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The Role of Eigenvalues of Parity Check Matrix in Low-Density Parity Check Codes

Dikshya Adhikari

Published Aug 1, 2020• 0 citations
Publisher

Abstract

The new developments in coding theory research have revolutionized the application of coding to practical systems. Low-Density Parity Check (LDPC) codes form a class of Shannon limit approaching codes opted for digital communication systems that require high reliability. This thesis investigates the underlying relationship between the spectral properties of the parity check matrix and LDPC decoding convergence. The bit error rate of an LDPC code is plotted for the parity check matrix that has different Second Smallest Eigenvalue Modulus (SSEM) of its corresponding Laplacian matrix. It is found that for a given (n,k) LDPC code, large SSEM has better error floor performance than low SSEM. The value of SSEM decreases as the sparseness in a parity-check matrix is increased. It was also found from the simulation that long LDPC codes have better error floor performance than short codes. This thesis outlines an approach to analyze LDPC decoding based on the eigenvalue analysis of the corresponding parity check matrix.

DOI
10.12794/metadc1707297

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The new developments in coding theory research have revolutionized the application of coding to practical systems. Low-Density Parity Check (LDPC) codes form a class of Shannon limit approaching codes opted for digital communication systems that require high reliability. This thesis investigates the underlying relationship between the spectral properties of the parity check matrix and LDPC decoding convergence. The bit error rate of an LDPC code is plotted for the parity check matrix that has different Second Smallest Eigenvalue Modulus (SSEM) of its corresponding Laplacian matrix. It is found that for a given (n,k) LDPC code, large SSEM has better error floor performance than low SSEM. The value of SSEM decreases as the sparseness in a parity-check matrix is increased. It was also found from the simulation that long LDPC codes have better error floor performance than short codes. This thesis outlines an approach to analyze LDPC decoding based on the eigenvalue analysis of the corresponding parity check matrix.
Low-density parity check (LDPC) codes are very popular among error correction codes because of their high-performance capacity. Numerous investigations have been carried out to analyze the performance and simplify the implementation of LDPC codes. Relatively slow convergence of iterative decoding algorithm affects the performance of LDPC codes. Faster convergence can be achieved by reducing the number of iterations during the decoding process. In this thesis, a new approach for faster convergence is suggested by choosing a systematic parity check matrix that yields lowest Second Smallest Eigenvalue Modulus (SSEM) of its corresponding Laplacian matrix. MATLAB simulations are used to study the impact of eigenvalues on the number of iterations of the LDPC decoder. It is found that for a given (n, k) LDPC code, a parity check matrix with lowest SSEM converges quickly as compared to the parity check matrix with high SSEM. In other words, a densely connected graph that represents the parity check matrix takes more iterations to converge than a sparsely connected graph.
Low-density parity check (LDPC) codes are very popular among error correction codes because of their high-performance capacity. Numerous investigations have been carried out to analyze the performance and simplify the implementation of LDPC codes. Relatively slow convergence of iterative decoding algorithm affects the performance of LDPC codes. Faster convergence can be achieved by reducing the number of iterations during the decoding process. In this thesis, a new approach for faster convergence is suggested by choosing a systematic parity check matrix that yields lowest Second Smallest Eigenvalue Modulus (SSEM) of its corresponding Laplacian matrix. MATLAB simulations are used to study the impact of eigenvalues on the number of iterations of the LDPC decoder. It is found that for a given (n, k) LDPC code, a parity check matrix with lowest SSEM converges quickly as compared to the parity check matrix with high SSEM. In other words, a densely connected graph that represents the parity check matrix takes more iterations to converge than a sparsely connected graph.
Low Density Parity Check Codes (LDPC) is a class of linear block codes best suited for forward error control in communication systems. When compared with other codes, LDPC codes have significant performance gain, which can be turned to an increase of the coverage range or a reduction of transmission power. Hence a convenient and practical receiver is obtained. The analysis done in this paper is based on the Bit Error Rate of the decoded output. The factors considered for the analysis are multiple code rate, different number of iterations and various decoding schemes. For lower code rate, soft decision decoding and increased number of iterations LDPC codes yields a better performance.
Low Density Parity Check (LDPC) codes are one of the block coding techniques that can approach the Shannon’s limit within a fraction of a decibel for high block lengths. In many digital communication systems these codes have strong competitors of turbo codes for error control. LDPC codes performance depends on the excellent design of parity check matrix and many diverse research methods have been used by different study groups to evaluate the performance. In this research work finite fields construction method is used and diverse techniques like masking, dispersion and array dispersion for the construction of parity check matrix H. These constructed codes are then simulated to make the performance evaluation. The key interest of this research work was to inspect the performance of LDPC codes based on finite fields. A comprehensive masking technique was also an objective of this research work. Simulation was done for different rates using additive white Gaussian noise (AWGN) channel. In MATLAB a library of different functions was created to perform the task for base matrix construction, dispersion, masking, generator matrix construction etc. Results give an idea that even at BER of 10-6 these short length LDPC codes performed much better and have no error floor. BER curves are also with 2-3dB from Shannon’s limit at BER of 10-6 for different rates.
After the introduction of iterative coding, Low Density Parity Check (LDPC) codes have received a lot of attention due to their remarkable near Shannon-limit performance and practical decoder implementation. However, when employing an LDPC code for various applications, multiple issues do arise at the simulation, algorithm and implementation levels. The first issue is the performance evaluation of the iterative LDPC decoder in the very low Bit Error Rate (BER) regions. So far, no analytical tool has been developed to evaluate the performance of LDPC codes. Unfortunately standard Monte Carlo (MC) simulations cannot be used for performance estimation in these low BER regions due to prohibitive execution times. The second issue is the efficiency of the decoding algorithm. Due to the presence of short cycles in the bipartite graph, sum-product decoding algorithms lead to considerable performance loss for short block-length LDPC codes and produce error floors for longer codes. In addition, the iterative processing and estimation of soft probabilistic values with these algorithms result in very high computational complexity, which in turn impedes the adoption of LDPC codes for power constrained systems. Another prime objective is to design a high throughput decoder architecture which is capable of supporting multiple rates and variable code lengths within minimal area. This work aims to address and provide solutions to the above problems. As a solution to the performance estimation problem of LDPC codes, a highly efficient and novel fast BER simulation technique is developed. At the algorithm level, a performance improvement and error floor avoidance method is introduced, and a computationally efficient algorithm is developed for the sum-product decoding of LDPC codes. And finally, at the architecture level, a high throughput multi-rate variable code length implementation is presented which, when implemented, efficiently minimizes device resources while alleviating the routing complexity of the decoder.
Introduction In this chapter we introduce low-density parity-check (LDPC) codes, a class of error correction codes proposed by Gallager in his 1962 PhD thesis 12 years after error correction codes were first introduced by Hamming (published in 1950). Both Hamming codes and LDPC codes are block codes: the messages are broken up into blocks to be encoded at the transmitter and similarly decoded as separate blocks at the receiver. While Hamming codes are short and very structured with a known, fixed, error correction ability, LDPC codes are the opposite, usually long and often constructed pseudo-randomly with only a probabilistic notion of their expected error correction performance. The chapter begins by presenting parity bits as a means to detect and, when more than one is employed, to correct errors in digital data. Block error correction codes are described as a linear combination of parity-check equations and thus defined by their parity-check matrix representation. The graphical representation of codes by Tanner graphs is presented and the necessary graph theoretic concepts introduced. In Section 2.4 iterative decoding algorithms are introduced using a hard decision algorithm (bit flipping), so that the topic is developed first without reference to probability theory. Subsequently the sum–product decoding algorithm is presented. This chapter serves as a self-contained introduction to LDPC codes and their decoding. It is intended that the material presented here will enable the reader to implement LDPC encoders and iterative decoders.