改进交织的单层极化码高阶编码调制系统

极化码作为一种新型编码方式,被采纳为5G通信中的短码方案。本文将极化码应用到比特交织编码调制（Bit-Interleaved Coded Modulation,BICM）系统,优化交织器的设计,提出了一种新型交织算法。相比于现有的交织算法,新型交织算法的提出是基于比特信道可靠性衡量参数,将高可靠性的比特信道与低可靠性的比特信道交错设计,按照高可靠性信道对低可靠性信道辅助译码的方式,提高极化码的纠错性能。由于新型交织算法只存在于比特信道可靠度参数的简单排序,在复杂度上没有明显增加。仿真结果表明:新型交织算法具有优异的性能,当误码率为10-5,码长为256时,采用新型交织算法的极化码BICM系统与LDPC码的BICM系统相比大约有1.51 dB的增益。      

Multilevel block codes for Rayleigh-fading channels

New multilevel block codes for Rayleigh-fading channels are presented. At high signal-to-noise ratios (SNRs), the proposed block codes can achieve better bit error performance over TCM codes, optimum for fading channels, with comparable decoder complexity and bandwidth efficiency. The code construction is based on variant length binary component block codes. As component codes for the 8-PSK multilevel block construction, the authors propose two modified forms of Reed-Muller codes giving a good trade-off between the decoder complexity and the effective code rates. Code design criteria are derived from the error performance analysis. Multistage decoding shows very slight degradation of bit error performance relative to the maximum likelihood algorithm     

On generalized low-density parity-check codes based on Hammingcomponent codes

In this paper we investigate a generalization of Gallager's (1963) low-density (LD) parity-check codes, where as component codes single error correcting Hamming codes are used instead of single error detecting parity-check codes. It is proved that there exist such generalized low-density (GLD) codes for which the minimum distance is growing linearly with the block length, and a lower bound of the minimum distance is given. We also study iterative decoding of GLD codes for the communication over an additive white Gaussian noise channel. The performance in terms of the bit error rate, obtained by computer simulations, is presented for GLD codes of different lengths     

Pre-Compensation for Rayleigh Fading Channels in Time Division Duplex OFDM Systems

Coherent demodulation results in good detection performance but requires channel estimation. Fading pre-compensation (precoding) at the transmitter can lead to low-complex receiver structures with good performance capabilities, without the need for channel estimation. Time division duplex systems based on orthogonal frequency division multiplex (OFDM) are particularly suited to this because intersymbol interference effects can be neglected, simplifying transmitter adjustments. Methods that involve amplitude and/or phase pre-compensation are compared in terms of resulting bit error rate and increase in peak-to-average power ratio. Dynamic channels degrade the performance as the block lengths get longer. For a certain block length the performance degrades below that of traditional differential decoding. A block length of up to 40 times that used in the Digital European Cordless Telephone system, DECT, is possible using channel estimation and ideal Wiener prediction.     

Coding for Parallel Channels: Gallager Bounds and Applications to Turbo-Like Codes

The transmission of coded communication systems is widely modeled to take place over a set of parallel channels. This model is used for transmission over block-fading channels, rate-compatible puncturing of turbo-like codes, multicarrier signaling, multilevel coding, etc. New upper bounds on the maximum-likelihood (ML) decoding error probability are derived in the parallel-channel setting. We focus on the generalization of the Gallager-type bounds and discuss the connections between some versions of these bounds. The tightness of these bounds for parallel channels is exemplified for structured ensembles of turbo codes, repeat-accumulate (RA) codes, and some of their recent variations (e.g., punctured accumulate-repeat-accumulate codes). The bounds on the decoding error probability of an ML decoder are compared to computer simulations of iterative decoding. The new bounds show a remarkable improvement over the union bound and some other previously reported bounds for independent parallel channels. This improvement is exemplified for relatively short block lengths, and it is pronounced when the block length is increased. In the asymptotic case, where we let the block length tend to infinity, inner bounds on the attainable channel regions of modern coding techniques under ML decoding are obtained, based solely on the asymptotic growth rates of the average distance spectra of these code ensembles.     

On the performance of high-rate TPC/SPC codes and LDPC codes overpartial response channels

This paper evaluates two-dimensional turbo product codes based on single-parity check codes (TPC/SPC) and low-density parity check (LDPC) codes for use in digital magnetic recording systems. It is first shown that the combination of a TPC/SPC code and a precoded partial response (PR) channel results in a good distance spectrum due to the interleaving gain. Then, density evolution is used to compute the thresholds for TPC/SPC codes and LDPC codes over PR channels. Analysis shows that TPC/SPC codes have a performance close to that of LDPC codes for large codeword lengths. Simulation results for practical block lengths show that TPC/SPC codes perform as well as LDPC codes in terms of bit error rate, but possess better burst error statistics which is important in the presence of an outer Reed-Solomon code. Further, the encoding complexity of TPC/SPC codes is only linear in the codeword length and the generator matrix does not have to be stored explicitly. Based on. the results in the paper and these advantages, TPC/SPC codes seem like a viable alternative to LDPC codes     

Hermitian codes for frequency-hop spread-spectrum packet radio networks

Hermitian codes are an attractive alternative to Reed-Solomon codes for use in frequency-hop spread-spectrum packet radio networks. For a given alphabet size, a Hermitian code has a much longer block length than a Reed-Solomon code. This and other considerations suggest that Hermitian codes may be superior for certain applications. Analytical results are developed for the evaluation of the packet error probability for frequency-hop transmissions using Hermitian coding. We find there are several situations for which Hermitian codes provide much lower packet error probabilities than can be obtained with Reed-Solomon codes. In general, as the code rate decreases or the symbol alphabet size increases, the relative performance of Hermitian codes improves with respect to Reed-Solomon codes. Performance evaluations are presented for an additive white Gaussian noise channel and for certain partial-band interference channels, and the packet error probability is evaluated for both errors-only and errors-and-erasures decoding.     

Discrete multiple tone modulation with coset coding for thespectrally shaped channel

A discrete approach to multiple tone modulation is developed for digital communication channels with arbitrary intersymbol interference (ISI) and additive Gaussian noise. Multiple tone modulation is achieved through the concatenation of a finite block length modulator based on discrete Fourier transform (DFT) code vectors, and high gain coset or trellis codes. Symbol blocks from an inverse DFT (IDFT) are cyclically extended to generate ISI-free channel-output symbols that decompose the channel into a group of orthogonal and independent parallel subchannels. Asymptotic performance of this system is derived, and examples of asymptotic and finite block length coding gain performance for several channels are evaluated at different values of bits per sample. This discrete multiple tone technique is linear in both the modulation and the demodulation, and is free from the effects of error propagation that often afflict systems employing bandwidth-optimized decision feedback plus coset codes     

A simple derivation of the coding theorem and some applications

Upper bounds are derived on the probability of error that can be achieved by using block codes on general time-discrete memoryless channels. Both amplitude-discrete and amplitude-continuous channels are treated, both with and without input constraints. The major advantages of the present approach are the simplicity of the derivations and the relative simplicity of the results; on the other hand, the exponential behavior of the bounds with block length is the best known for all transmission rates between0and capacity. The results are applied to a number of special channels, including the binary symmetric channel and the additive Gaussian noise channel.     

Universal decoding for finite-state channels

Universal decoding procedures for finite-state channels are discussed. Although the channel statistics are not known, universal decoding can achieve an error probability with an error exponent that, for large enough block length (or constraint length in case of convolutional codes), is equal to the random-coding error exponent associated with the optimal maximum-likelihood decoding procedure for the given channel. The same approach is applied to sequential decoding, yielding a universal sequential decoding procedure with a cutoff rate and an error exponent that are equal to those achieved by the classical sequential decoding procedure.     

Generalized LDPC codes and generalized stopping sets

A generalized low-density parity check code (GLDPC) is a low-density parity check code in which the constraint nodes of the code graph are block codes, rather than single parity checks. In this paper, we study GLDPC codes which have BCH or Reed-Solomon codes as subcodes under bounded distance decoding (BDD). The performance of the proposed scheme is investigated in the limit case of an infinite length (cycle free) code used over a binary erasure channel (BEC) and the corresponding thresholds for iterative decoding are derived. The performance of the proposed scheme for finite code lengths over a BEC is investigated as well. Structures responsible for decoding failures are defined and a theoretical analysis over the ensemble of GLDPC codes which yields exact bit and block error rates of the ensemble average is derived. Unfortunately this study shows that GLDPC codes do not compare favorably with their LDPC counterpart over the BEC. Fortunately, it is also shown that under certain conditions, objects identified in the analysis of GLDPC codes over a BEC and the corresponding theoretical results remain useful to derive tight lower bounds on the performance of GLDPC codes over a binary symmetric channel (BSC). Simulation results show that the proposed method yields competitive performance with a good decoding complexity trade-off for the BSC.     

A comparison of known codes, random codes, and the best codes

This paper calculates new bounds on the size of the performance gap between random codes and the best possible codes. The first result shows that, for large block sizes, the ratio of the error probability of a random code to the sphere-packing lower bound on the error probability of every code on the binary symmetric channel (BSC) is small for a wide range of useful crossover probabilities. Thus even far from capacity, random codes have nearly the same error performance as the best possible long codes. The paper also demonstrates that a small reduction k-k˜ in the number of information bits conveyed by a codeword will make the error performance of an (n,k˜) random code better than the sphere-packing lower bound for an (n,k) code as long as the channel crossover probability is somewhat greater than a critical probability. For example, the sphere-packing lower bound for a long (n,k), rate 1/2, code will exceed the error probability of an (n,k˜) random code if k-k˜>10 and the crossover probability is between 0.035 and 0.11=H^-1(1/2). Analogous results are presented for the binary erasure channel (BEC) and the additive white Gaussian noise (AWGN) channel. The paper also presents substantial numerical evaluation of the performance of random codes and existing standard lower bounds for the BEC, BSC, and the AWGN channel. These last results provide a useful standard against which to measure many popular codes including turbo codes, e.g., there exist turbo codes that perform within 0.6 dB of the bounds over a wide range of block lengths     

Universal space-time codes from demultiplexed trellis codes

In broadcast scenarios or in the absence of accurate channel probability distribution information, code design for consistent channel-by-channel performance, rather than average performance over a channel distribution, may be desirable. Root and Varaiya's compound channel theorem for linear Gaussian channels promises the existence of universal codes that operate reliably whenever the channel mutual information (MI) is above the transmitted rate. This paper presents two-dimensional trellis codes that provide such universal performance over the compound linear vector Gaussian channel when demultiplexed over two, three, and four transmit antennas. The presented trellis codes, found by exhaustive search, guarantee consistent performance on every matrix channel that supports the information transmission rate with an MI gap that is similar to the capacity gap of a well-designed additive white Gaussian noise (AWGN)-specific code on the AWGN channel. As a result of their channel-by-channel consistency, the universal trellis codes presented here also deliver comparable, or, in some cases, superior, frame-error rate and bit-error rate performance under quasi-static Rayleigh fading, as compared with trellis codes of similar complexity that are designed specifically for the quasi-static Rayleigh-fading scenario.     

DC-free coset codes

A class of block coset codes with disparity and run-length constraints are studied. They are particularly well suited for high-speed optical fiber links and similar channels, where DC-free pulse formats, channel error control, and low-complexity encoder-decoder implementations are required. The codes are derived by partitioning linear block codes. The encoder and decoder structures are the same as those of linear block codes with only slight modifications. A special class of DC-free coset block codes are derived from BCH codes with specified bounds on minimum distance, disparity, and run length. The codes have low disparity levels (a small running digital sum) and good error-correcting capabilities     

The performance of incremental redundancy schemes based on convolutional codes in the block-fading Gaussian collision channel

The throughput performance of incremental redundancy (INR) schemes, based on short constraint length convolutional codes, is evaluated for the block-fading Gaussian collision channel. Results based on simulations and union bound computations are compared to estimates of the achievable throughput performance with random binary and Gaussian coding in the limit of large block lengths, obtained through information outage considerations. For low channel loads, it is observed that INR schemes with binary convolutional codes and limited block length may provide throughput close to the achievable performance for binary random coding. However, for these low loads, compared to binary random coding, Gaussian random coding may provide significantly better throughput performance, which prompts the use of larger modulation constellations. For high channel loads, a relatively large gap in throughput performance between binary convolutional codes and binary random codes indicates a potential for extensive performance improvement by alternative coding strategies. Only small improvements of the throughput have been observed by increasing the complexity through increased state convolutional coding.     

Complexity reduction in subspace-based blind channel identificationfor DS/CDMA systems

Direct-sequence code-division multiple access is emerging as a potential multiple-access communication scheme for future digital wireless communications systems. Such wide-band systems usually operate in a frequency-selective fading channel that introduces intersymbol interference and thus potential performance degradation. Previously proposed subspace-based blind channel identification algorithms, which provide estimates of channel parameters for effective equalization, suffer from high numerical complexity for systems with large spreading gains. In this paper, it is shown that, through the use of matched filter outputs, reduction in numerical complexity can be obtained. The complexity reduction is considerable when the channel length is small and the system is moderately loaded. The results show that the new algorithm suffers a slight performance loss. Although the employed matched filter outputs do not form a set of sufficient statistics for the unknown channels, the difference between the matched filter outputs and the sufficient statistics becomes negligible for large observation lengths and the asymptotic normalized Fisher information does not change. Performance is evaluated through simulations, the derivation of a tight approximation of the mean-squared channel estimation error, and through comparisons to the Cramer-Rao bound for the estimation error variance. It is shown that the approximation of the mean-squared error can be obtained in terms of the correlation of the spreading codes and the channels. This representation of the error supplies a tool for investigating the relationship between performance and spreading sequence correlations     

Optimum cycle redundancy-check codes with 16-bit redundancy

Optimal binary cyclic redundancy-check codes with 16 parity bits (CRC-16 codes) are presented and compared to those in existing standards for minimum-distance, undetected-error probability on binary symmetric channels (BSCs) and properness. The codes in several cases are seen to be superior at block lengths of practical interest when they are used on low-noise BSCs. The optimum minimum distance obtainable by some CRC-16 codes is determined for all block lengths. For several typical low-noise BSCs the minimum undetected error probability achievable with some CRC-16 codes is given for all block lengths     

The art of signaling: fifty years of coding theory

In 1948 Shannon developed fundamental limits on the efficiency of communication over noisy channels. The coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero. Fifty years later, codes for the Gaussian channel have been discovered that come close to these fundamental limits. There is now a substantial algebraic theory of error-correcting codes with as many connections to mathematics as to engineering practice, and the last 20 years have seen the construction of algebraic-geometry codes that can be encoded and decoded in polynomial time, and that beat the Gilbert-Varshamov bound. Given the size of coding theory as a subject, this review is of necessity a personal perspective, and the focus is reliable communication, and not source coding or cryptography. The emphasis is on connecting coding theories for Hamming and Euclidean space and on future challenges, specifically in data networking, wireless communication, and quantum information theory