Finite-length Performance Research Articles

Finite-Length Analysis for Spatially Coupled LDPC Codes Based on Base Matrix.

Spatially coupled low density parity check (SC-LDPC) are prominent candidates for future communication standards due to their "threshold saturation" properties. To evaluate the finite-length performance of SC-LDPC codes, a general and efficient finite-length analysis from the perspective of the base matrix is proposed. We analyze the evolution of the residual graphs resulting at each iteration during the decoding process based on the base matrix and then derive the expression for the error probability. To verify the effectiveness of the proposed finite-length analysis, we consider the SC-LDPC code ensembles constructed by parallelly connecting multiple chains (PC-MSC-LDPC). The analysis results show that the predicted error probabilities obtained by using the derived expression for the error probability match the simulated error probabilities. The proposed finite-length analysis provides a useful engineering tool for practical SC-LDPC code design and for analyzing the effects of the code parameters on the performances.

A Unified, SNR-Aware SC-LDPC Code Design With Applications to Magnetic Recording

Spatially coupled (SC)-low-density parity-check (LDPC) codes are known to have outstanding error-correction performance and low decoding latency, which make them an excellent choice for high-density magnetic recording (MR) technologies. Whereas previous works on LDPC and SC-LDPC codes mostly take either an asymptotic or a finite-length design approach, we propose a unified framework for jointly optimizing the codes’ thresholds and cycle counts to address both regimes. We focus on circulant-based (CB) SC-LDPC code family as a representative, high-performance exemplar of structured SC-LDPC codes. The framework is based on efficient traversal and pruning of the code search space, building on the fact that the performance of a CB SC-LDPC code depends on some characteristics of the code’s partitioning matrix, which by itself is much smaller than the code’s full parity-check matrix. We then propose an algorithm that traverses all non-equivalent partitioning matrices and outputs a list of codes, each offering an attractive point on the trade-off between asymptotic and finite-length performance. Our simulations show that our framework results in SC-LDPC codes that outperform the state-of-the-art constructions, over both additive white Gaussian noise (AWGN) and partial response (PR) channel models, and that it offers the flexibility to choose low-signal-to-noise ratio (SNR), high-SNR, or in- between SNR region considering system requirements, e.g., that of the MR device.

Symbol-Level Detection With Matched Non-Binary LDPC Codes for Position Errors in Racetrack Memories

In racetrack memory (RM), insertion and deletion (ID) errors occur owing to alignment faults; such errors are referred to as position errors (PEs). Recently, several coding schemes for PEs have been proposed based on binary low-density parity-check (BI-LDPC) codes with a joint bit-wise (BW) detection and decoding algorithm. Although these schemes yield promising asymptotic performance, they suffer from unsatisfactory finite-length performance in short block sizes. To address this problem, we propose a novel coding scheme based on non-binary LDPC (NB-LDPC) codes and symbol-wise (SW) detection. The factor graph of the proposed scheme avoids generating cycles between the detector and decoder graphs by leveraging RM’s special property that multiple ID errors are caused by PEs simultaneously, which enables the attainment of good finite-length performance while providing fast convergence. With the aim of providing further performance improvement, we also provide an SW extrinsic information transfer chart analysis to design NB-LDPC codes for the proposed scheme.

Polar Coded Repetition

Constructing efficient low-rate error-correcting codes with low-complexity encoding and decoding has become increasingly important for applications involving ultra-low-power devices such as Internet-of-Things (IoT). To this end, schemes based on concatenating the state-of-the-art codes at moderate rates with repetition codes have emerged as practical solutions deployed in various standards. In this paper, we propose a novel mechanism for concatenating outer polar codes with inner repetition codes which we refer to as polar coded repetition. More specifically, we propose to transmit a slightly modified polar codeword by deviating from Arıkan’s standard <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2 \times 2$ </tex-math></inline-formula> Kernel in a certain number of polarization recursions at each repetition block. We show how this modification can improve the asymptotic achievable rate of the standard polar-repetition scheme, while ensuring that the overall encoding and decoding complexity is kept almost the same. The achievable rate is analyzed for the binary erasure channel (BEC) and additive white Gaussian noise (AWGN) channel. Moreover, we show that the finite-length performance of the polar coded repetition scheme under cyclic redundancy check (CRC) aided successive cancellation list (SCL) decoder over AWGN channel is better than the uncoded polar-repetition scheme at the cost of a slight increase in decoding complexity. We also compare the proposed scheme, in terms of performance and complexity, with other low-rate solution based on polar codes in the literature.

Optimization of SC-LDPC Codes for Window Decoding With Target Window Sizes

In this paper, we propose an optimization method for protograph-based spatially coupled low-density parity-check (SC-LDPC) codes under window decoding (WD). Previous works on constructing SC-LDPC codes for WD typically focused on optimizing asymptotic performance metrics such as the WD threshold. However, in this paper, it is observed that the WD threshold is not an appropriate metric to sufficiently explain the finite-length behavior of SC-LDPC codes under WD. Thus, we propose a new performance metric, called the window mean parameter, based on a scaling analysis to capture the WD performance more accurately and formulate a code optimization algorithm that optimizes the proposed performance metric. Since the proposed metric depends on the window size, the optimization algorithm can provide a code family of SC-LDPC codes optimized for various target window sizes. Simulation results confirm that the improvement in the proposed metric leads to a finite-length performance improvement, resulting in one to two orders of the frame error rate gain over the conventional SC-LDPC codes for a wide range of window sizes. Furthermore, we investigate structural characteristics of the proposed codes to provide a supplementary explanation for the performance improvement, which also promotes a better understanding of SC-LDPC codes for WD.

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.

Using List Decoding to Improve the Finite-Length Performance of Sparse Regression Codes

We consider sparse regression codes (SPARCs) over complex AWGN channels. Such codes can be efficiently decoded by an approximate message passing (AMP) decoder, whose performance can be predicted via so-called state evolution in the large-system limit. In this paper, we mainly focus on how to use concatenation of SPARCs and cyclic redundancy check (CRC) codes on the encoding side and use list decoding on the decoding side to improve the finite-length performance of the AMP decoder for SPARCs over complex AWGN channels. Simulation results show that such a concatenated coding scheme works much better than SPARCs with the original AMP decoder and results in a steep waterfall-like behavior in the bit-error rate performance curves. Furthermore, we apply our proposed concatenated coding scheme to spatially coupled SPARCs. Besides that, we also introduce a novel class of design matrices, i.e., matrices that describe the encoding process, based on circulant matrices derived from Frank or from Milewski sequences. This class of design matrices has comparable encoding and decoding computational complexity as well as very close performance with the commonly-used class of design matrices based on discrete Fourier transform (DFT) matrices, but gives us more degrees of freedom when designing SPARCs for various applications.

Optimizing Code Parameters of Finite-Length SC-LDPC Codes Using the Scaling Law

In this paper, we optimize code parameters of finite-length spatially coupled low-density parity-check (SC-LDPC) codes, represented by a set of code parameters $(l,r,w,L,M)$ . Although the finite-length scaling behavior of SC-LDPC codes was studied in the existing literature, the previous works impose a constraint such that the coupling width $w$ is equal to the variable node degree $l$ and they do not focus on optimizing the code parameters for given code and decoder specifications such as the code rate, frame size, and decoding complexity. In order to optimize the code parameters with the target specifications, we first extend the scaling law of SC-LDPC codes without the constraint $w=l$ . Using the scaling law formulated with a new variable $w$ , we show that the coupling width $w$ directly affects the slope of the performance curve and performance comparisons are given to investigate trade-offs inherent in the code parameters. It is shown that there are trade-offs for the code parameters in the perspective of the asymptotic performance limit, code rate, and scaling behaviors. In addition, the scaling law allows us to find the optimal code parameter set showing the best finite-length performance. Interestingly, the optimal code parameter set $(l,r,w)$ varies depending on the coupling length $L$ and uncoupled code length $M$ that determine the code and decoder specifications, which means there is no specific code parameter set prevailing over different kinds of applications. Finally, we illustrate this result using the investigated trade-offs on the code parameters, which gives us useful insight on how to choose the code parameters.

Noisy Density Evolution With Asymmetric Deviation Models

This paper considers low-density parity-check (LDPC) decoders affected by deviations introduced by the electronic device on which the decoder is implemented. Noisy density evolution (DE) that allows to theoretically study the performance of these LDPC decoders can only consider symmetric deviation models due to the all-zero codeword assumption. A novel DE method is proposed that admits the use of asymmetric deviation models, thus widening the range of faulty implementations that can be analyzed. DE equations are provided for three noisy decoders: belief propagation, Gallager B, and quantized min-sum (MS). Simulation results confirm that the proposed DE accurately predicts the performance of LDPC decoders with asymmetric deviations. Furthermore, asymmetric versions of the Gallager B and MS decoders are proposed to compensate the effect of asymmetric deviations. The parameters of these decoders are then optimized using the proposed DE, leading to better ensemble thresholds and improved finite-length performance in the presence of asymmetric deviations.

Finite-Length Performance Analysis of LDPC Coded Continuous Phase Modulation

Serial concatenation of LDPC codes and continuous phase modulation (CPM) has recently gained significant attention due to its capacity-approaching performance, efficient detection as well as owing to its constant-envelope nature. Most of the previous contributions on LDPC coded CPM were devoted to the design of LDPC codes and their asymptotic performance analysis. However, there is a paucity of work on the finite-length performance estimation of LDPC coded CPM, primarily because existing performance estimation techniques cannot be readily applied to the LDPC coded CPM. To fill this gap, we conceive an analytical bit error probability estimation technique for finite-length LDPC coded CPM in the waterfall region. Numerical results are provided both for regular and irregular LDPC codes having different codeword lengths, demonstrating that the estimated performances are closely matched by the simulated ones.

Finite-Length Scaling of Spatially Coupled LDPC Codes Under Window Decoding Over the BEC

We analyze the finite-length performance of spatially coupled low-density parity-check (SC-LDPC) codes under window decoding over the binary erasure channel. In particular, we propose a refinement of the scaling law by Olmos and Urbanke for the frame error rate (FER) of terminated SC-LDPC ensembles under full belief propagation (BP) decoding. The refined scaling law models the decoding process as two independent Ornstein-Uhlenbeck processes, in correspondence to the two decoding waves that propagate toward the center of the coupled chain for terminated SC-LDPC codes. We then extend the proposed scaling law to predict the performance of (terminated) SC-LDPC code ensembles under the more practical sliding window decoding. Finally, we extend this framework to predict the bit error rate (BER) and block error rate (BLER) of SC-LDPC code ensembles. The proposed scaling law yields very accurate predictions of the FER, BLER, and BER for both full BP and window decoding.

Numerical and Experimental Analysis of the Hydrodynamic Performance of a Three-Dimensional Finite-Length Rotating Cylinder

The hydrodynamic performance of a three-dimensional finite-length rotating cylinder is studied by means of a physical tank and numerical simulation. First, according to the identified influencing factors, a hydrodynamic performance test of the rotating cylinder was carried out in a circulating water tank. In order to explore the changing law of hydrodynamic performance with these factors, a particle image velocimetry device was used to monitor the flow field. Subsequently, a computational field dynamics numerical simulation method was used to simulate the flow field, followed by an analysis of the effects of speed ratio, Reynolds number, and aspect ratio on the flow field. The results show that the lift coefficient and drag coefficient of the cylinder increase first and then decrease with the increase of the rotational speed ratio. The trend of numerical simulation and experimental results is similar.

Fast Q-learning for Improved Finite Length Performance of Irregular Repetition Slotted ALOHA

Fast Q-learning for Improved Finite Length Performance of Irregular Repetition Slotted ALOHA

LDPC-Coded Modulation Performance Analysis and System Design

In this work, the low-density parity-check (LDPC)-coded modulation scheme which has been selected for inclusion in the next generation of International Telecommunication Union (ITU) broadband standard is investigated. The multi-level mapping of this scheme offers excellent performance along with a straightforward modulation which is separable into simple one-dimensional schemes. A subset of modulated bit positions are protected by the LDPC code, while the remaining positions benefit from improved Euclidean distance through assignment to distinct constellation regions. To date, there has been a lack of analytical treatment of the scheme. This is problematic given the number of tunable scheme and code parameters, necessitating time-consuming Monte Carlo simulation. This motivates the novel analytical work of this article, which derives overall error-rate and threshold performances through considering separately the effects of the channel on the coded and uncoded bits. First, new closed-form expressions are derived for the hard-decision performance of the bit mapping of the multi-level modulation scheme, for both coded and uncoded bits. Extrinsic information transfer (EXIT) analysis is then applied to the coded bits in the scheme. The derived theoretical performance of the uncoded bit positions is used in combination with the derived EXIT threshold to provide for the first time for this modulation scheme a method to evaluate analytically whether a designed system will offer capacity-approaching performance. Following this, an approximation to the capacity curve for the coded modulation scheme is produced, again based on the derived analytical performance. Finally, through the relationship between mutual information and error rate performance for error control codes, the derived analytical expressions are used to produce a semi-analytical finite-length performance predictor, whereby error rate results for a given code on the binary input additive white Gaussian noise (AWGN) channel can be mapped to the coded modulation scheme under consideration for any coded system parameters.

Fast Q-Learning for Improved Finite Length Performance of Irregular Repetition Slotted ALOHA

In this paper, we study the problem of designing adaptive Medium Access Control (MAC) solutions for wireless sensor networks (WSNs) under the Irregular Repetition Slotted ALOHA (IRSA) protocol. In particular, we optimize the degree distribution employed by IRSA for finite frame sizes. Motivated by characteristics of WSNs, such as the restricted computational resources and partial observability, we model the design of IRSA as a Decentralized Partially Observable Markov Decision Process (Dec-POMDP). We have theoretically analyzed our solution in terms of optimality of the learned IRSA design and derived guarantees for finding near-optimal policies. These guarantees are generic and can be applied in resource allocation problems that exhibit the waterfall effect , which in our setting manifests itself as a severe degradation in the overall throughput of the network above a particular channel load. Furthermore, we combat the inherent non-stationarity of the learning environment in WSNs by advancing classical Q-learning through the use of virtual experience (VE), a technique that enables the update of multiple state-action pairs per learning iteration and, thus, accelerates convergence. Our simulations confirm the superiority of our learning-based MAC solution compared to traditional IRSA and provide insights into the effect of WSN characteristics on the quality of learned policies.

Finite-length performance comparison of network codes using random vs Pascal matrices

In this letter, we evaluate the finite-length performance of network coding when using either random or structured encoding matrices. First, we present our novel construction of structured network codes over Fq(q=2m) using Pascal matrices. We present their encoding, re-encoding and decoding in matrix notation and derive their packet loss rate. Second, we propose a novel methodology to compute the optimal finite-length coding rate for representative and realistic traffic applications. Finally, our method allows to compare the performance of our codes with the performance of popular random codes. We show that our constructions always have better throughput and minimal overhead, which is more significant for short code lengths. Further, their larger decoding delay fulfils the delay constraints of realistic scenarios (e.g. 5G multihop networks).

Rate-Loss Mitigation of SC-LDPC Codes Without Performance Degradation

In research on spatially-coupled low-density parity-check (SC-LDPC) codes, rate-loss of SC-LDPC codes is one of the main issues to be addressed. One way to mitigate the rate-loss is to attach additional variable nodes with an irregular degree distribution, where the degree distribution is optimized with a constraint that the belief propagation (BP) threshold should not be degraded by attaching variable nodes. However, it is observed that the degree distribution obtained with the BP threshold constraint induces degradation of the finite-length performance. In order to address the problem, we propose new optimization methods to attach additional variable nodes while minimizing performance degradation. The proposed optimization methods are based on several design techniques including the scaling law, local threshold, expected graph evolution, differential evolution algorithms, the use of a protograph structure, and puncturing codewords. Using the optimized structure for additional variable nodes, the rate-loss of SC-LDPC codes can be reduced by more than 53% without sacrificing the finite-length performance. It is also shown that the rate-loss mitigation can be translated into a performance improvement if the proposed and the conventional SC-LDPC codes are compared at the same code rate.

On Channel Quantization for Spin-Torque Transfer Magnetic Random Access Memory

As emerging memories such as spin-torque transfer magnetic random access memory (STT-MRAM) suffer from reliability issues caused by process variations and thermal fluctuations, the design of channel quantizer with the minimum number of quantization bits is critical to support effective error correction coding for ensuring high-density and high-speed memory data storage. In this paper, we first propose a quantized channel model for STT-MRAM. Based on the quantized channel model, we derive various information theoretic bounds, including the mutual information, cutoff rate, and the Polyanskiy-Poor-Verdu (PPV) finite-length performance bound. By using these bounds as design criteria, we optimize the quantizer design for the polar-coded STT-MRAM channel. Moreover, we also propose a polar-code-specific quantization design with the successive cancellation decoding algorithm, by using the block error probability bound obtained from density evolution (DE). Simulation results show that all our proposed quantizers generally outperform the prior art greedy merging quantizer. In addition, both the cutoff rate and PPV bound based quantizers outperform the most widely applied mutual information based quantizer for short-length polar codes with 2-bit quantization. Furthermore, the DE quantizer designed specifically for polar codes achieves the best performance among all the proposed quantizers.

Code Design Based on Connecting Spatially Coupled Graph Chains

A novel code construction based on spatially coupled low-density parity-check (SC-LDPC) codes is presented. The proposed code ensembles are comprised of several protograph-based chains characterizing individual SC-LDPC codes. We demonstrate that the code ensembles obtained by connecting appropriately chosen individual SC-LDPC code chains at specific points have improved iterative decoding thresholds. In addition, the connected chain ensembles have a smaller decoding complexity required to achieve a specific bit error probability compared to the individual code chains. Moreover, we demonstrate that, like the individual component chains, the proposed constructions have a typical minimum distance that grows linearly with block length. Finally, we show that the improved asymptotic properties of the connected chain ensembles also translate into improved finite length performance.

Physical Layer Security for RF Satellite Channels in the Finite-Length Regime

Secure communications are becoming increasingly relevant in the development of space technology. Well-established cryptographic technology is already in place and is expected to continue to be so. On the other hand, information theoretical security emerges as a post-quantum versatile candidate to complement overall security strength. In order to prove such potential, performance analysis methods are needed that consider realistic legitimate and eavesdropper system assumptions and non-asymptotic coding lengths. In this paper, we propose the design of secure radio frequency (RF) satellite links with realistic system assumptions. Our contribution is three-fold. First, we propose a wiretap channel model for the finite-length regime. The model includes a stochastic wiretap encoding method using existing practical linear error correcting codes and hash codes. Secrecy is provided with privacy amplification, for which the finite-length secrecy metric is given that upper bounds semantic secrecy. Second, we derive a novel RF (broadcast) satellite wiretap channel model that parameterizes the stochastic degraded channel around the legitimate channel, a necessary condition to enable secure communication. Finally, we show the design of a secure satellite physical layer and finite-length performance evaluation. In doing so, we define as sacrifice rate the fixed fraction of the overall coding rate budget for reliability that needs to be allocated to secrecy. Our methodology does not make use of channel side information of the eavesdropper, it only makes assumptions on Eve’s noise power. We illustrate our proposed design method with numerical results using practical error correcting codes in current standards of satellite communication.