Progressive Edge-growth Algorithm Research Articles

Parity-Check Matrix Partitioning for Efficient Layered Decoding of QC-LDPC Codes

In this paper, we consider how to partition the parity-check matrices (PCMs) to reduce the hardware complexity and increase decoding throughput for the row layered decoding of quasi-cyclic low-density parity-check (QC-LDPC) codes. First, we formulate the PCM partitioning as an optimization problem, which targets to minimize the maximum column weight of each layer while maintaining a block cyclic shift property among different layers. As a result, we derive all the feasible solutions for the problem and propose a tight lower bound ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>LB</i></sub> on the minimum possible maximum column weight to evaluate a solution. Second, we define a metric called layer distance to measure the data dependency between consecutive layers and further illustrate how to identify the solutions with desired layer distance from those achieving the minimum value of ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>LB</i></sub> = 1, which is preferred to reduce computation delay. Next, we demonstrate that up-to-now, finding an optimal solution for the optimization problem with polynomial time complexity is unachievable. Therefore, both enumerative and greedy partition algorithms are proposed instead. After that, we modify the quasi-cyclic progressive edge-growth (QC-PEG) algorithm to directly construct PCMs that have a straightforward partition scheme to achieve ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>LB</i></sub> or the desired layer distance. Simulation results showed that the constructed codes have better error correction performance and achieve less average number of iterations than the underlying 5G LDPC codes.

Sparsity-Aware 3-D ToF Sensing

In this work, we propose several time-of-flight (ToF) sensing schemes which tackle the challenge of covering very-wide areas and long ranges in nearly real time, with relatively simple implementation and low associated computational load. We thoroughly describe two methodologies for the resolution of the inverse problem. First, we extend a greedy algorithm, such as orthogonal matching pursuit (OMP), by considering an initial refinement of the spatial domain in which the signal recovery is performed. Then, we propose various nonadaptive techniques for the construction of the sensing matrices, relying on the optimization of coherence and density. We further develop them by including an additional verification step which accounts for the noninstantaneous transitions from one element to another of the code and avoids any possible coincidence between rising and falling edges which may degrade the coherence. We also investigate the upper super-resolution limit, the over-sampling rate which yields unitary coherence, by considering the instrument response function (IRF) of the ToF sensor. Second, we expand an adaptive sensing scheme in which the rows of the sensing matrix are generated accounting for the information from previous measurements, such as adaptive progressive edge growth algorithm (APEG), by considering several groups of signals during the adaptation of the sensing matrices. The signals are then individually recovered for each pixel via OMP over the identified joint signal support. We validate the proposed methodologies by numerical simulations over datasets from stereo and ToF cameras.

Investigation of random-structure regular LDPC codes construction based on progressive edge-growth and algorithms for removal of short cycles

This paper investigates the construction of random-structure LDPC (low-density parity-check) codes using Progressive Edge-Growth (PEG) algorithm and two proposed algorithms for removing short cycles (CB1 and CB2 algorithm; CB stands for Cycle Break). Progressive Edge-Growth is an algorithm for computer-based design of random-structure LDPC codes, the role of which is to generate a Tanner graph (a bipartite graph, which represents a parity-check matrix of an error-correcting channel code) with as few short cycles as possible. Short cycles, especially the shortest ones with a length of 4 edges, in Tanner graphs of LDPC codes can degrade the performance of their decoding algorithm, because after certain number of decoding iterations, the information sent through its edges is no longer independent. The main contribution of this paper is the unique approach to the process of removing short cycles in the form of CB2 algorithm, which erases edges from the code's parity-check matrix without decreasing the minimum Hamming distance of the code. The two cycle-removing algorithms can be used to improve the error-correcting performance of PEG-generated (or any other) LDPC codes and achieved results are provided. All these algorithms were used to create a PEG LDPC code which rivals the best-known PEG-generated LDPC code with similar parameters provided by one of the founders of LDPC codes. The methods for generating the mentioned error-correcting codes are described along with simulations which compare the error-correcting performance of the original codes generated by the PEG algorithm, the PEG codes processed by either CB1 or CB2 algorithm and also external PEG code published by one of the founders of LDPC codes

Flexible construction of compressed sensing matrices with low storage space and low coherence

This paper considers the construction of measurement matrices with low storage space and good performance. Aiming at source-limited devices, this paper proposes a novel block sampling model based on the embedding operation. In the sampling procedure, the proposed model saves a lot of storage space by storing two smaller seed matrices. Meanwhile, the block sampling mechanism makes the proposed model have a lower sampling complexity. Since the embedding operation is oriented to binary matrices, the construction of binary matrices plays a crucial role in the proposed model. In order to construct binary matrices with low coherence, this paper proposes a modified progressive edge growth algorithm by removing some limited condition of the traditional progressive edge growth algorithm. Based on the modified progressive edge growth algorithm and the embedding operation, this paper proposes a flexible method to construct sparse measurement matrices with low coherence. Simulation experiments demonstrate that our measurement matrices give better performance than several typical measurement matrices.

Codebook-Based Training Beam Sequence Design for Millimeter-Wave Tracking Systems

In this paper, we propose a codebook-based beam tracking strategy for mobile millimeter-wave (mmWave) systems, where the temporal variation of the angle of departure (AoD) is considered. A closed-form upper bound of the average tracking error probability (ATEP) is derived and further optimized. We first consider a slow-varying scenario where narrow training beams implemented by single radio-frequency (RF) chain are employed. We show that the ATEP can be reduced by optimizing the power allocation strategy over these training beams, which is formulated and transformed into a second-order cone programming. The fast-varying scenario is further considered where the use of narrow training beams becomes inefficient due to the rapid variations of AoD. In order to reduce the training time, multiple RF chains generating wide beams are employed to track the AoD’s variations, and the associated beam pattern design problem is shown to be a 0 – 1 nonlinear optimization problem (NLP). A sequential quadratic programming method is used to solve this binary NLP. To reduce the complexity, a progressive edge-growth algorithm is further introduced by associating the binary NLP with a bipartite graph. Numerical results demonstrate significant gains of the proposed beam tracking strategy over existing benchmarks for both scenarios.

Extended construction of array‐based non‐binary LDPC codes

Array-based non-binary (NB) low-density parity-check (LDPC) codes are attractive in practice for their advantages in implementation and performance. However, existing construction of array-based NB LDPC codes is only with the prime number cases. In this Letter, the authors extend the array-based construction to composite numbers to give an explicit construction of structured array-based NB LDPC codes with column weights three, four, and five. The resulting codes have girth at least six. The simulation results show that the proposed structured codes perform as well as the random codes constructed by progressive edge-growth algorithm, when decoded with the Q-ary sum product algorithm and the low-complexity weighted bit-reliability-based decoding algorithm.

A Novel Weight Coefficient PEG Algorithm for Ultra-Reliable Short Length Analog Fountain Codes

The stringent requirements of ultra-reliable and low latency communication (URLLC) services is the most challenging feature of fifth generation systems. The design and optimization of efficient short length codes is essential for URLLC. This letter investigates a novel weight coefficient progressive edge-growth (WCPEG) algorithm to construct ultra-reliable analog fountain codes (AFCs) in the short length regime. Our WCPEG AFC can approach the normal approximation benchmark by eliminating the small girth cycles in the encoding Tanner graph. We first analyze the properties of short length AFC, and then present the comprehensive description and theoretical identification of the WCPEG AFC algorithm. Simulation results demonstrate the superior performance of WCPEG AFC in the short length regime.

Low-Rate PBRL-LDPC Codes for URLLC in 5G

In this letter, we propose a class of rate-compatible raptor-like protographs, which is aimed at the design of low-rate low-density parity-check (LDPC) codes with short length and fast convergence for ultra-reliable and low-latency communications in 5G. In the proposed protographs, the connection degrees can be progressively optimized by the protograph-based extrinsic information transfer analysis with a limited number of iterations. Following the protograph optimization, we construct the rate-compatible LDPC codes by a simplified error minimization progressive edge-growth algorithm and carry out simulations for performance comparisons. The simulation results show that our proposed LDPC codes have noticeable advantages over the existing low-rate LDPC codes with short length and limited numbers of decoding iterations.

Optimal Design of Cascade LDPC-CPM System Based on Bionic Swarm Optimization Algorithm

The direct-to-home satellite digital video broadcasting has been fast developed in the past decades, and the continuous phase modulation (CPM) is an alternative modulation scheme with the advantages of constant envelope and high spectral efficiency for the band-limited channel and non-linear amplifiers of the satellite transponder. In this paper, CPM cascaded with irregular low density parity check (LDPC) code is studied. A practical system model with global interleaving is proposed for the optimization of degree distribution of the LDPC codes, and code word design is obtained with a modified progressive edge-growth algorithm. Simulation results combined with theoretical analysis clarify the effect of signal-to-noise ratios and two important CPM parameters (modulation index ${h}$ and correlation length ${L}$ ), providing guidance for system design. Simulation on bit error rate performance show the validity of the proposed algorithm. This paper systematically solves the optimal system design of cascade irregular LDPC-CPM and is of great significance for future satellite broadcasting.

LDMC design for low complexity MIMO detection and efficient decoding

Low-density multiple-input multiple-output code (LDMC) can reduce the complexity of tree-search detection in MIMO systems. In this paper, we present a new modified progressive edge-growth (PEG) algorithm to construct large girth LDMCs, which are referred to as PEG-LDMCs. We analyze the complexity of the LDMC constrained sphere decoding (SD) and show that the LDMC constrained SD detection can be used for high reliability of the data transmission in MIMO systems. Furthermore, we propose two new efficient iterative decoding algorithms for LDMCs, which are high speed serial decoding and fast convergence shuffled decoding. Finally, we compare the bit error rate (BER) performance of PEG-LDMCs to that of the existing LDMCs. The simulation results show that the PEG-LDMCs can achieve better BER performance than that of the existing LDMCs.

PEG-Like Design of Binary QC-LDPC Codes Based on Detecting and Avoiding Generating Small Cycles

In this paper, we propose a new multi-edge metric-constrained quasi-cyclic progressive edge-growth algorithm (MM-QC-PEGA), which is suitable for constructing both single- and multi-weighted (binary) QC low-density parity-check (LDPC) codes with arbitrary lengths, rates, circulant sizes, and variable node (VN)-degree distributions. The MM-QC-PEGA is able to detect all cyclic-edge-set-minimum-virtual cycles (CMVCs), as it accurately computes the metric value of each CMVC with an a posteriori test. In addition, we propose a new greatest-common-divisor (GCD)-approximation for time efficiently approximating the metric value of a CMVC without a posteriori tests, and propose a new GCD-approximated MM-QC-PEGA (G-MM-QC-PEGA) by using the GCD-approximation to replace the a posteriori test involved in the MM-QC-PEGA. As a result, the G-MM-QC-PEGA is faster than the MM-QC-PEGA, and the CMVCs undetectable to the G-MM-QC-PEGA under multi- and single-weighted QC-LDPC code graphs have minimum lengths of eight and ten, respectively. Moreover, we propose a new masking technique that is efficient in masking the parity-check matrix consisting of an array of arbitrary circulants of the same size. Compared with the several existing works, our proposed algorithms could perform better or comparably in terms of avoiding generating small cycles and error performances. Our proposed algorithms somewhat perfect the works of QC-LDPC code construction.

Partly systematic Luby transform codes based on improved progressive edge-growth algorithm for Mars communications

The Luby transform code is a forward error correction code and was originally designed for the erasure channel. In this paper, the Luby transform codes over the Mars communication channel are investigated. To guarantee the reliable data transmission over the Mars communication channel which are usually affected by fading and noises, the progressive edge-growth algorithm is improved and used in for encoding the Luby transform codes. And then a partly systematic Luby transform code with an improved progressive edge-growth algorithm is designed to improve the error propagation phenomenon in Mars communications. Performance evaluations of the designed partly systematic Luby transform code based on the random graph, PEG and improved progressive edge-growth algorithms, respectively, are conducted through a series of simulations. Simulation results show that the performance of the partly systematic Luby transform code based on the improved progressive edge-growth algorithm outperforms those of the partly systematic Luby transform code based on two other algorithms and has the lower bit error rate and overhead.

A New Multi-Edge Metric-Constrained PEG Algorithm for Designing Binary LDPC Code With Improved Cycle-Structure

To obtain a better cycle-structure is still a challenge for the low-density parity-check (LDPC) code design. This paper formulates two metrics firstly so that the progressive edge-growth (PEG) algorithm and the approximate cycle extrinsic message degree (ACE) constrained PEG algorithm are unified into one integrated algorithm, called the metric-constrained PEG algorithm (M-PEGA). Then, as an improvement for the M-PEGA, the multi-edge metric-constrained PEG algorithm (MM-PEGA) is proposed based on two new concepts, the multi-edge local girth and the edge-trials. The MM-PEGA with the edge-trials, say a positive integer $r$, is called the $r$-edge M-PEGA, which constructs each edge of the non-quasi-cyclic (non-QC) LDPC code graph through selecting a check node whose $r$-edge local girth is optimal. In addition, to design the QC-LDPC codes with any predefined valid design parameters, as well as to detect and even to avoid generating the undetectable cycles in the QC-LDPC codes designed by the QC-PEG algorithm, the multi-edge metric constrained QC-PEG algorithm (MM-QC-PEGA) is proposed lastly. It is verified by the simulation results that increasing the edge-trials of the MM-PEGA/MM-QC-PEGA is expected to have a positive effect on the cycle-structures and the error performances of the LDPC codes designed by the MM-PEGA/MM-QC-PEGA.

Constructing Large Girth QC Protograph LDPC Codes Based on PSD-PEG Algorithm

For a given base graph, the lifted graph can be obtained by a copy-and-permute procedure. If the permutation is cyclic, the lifted graph corresponds to a quasi-cyclic (QC) protograph low-density parity-check (LDPC) code. The girth of the QC protograph LDPC code is determined by the girth of the base graph and the permutation shifts. In this paper, we first derive a lower bound on the lifting degree to achieve a large girth lifted graph. Then, motivated by the cycle searching and girth maximizing features of the progressive edge-growth (PEG) algorithm, we introduce the permutation shifts determining (PSD) PEG algorithm, which can construct large girth base graph and determine the optimal permutation shifts, simultaneously. It is shown that the computational complexity of PSD-PEG algorithm is much lower than that of the PEG algorithm and the PEG-QC algorithm for the same codeword length. Furthermore, we show that the PSD-PEG algorithm can also be used to construct nonbinary QC protograph LDPC codes without low weight codes. Simulation results show that the binary and nonbinary QC protograph LDPC codes constructed by the PSD-PEG algorithm have good bit error rate performance and frame error rate performance over the additive white Gaussian noise channel.

High-efficiency Gaussian key reconciliation in continuous variable quantum key distribution

Efficient reconciliation is a crucial step in continuous variable quantum key distribution. The progressive-edge-growth (PEG) algorithm is an efficient method to construct relatively short block length low-density parity-check (LDPC) codes. The qua-sicyclic construction method can extend short block length codes and further eliminate the shortest cycle. In this paper, by combining the PEG algorithm and qua-si-cyclic construction method, we design long block length irregular LDPC codes with high error-correcting capacity. Based on these LDPC codes, we achieve high-efficiency Gaussian key reconciliation with slice recon-ciliation based on multilevel coding/multistage decoding with an efficiency of 93.7%.

Security gap analysis of some LDPC coded transmission schemes over the flat and fast fading Gaussian wire-tap channels

It is known that the error rate can be used as a measure of reliability and security over the wire-tap channel when practical, finite length codes are used for transmission, and the security gap is an error rate based metric able to jointly treat these two aspects. In this paper, we consider several low-density parity-check (LDPC) coded transmissions, which represent the state of the art for transmissions over the wire-tap channel and we assess and compare their security gap performance. We consider two kinds of wire-tap channels: the flat and the fast fading wire-tap channels with additive white Gaussian noise. As a reference, we use the progressive edge growth (PEG) algorithm for the design of unstructured LDPC codes and compare its performance with that of four approaches for designing structured LDPC codes. We analyze both systematic and non-systematic transmissions and show that some structured code design techniques are able to achieve comparable or even better performance than the PEG algorithm over the considered channels, while taking advantage of their simpler encoding and decoding procedures.

Rate-compatible LDPC codes based on the PEG algorithm for relay communication systems

It is known that the progressive edge-growth (PEG) algorithm can be used to construct low-density parity-check (LDPC) codes at finite code lengths with large girths through the establishment of edges between variable and check nodes in an edge-by-edge manner. In [1], the authors derived a class of LDPC codes for relay communication systems by extending the full-diversity root-LDPC code. However, the submatrices of the parity-check matrix H corresponding to this code were constructed separately; thus, the girth of H was not optimized. To solve this problem, this paper proposes a modified PEG algorithm for use in the design of large girth and full-diversity LDPC codes. Simulation results indicated that the LDPC codes constructed using the modified PEG algorithm exhibited a more favorable frame error rate performance than did codes proposed in [1] over block-fading channels.

Progressive edge-growth algorithm for low-density MIMO codes

In low-density parity-check (LDPC) coded multiple-input multiple-output (MIMO) communication systems, probabilistic information are exchanged between an LDPC decoder and a MIMO detector. The MIMO detector has to calculate probabilistic values for each bit which can be very complex. In [1], the authors presented a class of linear block codes named low-density MIMO codes (LDMC) which can reduce the complexity of MIMO detector. However, this code only supports the outer-iterations between the MIMO detector and decoder, but does not support the inner-iterations inside the LDPC decoder. In this paper, a new approach to construct LDMC codes is introduced. The new LDMC codes can be encoded efficiently at the transmitter side and support both of the inner-iterations and outer-iterations at the receiver side. Furthermore they can achieve the design rates and perform very well over MIMO channels.

Efficient Progressive Edge-Growth Algorithm Based on Chinese Remainder Theorem

Progressive edge-growth (PEG) algorithm construction builds a Tanner graph, or equivalently a parity-check matrix, for an LDPC code by establishing edges between the symbol nodes and the check nodes in an edge-by-edge manner and maximizing the girth in a greedy fashion. This approach is simple but the complexity of the PEG algorithm scale is O(nm), where n is the number of symbol nodes and m is the number of check nodes. We deal with this problem by construct a base matrix H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> of size m <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> × n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> with the PEG algorithm and simultaneously expand this base matrix into a parity-check matrix H of size mx n via the the Chinese remainder theorem (CRT), where m ≫ m <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> and n ≥ n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> . The size of the base matrix is expanded without decreasing the girth. For convenience, the PEG and CRT combined algorithm is referred to as the PEG-CRT algorithm in this paper. Since a smaller matrix is constructed with the PEG algorithm and the complexity of the CRT computation is negligible compared to the PEG algorithm, the complexity of the whole code construction process is reduced. Furthermore, the proposed algorithm has a potential advantage of saving storage space by storing a smaller matrix H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> and expanding it to H "on-the-fly" in hardware. The expanded matrix H preserves the important properties of base matrix such as large girth, flexible code rate and low density. The complexity analysis shows that the complexity of the PEG-CRT algorithm does not grow with the code length n. Simulation results show that compared with the PEG LDPC codes of length n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> , the expanded PEG-CRT LDPC codes have better bit error rate (BER) performance with the iterative decoding. It is also shown that compared with PEG LDPC codes of length n, which constructed with higher complexities, the PEG-CRT codes have similar BER performance.

Improved construction of LDPC convolutional codes with semi-random parity-check matrices

In this paper, we investigate the construction of time-varying convolutional low-density paritycheck (LDPC) codes derived from block LDPC codes based on improved progressive edge growth (PEG) method. Different from the conventional PEG algorithm, the parity-check matrix is initialized by inserting certain patterns. More specifically, the submatrices along the main diagonal are fixed to be the identity matrix that ensures the fast encoding feature of the LDPC convolutional codes. Second, we insert a nonzero pattern into the secondary diagonal submatrices that ensures the encoding memory length of the time-varying LDPC convolutional codes as large as possible. With this semi-random structure, we have analyzed the code performance by evaluating the number of short cycles as well as the the bound of free distance. Simulation results show that the constructed LDPC convolutional codes perform well over additive white Gaussian noise (AWGN) channels.