Waterfall Performance Research Articles

Block Markov superposition transmission of LDPC codes

In this paper, based on the block Markov superposition transmission (BMST) technique, we present a new class of coupled low-density parity-check (LDPC) codes for the transport block (TB)-based transmission to improve the error-correcting performance. For encoding, the previous LDPC codewords corresponding to a TB (at prior time slot) are interleaved and superimposed onto the current LDPC codewords, resulting in the transmitted codewords. For decoding, the sliding window decoding algorithm with sum-product or min-sum implementations can be employed, inheriting a relatively low-latency decoding. A distinguished advantage of the proposed coded transmission over spatially coupled LDPC (SC-LDPC) codes is that the encoder/decoder of the proposed codes can be designed by reusing the encoder/decoder architecture of component block LDPC codes. To analyze the waterfall performance of BMST-LDPC code ensembles, we present the protograph-based EXIT chart analysis, which can efficiently predict the error-correcting performance in waterfall region. To analyze the error-floor performance of BMST-LDPC codes, we employ the genie-aided (GA) lower bound, which can efficiently predict the error-correcting performance in error-floor region. For ease of implementation, the BMST-LDPC codes are constructed by taking the (2, 4)-raptor-like LDPC codes or the 5G LDPC codes as the basic components. The numerical results reveal that the proposed codes can have capacity-approaching performance, exhibiting a gap of 0.007 ​dB away from the corresponding Shannon limit. They also reveal that, by using the proposed BMST construction, the error-correcting performance of the original 5G block LDPC codes can be significantly improved, achieving coding gains up to one dB over the AWGN channels and two dB over the fast fading channels.

Design and Analysis of Joint Source-Channel Code System with Fixed-Length Code

As the demands of multimedia and data services increase, efficient communication systems are being investigated to meet the high data rate requirements. Joint source-channel coding (JSCC) schemes were proposed for improving overall system performance. However, existing JSCC systems may suffer a symbol error rate (SER) performance loss when residual source redundancy is not fully exploited. This paper presents a novel, low-complexity JSCC system, which consists of a fixed-length source block code and an irregular convolutional channel code. A simple approach is proposed to design source codes that minimize the SER of source detection and guarantee the convergence of iterative source-channel decoding (ISCD). To improve the waterfall performance of ISCD, the channel code is optimized by using the extrinsic information transfer (EXIT) chart and the concept of irregular code. The channel code is constituted by recursive non-systematic convolutional (RNSC) subcodes. The weights of subcodes are optimized to make the EXIT curves of the channel decoder and the source decoder well-matched, and therefore, a near-capacity performance is achieved. Simulation results show that the proposed system achieves more than 1 dB gains and 0.3 dB gains compared to the separate source-channel code system and the other optimal JSCC systems, respectively. Additionally, the performance of the proposed system is within 1 dB deviation from the Shannon limit capacity.

Design of Code Pair for Protograph-LDPC Codes-Based JSCC System With Joint Shuffled Scheduling Decoding Algorithm

Although there are many studies on code optimization of the joint source-channel coding (JSCC) system based on double protograph low-density parity-check codes with the joint belief propagation (JBP) algorithm, but it is still unknown whether the source code and channel code (as a code pair) can perform well when the joint shuffled scheduling decoding (JSSD) algorithm is adopted. In this letter, two decoding threshold analysis algorithms, including joint shuffled protograph extrinsic information transfer (PEXIT) and source shuffled PEXIT algorithm, are proposed to calculate joint/source decoding thresholds for this system with the JSSD algorithm. With the proposed algorithms, it is found that the optimized code pairs for this system with the JBP algorithm may not perform well with the JSSD algorithm, implying that code pair with the JSSD algorithm needs to be redesigned. Then, a two-stage optimized framework is proposed to design the code pair for this system with the JSSD algorithm. Simulations and decoding threshold analysis both show that the proposed code pair for this system with the JSSD algorithm can obtain lower error floor and better waterfall performance than the existing code pairs.

Systematic Convolutional Low Density Generator Matrix Code

In this paper, we propose a systematic low density generator matrix (LDGM) code ensemble, which is defined by the Bernoulli process. We prove that, under maximum likelihood (ML) decoding, the proposed ensemble can achieve the capacity of binary-input output symmetric (BIOS) memoryless channels in terms of bit error rate (BER). The proof technique reveals a new mechanism, different from lowering down frame error rate (FER), that the BER can be lowered down by assigning light codeword vectors to light information vectors. The finite length performance is analyzed by deriving an upper bound and a lower bound, both of which are shown to be tight in the high signal-to-noise ratio (SNR) region. To improve the waterfall performance, we construct the systematic convolutional LDGM (SysConv-LDGM) codes by a random splitting process. The SysConv-LDGM codes are easily configurable in the sense that any rational code rate can be realized without complex optimization. As a universal construction, the main advantage of the SysConv-LDGM codes is their near-capacity performance in the waterfall region and predictable performance in the error-floor region that can be lowered down to any target as required by increasing the density of the uncoupled LDGM codes. Numerical results are also provided to verify our analysis.

Construction of Irregular Protograph-Based QC-LDPC Codes With Low Error Floor

In this article, we design finite-length irregular protograph-based quasi-cyclic (QC) low-density parity-check (LDPC) codes with good waterfall performance and low error floor. To achieve a low error floor, we eliminate a targeted set of dominant elementary trapping sets (ETS) ${\mathcal{ L}}$ in the Tanner graph of the code. For a given rate and girth, the codes are designed to be free of the largest set of problematic ETSs for a given block length, or to have the shortest block length while a given set of ETSs is avoided. The design is based on a search algorithm that identifies whether any instance of any structure within ${\mathcal{ L}}$ exists in the Tanner graph of the constructed code or not. The search algorithm performs this task with minimal complexity, making it feasible to construct practical codes by running the search algorithm a large number of times. Simulation results are provided to demonstrate the superior performance of designed codes compared to similar state-of-the-art irregular QC-LDPC codes.

Performance Analysis of Finite-Length LDPC Codes Over Asymmetric Memoryless Channels

In this paper, we analyze the waterfall performance of finite-length low-density parity-check (LDPC) codes for asymmetric memoryless channels by considering variations of channel parameters: the observed bit error rate $P_{\text{obs}}$ and mutual information $I_{\text{obs}}$ during the real-time transmission. Moreover, we propose to use the variation of a new channel parameter: the maximal achievable channel coding rate $R_{\text{obs}}$ at finite blocklengths to estimate the waterfall performance. Simulation results demonstrate that our estimations can well predict the waterfall performance of LDPC codes. In particular, the estimation based on $R_{\text{obs}}$ can fit the waterfall performance for various code lengths. For estimations based on $P_{\text{obs}}$ and $I_{\text{obs}}$ , their accuracy increases as the codeword length increases and no curve fitting is required.

Adaptive Rate-Compatible Non-Binary LDPC Coding Scheme for the B5G Mobile System.

This paper studies an adaptive coding scheme for B5G (beyond 5th generation) mobile system-enhanced transmission technology. Different from the existing works, the authors develop a class of rate-compatible, non-binary, low-density parity check (RC-NB-LDPC) codes, which expresses the strong connection between the algebra-based and graph-theoretic-based constructions. The constructed codes can not only express rate-compatible (RC) features, but also possess a quasi-cyclic (QC) structure that facilitates the encoding implementation. Further, in order to achieve the code rate-adaptive allocation scheme, the authors propose using the K-means++ clustering algorithm to cluster different channel environments, considering various factors that affect channel characteristics. Finally, in order to present the advantages of the adaptive coding scheme, the authors construct a coding scheme for image transmission. The numerical results demonstrate that the developed code can obtain better waterfall performance in a larger code rate range, which is more suitable for data transmission; the adaptive coding transmission scheme can obtain higher reconstructed image quality compared to the fixed code rate-coding scheme. Moreover, when considering unequal error protection (UEP), the proposed scheme can further improve the reconstructed image quality.

Block Markov Superposition Transmission of BCH Codes With Iterative Erasures-and-Errors Decoders

In this paper, we present the block Markov superposition transmission of BCH (BMST-BCH) codes, which can be constructed to obtain a very low error floor. To reduce the implementation complexity, we design a low complexity iterative sliding-window decoding algorithm, in which only binary and/or erasure messages are processed and exchanged between processing units. The error floor can be predicted by the proposed genie-aided lower bounds, while the waterfall performance can be analyzed by the density evolution method. To evaluate the error floor of the constructed BMST-BCH codes at a very low bit error rate (BER) region, we propose a fast simulation approach. Numerical results show that, at a target BER of $10^{-15}$ , the proposed BMST-BCH code with hard-decision can achieve a net coding gain (NCG) of 10.55 dB with 25% overhead, while a soft-decision design can yield an NCG of 10.74 dB. The construction of BMST-BCH codes is flexible to trade off latency against performance at all overheads of interest and may find applications in optical transport networks as an attractive candidate.

Design of Linking Matrix in JSCC Scheme Based on Double Protograph LDPC Codes

A joint source channel coding (JSCC) scheme based on source and channel low density parity check (LDPC) codes, which is named as the D-LDPC system, has attracted a lot of attention recently. However, it suffers from a high error-floor. By introducing a linking matrix connecting variable nodes of source LDPC and check nodes of channel LDPC, the error-floor can be lowered effectively. However, the water-fall performance was lost with the increasing of the size of linking matrix. In this paper, a detailed analysis about the impact of this linking matrix on the water-fall region is conducted. Some design principles for the linking matrix are proposed, by which several linking matrices for different source statistics will be designed to improve the water-fall performance from the perspective of both the source and channel LDPC codes. The extrinsic information transfer (EXIT) analysis and numerical simulations will be compared to verify the superiority of the proposed linking matrices.

Performance Analysis of LDPC-Coded Diversity Combining on Rayleigh Fading Channels With Impulsive Noise

Spatial diversity is an effective method to mitigate the effects of fading, and when used in conjunction with low-density parity-check (LDPC) codes, it can achieve excellent error-correcting performance. Noise added at each branch of the diversity combiner is generally assumed to be additive white Gaussian noise, but there are many applications where the received signal is impaired by noise with a non-Gaussian distribution. In this paper, we derive the exact bit-error probability of different linear combining techniques on Rayleigh fading channels with impulsive noise, which is modeled using symmetric alpha-stable distributions. The relationship for the signal-to-noise ratios of these linear combiners is derived and then different non-linear detectors are presented. A detector based on the bi-parameter Cauchy–Gaussian mixture model is used and shows near-optimal performance with a significant reduction in complexity when compared with the optimal detector. Furthermore, the threshold signal-to-noise ratio of LDPC codes for different combining techniques on these channels is derived using density evolution and an estimation of the waterfall performance of LDPC codes is derived that reduces the gap between simulated and asymptotic performance.

On the Waterfall Performance of Finite-Length SC-LDPC Codes Constructed From Protographs

An analysis of spatially coupled low-density parity-check (SC-LDPC) codes constructed from protographs is proposed. Given the protograph used to generate the SC-LDPC code ensemble, a set of scaling parameters to characterize the average finite-length performance in the waterfall region is computed. The error performance of structured SC-LDPC code ensembles is shown to follow a scaling law similar to that of unstructured randomly constructed SC-LDPC codes. Under a finite-length perspective, some of the most relevant SC-LDPC protograph structures proposed to date are compared. The analysis reveals significant differences in their finite-length scaling behavior, which is corroborated by simulation. Spatially coupled repeat-accumulate codes present excellent finite-length performance, as they outperform in the waterfall region SC-LDPC codes of the same rate and better asymptotic thresholds.

Non-Binary LDPC Erasure Codes With Separated Low-Degree Variable Nodes

The code design of non-binary low-density parity-check codes for the erasure channel, under maximum a posteriori decoding, is addressed. In particular, a partially structured ensemble of codes, characterized by a careful control of the amount and of the connectivity of the variable nodes of small degrees, is proposed. The identified ensemble of codes is analyzed in terms of asymptotic thresholds and weight distribution and it is shown that codes from the ensemble provide a remarkable trade-off between waterfall performance, error floor, and decoding complexity. As an example, the performance curve of a short (256,128) code on the memoryless 16-ary erasure channel tightly approaches the Singleton bound at least down to a codeword error rate of $10^{-9}$ , at low decoding complexity.

Construction of Irregular QC-LDPC Codes via Masking with ACE Optimization

Quasi-cyclic low-density parity-check (QC-LDPC) codes constructed by using algebraic approaches, such as finite geometry and finite field, generally have good structural properties and very low error-floors, and facilitate hardware implementation. Irregular QC-LDPC codes constructed from such QC-LDPC codes by using the masking technique, when decoded with the sum-product algorithm (SPA), have low decoding complexity, but often show early error-floors. In this paper, the relationship of cycle, girth and approximate cycle EMD (ACE) between the masking matrix and masked matrix is investigated, and the ACE algorithm is modified and used to construct the masking matrix for irregular QC-LDPC codes. Simulations demonstrate that the codes constructed by masking with ACE optimization exhibit much improved waterfall performance and lower error floors.

Computing the Minimum Distance of Nonbinary LDPC Codes

Finding the minimum distance of low-density-parity-check (LDPC) codes is an NP-hard problem. Different from all existing works that focus on binary LDPC codes, we in this paper aim to compute the minimum distance of nonbinary LDPC codes, motivated by the fact that operating in a large Galois field provides one important degree of freedom to achieve both good waterfall and error-floor performance. Our method is based on the existing nearest nonzero codeword search (NNCS) method, but several modifications are incorporated for nonbinary LDPC codes, including the modified error impulse pattern, the dithering method, and the nonbinary decoder. Numerical results on the estimated minimum distances show that a code's minimum distance can be increased by careful selection of nonzero elements of the parity check matrix, or by increasing the mean column weight, or by increasing the size of the Galois field. These results support observations that have been made based on simulated performance in the literature. Finally, we provide an upper bound on the minimum distance for nonbinary quasi-cyclic LDPC codes.

Interleaved Product LDPC Codes

Product LDPC codes take advantage of LDPC decoding algorithms and the high minimum distance of product codes. We propose to add suitable interleavers to improve the waterfall performance of LDPC decoding. Interleaving also reduces the number of low weight codewords, that gives a further advantage in the error floor region.

Generalized and Doubly Generalized LDPC Codes With Random Component Codes for the Binary Erasure Channel

In this paper, a method for the asymptotic analysis of generalized low-density parity-check (GLDPC) codes and doubly generalized low-density parity-check (D-GLDPC) codes over the binary erasure channel (BEC), based on extrinsic information transfer (EXIT) chart, is described. This method overcomes the problem consisting of the impossibility to evaluate the EXIT function for the check or variable component codes, in situations where the information functions or split information functions for component codes are unknown. According to the proposed technique, GLDPC codes and D-GLDPC codes where the generalized check and variable component codes are <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">random</i> codes with minimum distance at least 2, are considered. A technique is then developed which finds the EXIT chart for the overall GLDPC or D-GLDPC code, by evaluating the expected EXIT function for each check and variable component code. This technique is finally combined with the differential evolution algorithm in order to generate some good GLDPC and D-GLDPC edge distributions. Numerical results of long, random codes, are presented which confirm the effectiveness of the proposed approach. They also reveal that D-GLDPC codes can outperform standard LDPC codes and GLDPC codes in terms of both waterfall performance and error floor.

Waterfall Performance Analysis of Finite-Length LDPC Codes on Symmetric Channels

An efficient method for analyzing the performance of finite-length low-density parity-check (LDPC) codes in the waterfall region, when transmission takes place on a memoryless binary-input output-symmetric channel is proposed. This method is based on studying the variations of the channel quality around its expected value when observed during the transmission of a finite-length codeword. We model these variations with a single parameter. This parameter is then viewed as a random variable and its probability distribution function is obtained. Assuming that a decoding failure is the result of an observed channel worse than the codeiquests decoding threshold, the block error probability of finite-length LDPC codes under different decoding algorithms is estimated. Using an extrinsic information transfer chart analysis, the bit error probability is obtained from the block error probability. Different parameters can be used for modeling the channel variations. In this work, two of such parameters are studied. Through examples, it is shown that this method can closely predict the performance of LDPC codes of a few thousand bits or longer in the waterfall region.

Improvements in belief-propagation decoding based on averaging information from decoder and correction of clusters of nodes

In this letter we provide several methods to improve belief-propagation (BP) decoding of low-density parity-check codes. The methods use additional processing of the decoded information during the decoding process and are grouped into two categories. The first category is useful in the waterfall performance region and is based on processing averaged information obtained from the decoder. Several choices of the feedback information utilized in additional processing are introduced. The second category is useful in the error-floor performance region and is based on fixing clusters of symbols simultaneously, again utilizing the information obtained by monitoring the decoder. We demonstrate on a code example that our methods achieve sizeable gains compared to the original BP decoder