Irregular low-density parity-check (LDPC) codes constructed from small protographs are one of the most powerful LDPC block codes. In this paper, we introduce the convolutional version of these codes. LDPC convolutional codes constructed from protographs have some advantages in comparison to LDPC block codes constructed from protographs, e.g., an effective pipeline decoding. We show that LDPC convolutional codes constructed from protographs have much lower error rates than analogous LDPC block codes with the same complexity and that they operate with low error rates near the Shannon limits even for a relatively small memory size
Irregular low-density parity-check (LDPC) codes constructed from small protographs are one of the most powerful LDPC block codes. In this paper, we introduce the convolutional version of these codes. LDPC convolutional codes constructed from protographs have some advantages in comparison to LDPC block codes constructed from protographs, e.g., an effective pipeline decoding. We show that LDPC convolutional codes constructed from protographs have much lower error rates than analogous LDPC block codes with the same complexity and that they operate with low error rates near the Shannon limits even for a relatively small memory size
ARA- and protograph-based LDPC codes are capable of achieving error performance similar to randomly constructed codes while enjoying several implementation advantages as a result of their structure. LDPC convolutional codes can be derived from these codes through an unwrapping process. In this paper, we review the unwrapping process as well as the pipeline decoder that allows continuous decoding of LDPC convolutional codes. Computer simulations are then used to demonstrate that the unwrapped convolutional codes achieve a "convolutional gain" in error performance. We conjecture that this is due to the concatenation of many constraint lengths worth of received symbols in the pipeline decoding process. The consequences of this improved performance are examined in terms of factors related to decoder implementation: processor size, memory requirements, and decoding delay (latency). Finally, given identical protograph kernels, we compare derived block and convolutional codes based on the above measures.
Low-density parity-check (LDPC) block codes invented by Gallager in 1962 achieve exceptional error performance on a wide variety of communication channels. LDPC convolutional codes are the convolutional counterparts of LDPC block codes. This dissertation describes techniques for the design of LDPC convolutional codes, and analyzes their distance properties and iterative decoding convergence thresholds. The construction of time invariant LDPC convolutional codes by unwrapping the Tanner graph of algebraically constructed quasi-cyclic LDPC codes is described. The convolutional codes outperform the quasi-cyclic codes from which they are derived. The design of parity-check matrices for time invariant LDPC convolutional codes by the polynomial extension of a base matrix is proposed. An upper bound on free distance, proving that time invariant LDPC codes are not asymptotically good, is obtained. The Tanner graph is used to describe a pipelined message passing based iterative decoder for LDPC convolutional codes that outputs decoding results continuously. The performance of LDPC block and convolutional codes are compared for fixed decoding parameters like computational complexity, processor complexity, and decoding delay. In each case, the LDPC convolutional code performance is at least as good as that of LDPC block codes. An analog circuit to implement pipelined decoding of LDPC convolutional codes is proposed. The distance properties of a permutation matrix based (time varying) ensemble of (J,K) regular LDPC convolutional codes is investigated. It is proved that these codes (for J > 2) have free distance increasing linearly with constraint length, i.e., they are asymptotically good. Further, the asymptotic free distance to constraint length ratio for the convolutional codes is several times larger than the minimum distance to block length ratio for corresponding LDPC block codes. Iterative decoding of terminated LDPC convolutional codes, based on the ensemble mentioned above, is analyzed, assuming transmission over a binary erasure channel. The structured irregularity of the codes leads to significantly better convergence thresholds compared to corresponding LDPC block codes. At the calculated thresholds, both the bit and block error probability can be made arbitrarily small. The results obtained suggest that the thresholds approach capacity with increasing J.
Forward error correcting (FEC) codes are used in many communication standards with a wide range of requirements. FEC codes should work close to capacity, achieve low error floors, and have low decoding complexity. In this paper, we propose a novel category of low-density parity-check (LDPC) codes, based on protograph codes with local irregularity. This new code family generalizes conventional protograph-based LDPC codes and is capable of reducing the iterative decoding threshold of the conventional counterpart. We introduce an adapted version of the protograph extrinsic information transfer (PEXIT) algorithm to estimate decoding thresholds on the binaryinput additive white Gaussian noise channel, perform optimizations on the local irregularity, and simulate the performance of some constructed codes.
This paper provides a construction method for low-rate low density parity check codes. Inspired by recently proposed accumulate-repeat-accumulate (ARA) codes, and hybrid concatenated codes, in this paper we extend the construction to low rates. Such codes can be viewed as hybrid concatenations of simple modules such as accumulators, repetition codes, differentiators, and punctured single parity check codes. These codes constitute a subclass of LDPC codes with very fast encoder structure. They also have a projected graph or protograph representation that allows for high-speed decoder implementation. Based on density evolution, we show through some examples that low iterative decoding thresholds close to the channel capacity limits can be achieved, as the block size goes to infinity. Iterative decoding simulation results for short blocks are provided for a few examples that show near-capacity performance and very low error floor
Irregular low-density parity-check (LDPC) codes constructed from small protographs are one of the most powerful LDPC codes. In this paper, we show that the performance of LDPC codes based on small protographs can be further improved by using different transmission energies for every variable node of the protograph. Thus, a protograph is now described by a set of variable nodes, a set of check nodes, edges connecting variable nodes and check nodes, and the transmission energy used for every variable node, which is called the energy distribution. We optimize the energy distribution of protographs by choosing the energy distribution with the highest threshold calculated with a generalization of the discretized density evolution. Furthermore, we show by simulations that the performance of long LDPC codes based on protographs can be improved as expected from the thresholds.
Low-density parity-check (LDPC) convolutional codes have been shown to be capable of achieving capacity-approaching performance with iterative message-passing decoding. In the first part of this paper, using asymptotic methods to obtain lower bounds on the free distance to constraint length ratio, we show that several ensembles of regular and irregular LDPC convolutional codes derived from protograph-based LDPC block codes have the property that the free distance grows linearly with respect to the constraint length, i.e., the ensembles are asymptotically good. In particular, we show that the free distance to constraint length ratio of the LDPC convolutional code ensembles exceeds the minimum distance to block length ratio of the corresponding LDPC block code ensembles. A large free distance growth rate indicates that codes drawn from the ensemble should perform well at high signal-to-noise ratios under maximum-likelihood decoding. When suboptimal decoding methods are employed, there are many factors that affect the performance of a code. Recently, it has been shown that so-called trapping sets are a significant factor affecting decoding failures of LDPC codes over the additive white Gaussian noise channel with iterative message-passing decoding. In the second part of this paper, we study the trapping sets of the asymptotically good protograph-based LDPC convolutional codes considered earlier. By extending the theory presented in part one and using similar bounding techniques, we show that the size of the smallest non-empty trapping set grows linearly with the constraint length for these ensembles.