Performance of medium length LDPC codes is directly proportional to error floor .In these codes, error floor can be mainly improved through elimination of short cycles.Hence, for effective elimination of short cycles, it is customary to use a girth conditioning in random construction process.Various accurate techniques, such as, density evolution and EXIT chart are developed for performance prediction of large block length, iteratively decoded LDPC codes.However, for medium block length, this method evolved with erroneous results.Even though, the medium block length LDPC codes have many application, short cycles degrades the performance and adds computational complexity in decoding process.
Performance of medium length LDPC codes is directly proportional to error floor .In these codes, error floor can be mainly improved through elimination of short cycles.Hence, for effective elimination of short cycles, it is customary to use a girth conditioning in random construction process.Various accurate techniques, such as, density evolution and EXIT chart are developed for performance prediction of large block length, iteratively decoded LDPC codes.However, for medium block length, this method evolved with erroneous results.Even though, the medium block length LDPC codes have many application, short cycles degrades the performance and adds computational complexity in decoding process.
Gallager proposed and developed low density parity check (LDPC) codes in the early 1960s. LDPC codes were rediscovered in the early 1990s and shown to be capacity approaching over the additive white Gaussian noise (AWGN) channel. Subsequently, density evolution (DE) optimized symbol node degree distributions were used to significantly improve the decoding performance of short to medium length irregular LDPC codes. Currently, the short to medium length LDPC codes with the lowest error floor are DE optimized irregular LDPC codes constructed using progressive edge growth (PEG) algorithm modifications which are designed to increase the approximate cycle extrinsic message degrees (ACE) in the LDPC code graphs constructed. The aim of the present work is to find efficient means to improve on the error floor performance published for short to medium length irregular LDPC codes over AWGN channels in the literature. An efficient algorithm for determining the girth and ACE distributions in short to medium length LDPC code Tanner graphs has been proposed. A cyclic PEG (CPEG) algorithm which uses an edge connections sequence that results in LDPC codes with improved girth and ACE distributions is presented. LDPC codes with DE optimized/’good’ degree distributions which have larger minimum distances and stopping distances than previously published for LDPC codes of similar length and rate have been found. It is shown that increasing the minimum distance of LDPC codes lowers their error floor performance over AWGN channels; however, there are threshold minimum distances values above which there is no further lowering of the error floor performance. A minimum local girth (edge skipping) (MLG (ES)) PEG algorithm is presented; the algorithm controls the minimum local girth (global girth) connected in the Tanner graphs of LDPC codes constructed by forfeiting some edge connections. A technique for constructing optimal low correlated edge density (OED) LDPC codes based on modified DE optimized symbol node degree distributions and the MLG (ES) PEG algorithm modification is presented. OED rate-½ (n, k)=(512, 256) LDPC codes have been shown to have lower error floor over the AWGN channel than previously published for LDPC codes of similar length and rate. Similarly, consequent to an improved symbol node degree distribution, rate ½ (n, k)=(1024, 512) LDPC codes have been shown to have lower error floor over the AWGN channel than previously published for LDPC codes of similar length and rate. An improved BP/SPA (IBP/SPA) decoder, obtained by making two simple modifications to the standard BP/SPA decoder, has been shown to result in an unprecedented generalized improvement in the performance of short to medium length irregular LDPC codes under iterative message passing decoding. The superiority of the Slepian Wolf distributed source coding model over other distributed source coding models based on LDPC codes has been shown.
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.
There is an important breakthrough in the field of forward error correction techniques, low-density parity-check (LDPC) codes. These codes have potential to approach the Shannon limit with reliable performance on a given channel, known as the channel capacity. Recently, researchers pay much more attention towards LDPC codes because its performance is better than Turbo codes. However, the LDPC code is the best when it is an irregular code and uses with a very large block-length. Although, the short block-length LDPC codes perform not so well, they are easy to implement in various practical applications. Therefore, there is still a demand of new development on the encoding side of LDPC codes for various ranges of code rate and code length so as to capable of using in many applications.This thesis proposes the development of quasi-cyclic (QC) LDPC codes for column weight 3. In the first part of this thesis, we propose a new construction algorithm of QC-LDPC codes for medium to large block-length by combining QC-LDPC codes of small block-length as their component codes, via Chinese remainder theorem (CRT). Such component codes were constructed by permuting each column block sequentially to attain the desire local girth. After combining all component codes to generate an expanded parity-check matrix, the resulting girth is greater than or at least equal to the maximum girth of the component codes. Simulation results show that our proposed construction method of the parity-check matrix significantly outperforms the other well-known existing methods in terms of low error-floor, simple structure, high performance, and can reduce encoding complexity.In addition, this thesis also proposes two new construction methods for QC-LDPC codes, namely a base matrix method and a subtraction based method. A base matrix based method is a simple, less computational complexity method for constructing the exponent matrix (3, K) of girth 8, 10 and 12 of QC-LDPC codes. Another method is a subtraction method, which has a similar exponent matrix as a base matrix method and has a girth at least 8. Results indicate that the LDPC codes constructed from these two methods have flexibility for arbitrary block-column length K and have a similar BER performance if compared to existing methods.
Error control coding is an essential part of modern communications systems. LDPC codes have been demonstrated to offer performance near the fundamental limits of channels corrupted by random noise. Optimal maximum likelihood decoding of LDPC codes is too complex to be practically useful even at short block lengths and so a graph-based message passing decoder known as the belief propagation algorithm is used instead. In fact, on graphs without closed paths known as cycles the iterative message passing decoding is known to be optimal and may converge in a single iteration, although identifying the message update schedule which allows single-iteration convergence is not trivial. At finite block lengths graphs without cycles have poor minimum distance properties and perform poorly even under optimal decoding. LDPC codes with large block length have been demonstrated to offer performance close to that predicted for codes of infinite length, as the cycles present in the graph are quite long. In this thesis, LDPC codes of shorter length are considered as they offer advantages in terms of latency and complexity, at the cost of performance degradation from the increased number of short cycles in the graph. For these shorter LDPC codes, the problems considered are: \n \nFirst, improved construction of structured and unstructured LDPC code graphs of short length with a view to reducing the harmful effects of the cycles on error rate performance, based on knowledge of the decoding process. Structured code graphs are particularly interesting as they allow benefits in encoding and decoding complexity and speed. Secondly, the design and construction of LDPC codes for the block fading channel, a particularly challenging scenario from the point of view of error control code design. Both established and novel classes of codes for the channel are considered. Finally the decoding of LDPC codes by the belief propagation algorithm is considered, in particular the scheduling of messages passed in the iterative decoder. A knowledge-aided approach is developed based on message reliabilities and residuals to allow fast convergence and significant improvements in error rate performance.
Low-density parity-check (LDPC) codes can approach the Shannon limit performance closely, and are becoming one of the most promising channel codes in the Error Control Coding area.The performance of an LDPC code with given length is mainly affected by the sizes of some combinatorial characteristics of its corresponding bipartite graph and the distribution of variable degrees and check node degrees.With the help of the density evolution algorithm, randomly designed LDPC codes, with carefully chosen degree distribution pairs, have been shown to achieve better Shannon capacity performance than their regular counterparts when decoded using the iterative belief propagation (BP) decoding algorithm.However, regular LDPC codes outperform their irregular counterparts in term of their error floors.Therefore, construction of LDPC codes which have both attractive error rate performance and low error floor performance has become an attractive topic.Currently, optimization of variable node degree distribution and check node degree distribution for an LDPC code with given length, which could help the code to achieve good Shannon limit performance, has been extensively studied.In this dissertation, we demonstrate some approaches of improving the error floor performance for a given LDPC code with desired degree distribution pair.The contribution mainly includes three parts.In the first part, the relationship between cycles and unfavorable combinatorial
Recently, short block codes are in great demand due to the emergent applications requiring the transmission of a short data unit and can guarantee speedy communication, with a minimum of latency and complexity which are among the technical challenges in today’s wireless services and systems. In the context of channel coding using low density parity check (LDPC) codes, the shorter length LDPC block codes are more likely to have short cycles with lengths of 4 and 6. The effect of the cycle with the minimum size is that this one prevents the propagation of the information in the Tanner graph during the iterative process. Therefore, the message decoded by short block code is assumed to be of poor quality due to short cycles. In this work, we present a study of the evolution of the messages on check nodes during the iterative decoding process when using the LDPC decoding algorithm normalized min sum (NMS), to see the destructive effect of short cycles and justify the effectiveness of the girth aware normalized min sum (GA-NMS) decoding LDPC codes algorithm in terms of correction of the errors, particularly for the codes with short cycles 4 and 6. In addition to this, the GA-NMS algorithm is evaluated in terms of bit error rate performance and convergence behavior, using wireless regional area networks (WRAN) LDPC code, which is considered as a short block code.