A class of error correction codes called ‘low-density parity-check (LDPC)’ codes have been the subject of a great deal of recent study in the coding community as a result of their ability to approach Shannon's fundamental capacity limit. Crucial to the performance of these codes is the use of an ‘iterative’ decoder. These iterative decoders are effectively high-dimensional, nonlinear dynamical systems and, consequently, control-theoretic tools are useful in analysing such decoders. The authors describe LDPC codes and the decoding algorithm and make a connection between the fixed points of the decoding algorithm and the well-known root locus plot. Through two examples of LDPC codes, the authors show that the root locus plot visually captures the bifurcation behaviour of iterative decoding that occurs at the signal-to-noise ratio predicted by Shannon's noisy channel coding theorem.
A class of error correction codes called ‘low-density parity-check (LDPC)’ codes have been the subject of a great deal of recent study in the coding community as a result of their ability to approach Shannon's fundamental capacity limit. Crucial to the performance of these codes is the use of an ‘iterative’ decoder. These iterative decoders are effectively high-dimensional, nonlinear dynamical systems and, consequently, control-theoretic tools are useful in analysing such decoders. The authors describe LDPC codes and the decoding algorithm and make a connection between the fixed points of the decoding algorithm and the well-known root locus plot. Through two examples of LDPC codes, the authors show that the root locus plot visually captures the bifurcation behaviour of iterative decoding that occurs at the signal-to-noise ratio predicted by Shannon's noisy channel coding theorem.
A well known class of error correction codes called low-density parity-check (LDPC) codes have been the subject of a great deal of recent study in the coding community as a result of their ability to approach Shannon's fundamental capacity limit. Crucial to the performance of these codes is the use of an iterative decoder. We describe LDPC codes and the decoding algorithm and make a connection between the fixed points of the decoding algorithm and the well-known root locus plot. Via two example LDPC codes, we describe the insights afforded by the root locus plot
In this paper we present an abstraction of the extrinsic information transfer (EXIT) chart as the interconnection of two nonlinear systems in feedback with each other. We present results on the stability of fixed points for such a dynamical system and use this framework to rederive the well-known stability condition, connecting this to the one-dimensional dynamical system describing the fractions of erasure for low-density parity-check (LDPC) codes on the binary erasure channel (BEC). We observe that the error threshold corresponds to a fixed point bifurcation for this one-dimensional system, and show that this information can be visualized using a well-known tool from control theory: the root locus plot. We further show that these bifurcations can be seen by examining the EXIT chart
The goal of error control coding is to encode information in such a way, that in the event that errors occur during transmission over a noisy communication channel or during storage in an unreliable memory, the receiver can correct the errors and recover the original transmitted information. Low-density parity-check (LDPC) codes are a kind of high performance linear block code, which are already used in many recent communication systems. Information rates guaranteed by these codes approach the theoretical Shannon capacity limit. In addition, LDPC codes are now widely used in practice and included in many communication standards. Therefore study of these codes is very important and practical approaches to their design and decoding techniques are of great interest. There exist very good decoding algorithms for LDPC codes with respect to different channel models. However, LDPC codes suffer from the infamous error floor problem at high signal-to-noise ratios. This problem is attributed to the trapping sets by Richardson. It is shown that the dominant trapping sets of regular LDPC codes, so called absorption sets, undergo a two-phase dynamic behavior in the iterative message-passing decoding algorithm. We present a linear dynamic model for the iteration behavior of these sets. It can be seen that they undergo an initial geometric growth phase which stabilizes in a final bit-flipping behavior where the algorithm reaches a fixed point. Our analysis is shown to lead to very accurate numerical calculations of the error floor bit error rates down to error rates that are inaccessible by simulation. The topologies of the dominant absorption sets of two example codes, the IEEE 802.3an [2048,1723] regular (6,32) LDPC code and the Tanner [155,64,20] regular (3,5) LDPC code, are identified and tabulated by using topological features in combination with search algorithms, respectively. To make our analysis more complete, we provide more solid evidence showing that trapping sets are equivalent to absorption sets. In other words, we argue that absorption sets characterize all failure mechanisms. Some insights from our linear analysis can be borrowed to approach this problem. Another insight that this formula provides to us is the means to reduce the error floor, which is another important part of what we hoped to achieve by developing the formula in the first place. By allowing the log-likelihood ratios (LLRs) utilized by the message-passing decoding to grow bigger in precision length, the absorption sets as a trouble-causing structure will be successfully corrected. Therefore, the absorption sets, generically born with the code design, will no longer threaten the error correcting performance as long as the messages have enough room and time to grow. This translates to greater LLR clipping thresholds and more iterations, both of which are preset at the decoder. However, the actual settings are dependent on how much the subgraph of the dominant absorption set resembles that of a non-zero minimum-weight codeword, therefore the settings are empirical. In general more likeness in their topologies, the more effort it costs to correct the absorption set.
Ever since the publication of Shannon's seminal work in 1948, the search for capacity achieving codes has led to many interesting discoveries in channel coding theory. Low-density parity-check (LDPC) codes originally proposed in 1963 were largely forgotten and rediscovered recently. The significance of LDPC codes lies in their capacity approaching performance even when decoded using low complexity sub-optimal decoding algorithms. Iterative decoders are one such class of decoders that work on a graphical representation of a code known as the Tanner graph. Their properties have been well understood in the asymptotic limit of the code length going to infinity. However, the behavior of various decoders for a given finite length code remains largely unknown.An understanding of the failures of the decoders is vital for the error floor analysis of a given code. Broadly speaking, error floor is the abrupt degradation in the frame error rate (FER) performance of a code in the high signal-to-noise ratio domain. Since the error floor phenomenon manifests in the regions not reachable by Monte-Carlo simulations, analytical methods are necessary for characterizing the decoding failures. In this work, we consider hard decision decoders for transmission over the binary symmetric channel (BSC).For column-weight-three codes, we provide tight upper and lower bounds on the guaranteed error correction capability of a code under the Gallager A algorithm by studying combinatorial objects known as trapping sets. For higher column weight codes, we establish bounds on the minimum number of variable nodes that achieve certain expansion as a function of the girth of the underlying Tanner graph, thereby obtaining lower bounds on the guaranteed error correction capability. We explore the relationship between a class of graphs known as cage graphs and trapping sets to establish upper bounds on the error correction capability.We also propose an algorithm to identify the most probable noise configurations, also known as instantons, that lead to error floor for linear programming (LP) decoding over the BSC. With the insight gained from the above analysis techniques, we propose novel code construction techniques that result in codes with superior error floor performance.
Low-density-parity-check (LDPC) codes have aroused much research interest because of their excellent bit error performance. The behaviour of the iterative LDPC decoders, however, has not been fully investigated at all signal-to-noise ratios (SNRs). By considering the LDPC decoders as high-dimensional nonlinear dynamical systems, we attempt to study the corresponding phase trajectories at different SNR values. By having an in-depth understanding of the decoder behaviour, engineers should be able to design more effective and efficient decoders.
4 ABSTRACT Tremendous amount of research has been conducted in modern coding theory in the past few years and much of the work has been done in developing new coding techniques. Low density parity check (LDPC) codes are class of linear block error correcting codes which provide capacity performance on a large collection of data transmission and storage channels while Root LDPC codes in this thesis work are admitting implementable decoders with manageable complexity. Furthermore, work has been conducted to develop graphical methods to represent LDPC codes. This thesis implement one of the LDPC kind “Root LDPC code” using iterative method and calculate its threshold level for binary and non-binary Root LDPC code. This threshold value can serve as a starting point for further study on this topic. We use C++ as tool to simulate the code structure and parameters. The results show that non-binary Root LDPC code provides higher threshold value as compare to binary Root LDPC code.