Errors

We show that for low-density parity-check (LDPC) codes whose Tanner graphs have sufficient expansion, the linear programming (LP) decoder of Feldman, Karger, and Wainwright can correct a constant fraction of errors. A random graph will have sufficient expansion with high probability, and recent work shows that such graphs can be constructed efficiently. A key element of our method is the use of a dual witness: a zero-valued dual solution to the decoding linear program whose existence proves decoding success. We show that as long as no more than a certain constant fraction of the bits are flipped by the channel, we can find a dual witness. This new method can be used for proving bounds on the performance of any LP decoder, even in a probabilistic setting. Our result implies that the word error rate of the LP decoder decreases exponentially in the code length under the binary-symmetric channel (BSC). This is the first such error bound for LDPC codes using an analysis based on "pseudocodewords." Recent work by Koetter and Vontobel shows that LP decoding and min-sum decoding of LDPC codes are closely related by the "graph cover" structure of their pseudocodewords; in their terminology, our result implies that that there exist families of LDPC codes where the minimum BSC pseudoweight grows linearly in the block length     

A new concatenated matrix code with cluster,burst and random error correcting abilities

A class of linear matrix codes for compound channels with memory (i.e. channels on which burst, cluster and random errors occur) ia given. Explicit formulas are given for the number of burst, cluster and random errors that can be corrected with tlicso codes. Decoding nchcmes and analysis techniques to study error propagation in the proposed codoa are given. In particular, a new decoding algorithm for a concatenated matrix codo is given. The algorithm utilizes the decoding algorithm for the corresponding concatenated matrix code. A coding–decoding procedure is first, followed. The codes can bo used to transmit information from fixed–rate sources through fixed–rate compound channels. Tho channel probabilities are related to channel error propagation.     

LDPC Decoding by Failed Check Nodes for Serial Concatenated Code

The use of serial concatenated codes is an effective technique for alleviating the error floor phenomenon of low‐density parity‐check (LDPC) codes. An enhanced sum–product algorithm (SPA) for LDPC codes, which is suitable for serial concatenated codes, is proposed in this paper. The proposed algorithm minimizes the number of errors by using the failed check nodes (FCNs) in LDPC decoding. Hence, the error‐correcting capability of the serial concatenated code can be improved. The number of FCNs is simply obtained by the syndrome test, which is performed during the SPA. Hence, the decoding procedure of the proposed algorithm is similar to that of the conventional algorithm. The error performance of the proposed algorithm is analyzed and compared with that of the conventional algorithm. As a result, a gain of 1.4 dB can be obtained by the proposed algorithm at a bit error rate of 10^?8. In addition, the error performance of the proposed algorithm with just 30 iterations is shown to be superior to that of the conventional algorithm with 100 iterations.     

Exploit Network Coding Over GF(2 q ) for Multi-user Cooperative Wireless Networks

In this paper, we present a decode-and-forward network coded (DFNC) scheme over GF(2 ^q ) for the multi-user cooperative communication systems. In particular, we consider a cooperative network with m users transmitting independent packets to the same destination. These users form a cooperation set to help each other by using linear network coding. We propose a coding coefficients construction method which can efficiently reduce the transmission overhead from m(q + log₂ m) to m bits compared with conventional random network coding. Furthermore, we propose a novel decoding algorithm—credit-based updating algorithm in order to improve the solvability of decoding set of equations at the destination. The proposed decoding algorithm is combined with channel decoding and is applied on symbol-level. It can fully make use of the error recovery property of network coding while conventional decoding algorithms (e.g., Gaussian elimination) overlook it. We theoretically analyze the diversity performance in terms of information outage probability, and the results show that diversity order of m + 1 can be achieved for a m-user cooperation system. Moreover, we conduct extensive simulations to show that DFNC outperforms other transmission schemes in terms of symbol error rate and achieves higher diversity order. We also demonstrate that the proposed decoding algorithm provides significant performance gain over conventional decoding algorithm.     

A folding technique for reducing circuit complexity of flash ADC decoders

The performance of a decoder is one of the factors that dominate the performance of a flash ADC. In this paper a folding technique is proposed to reduce the decoder circuit complexity. After folding, a k-bit decoder is replaced with two sub-decoders. The decoding of the upper k/2 bits and the lower k/2 bits can be accomplished respectively. Consequently, the number of inputs to the decoder is reduced to the square root of the original. Analytic results show that for different decoder structures, more than 17% of hardware and 13% of time delay can be saved. Moreover, the tolerance of bubble induced errors is enhanced. A 6-bit flash ADC has been implemented in 0.18-μm CMOS that occupies 0.37 mm × 0.35 mm active area. Simulations show that the figure-of-merit number is as low as 1.03 pJ/convsetp at 1G Sample/s and the maximum bubble induced error is limited to the number of bubbles.     

Performance of CO-OFDM system with RS-Turbo concatenated code

In this paper, the RS-Turbo concatenated code is applied to coherent optical orthogonal frequency division multiplexing (CO-OFDM) system. RS(186,166,8) and Turbo code with code rate of 1/2 are employed for RS-Turbo concatenated code. Two decoding algorithms, which are Max-Log-MAP algorithm and Log-MAP algorithm, are adopted for Turbo decoding, and the iteration Berlekamp-Massey (BM) algorithm is adopted for RS decoding. The simulation results show that the bit error rate (BER) performance of CO-OFDM system with RS-Turbo concatenated code is significantly improved at high optical signal to noise ratio (OSNR), and the iteration number is reduced compared with that of the Turbo coded system. Furthermore, when the Max-Log-MAP algorithm is adopted for Turbo decoding, the transmission distance of CO-OFDM system with RS-Turbo concatenated code can reach about 400 km without error, while that of the Turbo coded system can only reach about 240 km when BER is lower than 10-4 order of magnitude.     

Estimation of the error performance objectives for satellite communication systems

We consider the physical layer error performance parameters and design criteria for digital satellite systems established by ITU‐R Recommendation S.1062, where the performance objectives are given in terms of the bit error rate (BER) divided by the average number of errors within a cluster. It is well known that errors on satellite links employing forward error correction (FEC) schemes tend to occur in clusters. The resulting block error rate is the same as if it was caused by randomly occurring bit errors with an error‐event ratio of BER/α, where α is the average number of errors within a cluster. The factor, α, accounts for the burstiness of the errors and also represents the ratio between the BER and the error‐event ratio. This paper proposes theoretical methods to estimate the factor, α. Using the weight distributions of the FEC codes, we derive a set of expressions for the factor, α, as well as their compact lower bounds. We present lower bounds for various FEC schemes including binary BCH codes, block turbo codes, convolutional codes, and turbo codes. The simulation results show that the proposed lower bounds are good estimates in the high signal‐to‐noise ratio region. Copyright © 2008 John Wiley & Sons, Ltd.     

Burst-trapping techniques for a compound channel

An adaptive decoding technique called burst trapping is presented to correct both random and burst errors. Two decoding algorithms are used, one for random errors, and the other for bursts. The former is based on a conventional correction technique, the latter utilizes an encoding procedure for which each information digit appears twice in the data stream, first unchanged, and second combined with (addition modulo2) a check digit of a widely separated later block. Whenever the number of errors within a code block are detected to be too large to correct with the random-error-correcting algorithm, the burst-correcting algorithm corrects these errors by recovering the information from later blocks where it appears in combination with check digits. It is shown that the scheme requires very limited guard space and has limited error propagation. Furthermore, the storage requirement is even smaller than the guard space. This is the only known coding system that has this desirable feature. Results of simulation of such codes over telephone channels indicate that the performance of such codes, when compared with interleaved block codes, offers better results at significantly lower cost.     

Partitioned scheduling for sphere decoding with runtime constraints for practical MIMO communication systems

This paper presents an effective scheduling scheme for sphere decoding (SD) with runtime constraints, targeting the practical multiple‐input multiple‐output (MIMO) communication systems where neither the interleaving scheme nor its block size cannot be designed freely. The proposed scheme imposes runtime constraints on SD to distribute the errors due to the early termination of SD. Because the distributed errors may be corrected effectively by forward error correction, the error‐rate performance can be improved; experimental results show that the performance improvement is approximately 2dB in terms of the signal‐to‐noise ratio to achieve a bit‐error rate of 10^?4 in 4 × 4 16‐QAM MIMO systems. Copyright © 2015 John Wiley & Sons, Ltd.