Previous studies on low-density parity-check convolutional codes (LDPC-CC) reveal that LDPC-CC with rational parity-check matrices (RPCM) suffer from the unaffordable decoding latency/complexity due to the infinite memory order and the poor bit-error-rate performance due to the existence of length-4 cycles in the Tanner graph. However, in this paper, we show that every LDPC-CC with RPCM can be associated with an equivalent Tanner graph which can avoid the infinite memory order and undesired short length cycles but still implements the same constraints specified by the RPCM. Together with the iterative decoding based on belief propagation with proper scheduling, simulation results indicate that LDPC-CC with RPCM can also provide satisfactory decoding performance.
Previous studies on low-density parity-check convolutional codes (LDPC-CC) reveal that LDPC-CC with rational parity-check matrices (RPCM) suffer from the unaffordable decoding latency/complexity due to the infinite memory order and the poor bit-error-rate performance due to the existence of length-4 cycles in the Tanner graph. However, in this paper, we show that every LDPC-CC with RPCM can be associated with an equivalent Tanner graph which can avoid the infinite memory order and undesired short length cycles but still implements the same constraints specified by the RPCM. Together with the iterative decoding based on belief propagation with proper scheduling, simulation results indicate that LDPC-CC with RPCM can also provide satisfactory decoding performance.
In this paper, decoding of LDPC convolutional codes with rational parity-check matrices (LDPC-CC-RPCM) is investigated. We show that Tanner graph of every LDPC-CC-RPCM can always be transformed into an equivalent one with enlarged girth and finite memory order suitable for practical pipeline decoder. Based on the transformed graph, a dynamic scheduling-aided decoding scheme with the enhancement of signal perturbation and error cancellation is presented to improve the convergence speed and bit-error-rate performance in both of the waterfall and error-floor regions. Simulation results also reveal that LDPC-CC-RPCM may outperform ordinary LDPC-CC with polynomial parity-check matrices in some cases under the same code rate and decoding complexity.
The best known practical algorithm for the decoding of low-density parity-check (LDPC) codes is the iterative sum-product or belief propagation algorithm, operating on a Tanner graph (TG) of the code. Conventionally, in each iteration, all the symbol nodes and subsequently all the check nodes in the TG pass new messages to their neighbors (the so-called "flooding schedule"). We propose a new message-passing schedule, called "probabilistic schedule". Unlike flooding, the probabilistic schedule operation is based on the structure of the TG, and probabilistically balances the frequency at which different symbol nodes update their outgoing messages in accordance with their girths. Our results show that, particularly for short block lengths, the new schedule offers a much better performance/complexity trade-off. For a particular example of a (1268, 456) code, at no cost in complexity, the probabilistic schedule not only decreases both the bit and message error rates by an order of magnitude, but also reduces the number of undetected errors considerably. This work shows the importance of scheduling in the performance of iterative decoding algorithms and suggests that a schedule which matches the structure of the TG can substantially improve the performance/complexity trade-off in the decoding of short LDPC codes.
A novel belief propagation(BP) decoding algorithm for convolutional low-density parity-check codes is proposed.The proposed algorithm raises the efficiency of updating the variable information by applying feedback information at each decoding iteration and employs the weighting factor to reduce the error propagation caused by the cycles in the Tanner graph,thus yielding a faster convergence of the decoding.Simulation results show that an error performance better than that of the existing belief propagation algorithm can be achieved,while the 5/8 decoding delay and the computation complexity are effectively reduced.Compared with the existing BP,the proposed algorithm achieves a gain of 0.16dB at the BER of 10-6 with the same number of iterations.
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.
The best known practical algorithm for the decoding of low-density parity-check (LDPC) codes is the iterative sum-product or belief propagation algorithm, operating on a Tanner graph (TG) of the code. Conventionally, in each iteration, all the symbol nodes and subsequently all the check nodes in the TG pass new messages to their neighbors (the so-called "flooding schedule"). We propose a new message-passing schedule, called "probabilistic schedule." Unlike flooding, the probabilistic schedule operates based on the structure of the TG, and probabilistically balances the frequency at which different symbol nodes update their outgoing messages in accordance with their girths. Our results show that, particularly for short block lengths, the new schedule offers a much better performance/complexity trade-off. For a particular example of a (1268,456) code, at no cost in complexity, the probabilistic schedule not only decreases both the bit and message error rates by an order of magnitude, but also reduces the number of undetected errors considerably. This work shows the importance of scheduling in the performance of iterative decoding algorithms and suggests that a schedule which matches the structure of the TG can substantially improve the performance/complexity trade-off in the decoding of short LDPC codes.
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.