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.
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.
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
LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message passing decoding. In this dissertation, we present several methods of deriving families of time-varying and time-invariant LDPC convolutional codes from LDPC block codes. We demonstrate that the derived LDPC convolutional codes significantly outperform the underlying LDPC block codes, and we investigate the reasons for these 'convolutional gains'. It is well known that cycles in the Tanner graph representation of a sparse code affect the iterative decoding algorithm, with short cycles generally pushing its performance further away from optimum. Hence it is common practice to design codes that do not contain short cycles, so as to obtain independent messages in at least the initial iterations of the decoding process. We show that the derived LDPC convolutional codes have better graph-cycle properties than their block code counterparts. In particular, we show that the unwrapping process that is used to derive LDPC convolutional codes from LDPC block codes can 'break' some cycles of the underlying LDPC block code, but cannot create any shorter cycles. We prove that any cycle in an LDPC convolutional code always maps to a cycle of the same or smaller length in the underlying LDPC block code. Minimum distance is an important code design parameter when the channel quality is high, since codewords that are minimum distance apart are the most likely to cause errors with ML or near-ML decoding. In this case, the minimum distance determines the so-called error floor behavior in the performance curve corresponding to high channel signal-to-noise ratios. Thus studying the minimum distance properties of a code family gives insight into its error floor behavior. We use asymptotic methods to calculate a lower bound on the free distance of several ensembles of asymptotically good LDPC convolutional codes derived from protograph-based LDPC block codes. Further, we show that the free distance to constraint length ratio of the LDPC convolutional codes exceeds the minimum distance to block length ratio of the corresponding LDPC block codes. Message-passing iterative decoders for LDPC block codes are known to be subject to decoding failures due to so-called pseudo-codewords. These failures can cause the large signal-to-noise ratio performance of message-passing iterative decoding to be worse than that predicted by the maximum-likelihood decoding union bound. We address the pseudo-codeword problem from the convolutional code perspective. In particular, we show that the minimum pseudo-weight of an LDPC convolutional code is at least as large as the minimum pseudo-weight of an underlying LDPC block code. This result, which parallels a well-known relationship between the minimum Hamming weight of convolutional codes and the minimum Hamming weight of their quasi-cyclic counterparts, is due to the fact that every pseudo-codeword in the LDPC convolutional code induces a pseudo-codeword in the LDPC block code with pseudo-weight no larger than that of the convolutional pseudo-codeword. More generally, we demonstrate a difference in the weight spectra of LDPC block and convolutional codes that leads to improved performance at low-to-moderate signal-to-noise ratios for the convolutional codes, a conclusion supported by simulation results.
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.
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.
LDPC convolutional research has far-reaching significance. Its parity check matrix is periodic, which greatly reduces the encoding process and is easy to implement. And different convolutional LDPC codes can carry on the unified coding to the information blocks of the length transformation. The information transfer efficiency can be improved obviously. In the paper, based on the key problem about the construction of the protograph LDPC convolutional codes, the paper studies boundary extension method and matrix extension algorithm. A series of termination convolutional codes are constructed and the performance simulation is carried out. The experimental results show that the structure of terminated LDPC convolutional codes has a better performance than the protograph LDPC codes in the bit error rate.
It has been suggested that ¿near-codewords¿ may be a significant factor affecting decoding failures of LDPC codes over the AWGN channel. A near-codeword is a sequence that satisfies almost all of the check equations. These near-codewords can be associated with so-called `trapping sets' that exist in the Tanner graph of a code. In this paper, we analyse the trapping sets of protograph-based LDPC convolutional codes. LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message-passing decoding. Further, it has been shown that some ensembles of LDPC convolutional codes are asymptotically good, in the sense that the average free distance grows linearly with constraint length. Here, asymptotic methods are used to calculate a lower bound on the trapping set growth rates for two ensembles of asymptotically good protograph-based LDPC convolutional codes. This can be used to predict where the error floor will occur for these codes under iterative message-passing decoding.