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한국통신학회논문지

A Modified Sum-Product Algorithm for Error Floor Reduction in LDPC Codes

Un-Hee Park, Dae-Sub Oh, Youngmin Kim, Sooyoung Kim

Published Jan 1, 2010• 4 citations
OpenAlex

Abstract

In this paper, a modified sum-product algorithm to correct bit errors captured within the trapping sets, which are produced in decoding of low-density parity-check (LDPC) codes, is proposed. Unlike the original sum-product algorithm, the proposed decoding method consists of two stages. Whether the main cause of decoding failure is the trapping sets or not is determined at the first stage. And the bit errors within the trapping sets are corrected at the second stage. In the modified algorithm, the set of failed check nodes and the transition patterns of hard-decision bits are exploited to search variable nodes in the trapping sets. After inverting information of the variable nodes, the sum-product algorithm is carried out to correct the bit errors. As a result of simulation, the proposed algorithm shows continuously improved error performance with increase in the signal-to-noise ratio. It is, therefore, considered that the modified sum-product algorithm significantly reduces or possibly eliminates the error floor in LDPC codes.

Related papers

Design and Implementation of Low Bit Error Rate of LDPC Decoder

Ashlesha P. Kshirsagar, Sandeep Kakde, Manish Devendra Chawhan +1 more

Many classes of high-performance Low-density parity check codes are based on parity check matrices composed of permutation sub matrices. The emulation-simulation framework further allows the algorithm and implementation to be iteratively redefined to improve the error floor performance of message passing decoder. Log-Like hood-Ratio (LLR) based Belief-Propagation (BP) algorithm is presented for Low Density Parity Check codes. Numerically accurate representation of check node update computation used in LLR-BP decoding is described. The implementation of Sum-Product algorithm (SPA) within Low Density Parity Check Code (LDPC) decoder is described in this paper and the correction term is used to improve the decoding performance of min-sum algorithm (MSA). Quantization and log-tanh function approximation in sum-product decoder strongly affect which absorbing set dominates in error floor region. For LDPC decoder, bit error rate (BER) decreases with increase in the signal to noise ratio.

International Conference on Communication Systems and Network Technologies•Apr 1, 2015•1 citation

A Modified Sum-Product Algorithm over Graphs with Short Cycles

Nithin Raveendran

We investigate into the limitations of the sum-product algorithm for binary low density parity check (LDPC) codes having isolated short cycles. Independence assumption among messages passed, assumed reasonable in all configurations of graphs, fails the most in graphical structures with short cycles. This research work is a step forward towards understanding the effect of short cycles on error floors of the sum-product algorithm. We propose a modified sum-product algorithm by considering the statistical dependency of the messages passed in a cycle of length 4. We also formulate a modified algorithm in the log domain which eliminates the numerical instability and precision issues associated with the probability domain. Simulation results show a signal to noise ratio (SNR) improvement for the modified sum-product algorithm compared to the original algorithm. This suggests that dependency among messages improves the decisions and successfully mitigates the effects of length-4 cycles in the Tanner graph. The improvement is significant at high SNR region, suggesting a possible cause to the error floor effects on such graphs. Using density evolution techniques, we analysed the modified decoding algorithm. The threshold computed for the modified algorithm is higher than the threshold computed for the sum-product algorithm, validating the observed simulation results. We also prove that the conditional entropy of a codeword given the estimate obtained using the modified algorithm is lower compared to using the original sum-product algorithm.

•Jan 1, 2018•0 citations

Improving the Error Floor Performance of LDCP Codes with Better Codes and Better Decoders

Dung Viet Nguyen

Error correcting codes are used in virtually all communication systems to ensure reliable transmission of information. In 1948, Shannon established an upper-bound on the maximum rate at which information can be transmitted reliably over a noisy channel. Reliably transmitting information with a rate close to this theoretical limit, known as the channel capacity, has been the goal of channel coding scientists for the last five decades. The rediscovery of low-density parity-check (LDPC) codes in the 1990s added much-renewed excitement in the coding community. LDPC codes are interesting because they can approach channel capacity under sub-optimum decoding algorithms whose complexity is linear in the code length. Unsurprisingly, LDPC codes quickly attained their popularity in practical applications such as magnetic storage, wireless and optical communications. One, if not the most, important and challenging problem in LDPC code research is the study and analysis of the error floor phenomenon. This phenomenon is described as an abrupt degradation in the frame error rate performance of LDPC codes in the high signal-to-noise ratio region. Error floor is harmful because its presence prevents the LDPC decoder from reaching very low probability of decoding failure, an important requirement for many applications. Not long after the rediscovery of LDPC codes, scientists established that error floor is caused by certain harmful structures, most commonly known as trapping sets, in the Tanner representation of a code. Since then, the study of error floor mostly consists of three major problems: 1) estimating error floor; 2) constructing LDPC codes with low error floor and 3) designing decoders that are less susceptible to error floor. Although some parts of this dissertation can be used as important elements in error floor estimation, our main contributions are a novel method for constructing LDPC codes with low error floor and a novel class of low complexity decoding algorithms that can collectively alleviate error floor. These contributions are summarized as follows. A method to construct LDPC codes with low error floors on the binary symmetric channel is presented. Codes are constructed so that their Tanner graphs are free of certain small trapping sets. These trapping sets are selected from the Trapping Set Ontology for the Gallager A/B decoder. They are selected based on their relative harmfulness for a given decoding algorithm. We evaluate the relative harmfulness of different trapping sets for the sum-product algorithm by using the topological relations among them and by analyzing the decoding failures on one trapping set in the presence or absence of other trapping sets. We apply this method to construct structured LDPC codes. To facilitate the discussion, we give a new description of structured LDPC codes whose parity-check matrices are arrays of permutation matrices. This description uses Latin squares to define a set of permutation matrices that have disjoint support and to derive a simple necessary and sufficient condition for the Tanner graph of a code to be free of four-cycles. A new class of bit flipping algorithms for LDPC codes over the binary symmetric channel is proposed. Compared to the regular (parallel or serial) bit flipping algorithms, the proposed algorithms employ one additional bit at a variable node to represent its "strength." The introduction of this additional bit allows an increase in the guaranteed error correction capability. An additional bit is also employed at a check node to capture information which is beneficial to decoding. A framework for failure analysis and selection of two-bit bit flipping algorithms is provided. The main component of this framework is the (re)definition of trapping sets, which are the most "compact" Tanner graphs that cause decoding failures of an algorithm. A recursive procedure to enumerate trapping sets is described. This procedure is the basis for selecting a collection of algorithms that work well together. It is demonstrated that decoders which employ a properly selected group of the proposed algorithms operating in parallel can offer high speed and low error floor decoding.

•Jan 1, 2012•1 citation

An efficient post processing scheme to lower the error floor of LDPC decoders

Wenjie Li, Jun Ting Lin, Zhongfeng Wang

Low-Density Parity-Check (LDPC) codes have extensive applications in numerous communication systems. However, iterative LDPC decoders may suffer from error floor. In this paper, a low complexity post processing scheme (PPS) is proposed to significantly improve the error correction performance of a Min-Sum based LDPC decoder in the error floor region. Instead of using trapping set elimination algorithms, the proposed PPS introduces perturbations so that a decoder can escape from undesired local maximums. After decoding failure, for these unreliable code bits, the proposed PPS generates carefully modified soft messages, which are employed in the following decoding iterations. Numerical results demonstrate that the presented PPS significantly lowers the error floor for simulated LDPC codes.

IEEE International Conference on Communication Technology (ICCT)•Oct 1, 2017•4 citations

Optimized Min-Sum Decoding Algorithm for Low Density Parity Check Codes

Mohammad Rakibul, Dewan Siam Shafiullah, Muhammad Mostafa +1 more

Low Density Parity Check (LDPC) code approaches Shannon–limit performance for binary field and long code lengths. However, performance of binary LDPC code is degraded when the code word length is small. An optimized min-sum algorithm for LDPC code is proposed in this paper. In this algorithm unlike other decoding methods, an optimization factor has been introduced in both check node and bit node of the Min-sum algorithm. The optimization factor is obtained before decoding program, and the same factor is multiplied twice in one cycle. So the increased complexity is fairly low. Simulation results show that the proposed Optimized Min-Sum decoding algorithm performs very close to the Sum-Product decoding while preserving the main features of the Min-Sum decoding, that is low complexity and independence with respect to noise variance estimation errors.

International Journal of Advanced Computer Science and Applications•Jan 1, 2011•19 citations

Optimized Min-Sum Decoding Algorithm for Low Density Parity Check Codes

Mohammad Rakibul Islam, Dewan Siam Shafiullah, Muhammad Mostafa +2 more

Abstract ― Low Density Parity Check (LDPC) code approaches Shannon–limit performance for binary field and long code lengths. However, performance of binary LDPC code is degraded when the code word length is small. An optimized min-sum algorithm for LDPC code is proposed in this paper. In this algorithm unlike other decoding methods, an optimization factor has been introduced in both check node and bit node of the Min-sum algorithm. The optimization factor is obtained before decoding program, and the same factor is multiplied twice in one cycle. So the increased complexity is fairly low. Simulation results show that the proposed Optimized Min-Sum decoding algorithm performs very close to the Sum-Product decoding while preserving the main features of the Min-Sum decoding, that is low complexity and independence with respect to noise variance estimation errors.

•Oct 31, 2015•2 citations
In this paper, a modified sum-product algorithm to correct bit errors captured within the trapping sets, which are produced in decoding of low-density parity-check (LDPC) codes, is proposed. Unlike the original sum-product algorithm, the proposed decoding method consists of two stages. Whether the main cause of decoding failure is the trapping sets or not is determined at the first stage. And the bit errors within the trapping sets are corrected at the second stage. In the modified algorithm, the set of failed check nodes and the transition patterns of hard-decision bits are exploited to search variable nodes in the trapping sets. After inverting information of the variable nodes, the sum-product algorithm is carried out to correct the bit errors. As a result of simulation, the proposed algorithm shows continuously improved error performance with increase in the signal-to-noise ratio. It is, therefore, considered that the modified sum-product algorithm significantly reduces or possibly eliminates the error floor in LDPC codes.
Many classes of high-performance Low-density parity check codes are based on parity check matrices composed of permutation sub matrices. The emulation-simulation framework further allows the algorithm and implementation to be iteratively redefined to improve the error floor performance of message passing decoder. Log-Like hood-Ratio (LLR) based Belief-Propagation (BP) algorithm is presented for Low Density Parity Check codes. Numerically accurate representation of check node update computation used in LLR-BP decoding is described. The implementation of Sum-Product algorithm (SPA) within Low Density Parity Check Code (LDPC) decoder is described in this paper and the correction term is used to improve the decoding performance of min-sum algorithm (MSA). Quantization and log-tanh function approximation in sum-product decoder strongly affect which absorbing set dominates in error floor region. For LDPC decoder, bit error rate (BER) decreases with increase in the signal to noise ratio.
We investigate into the limitations of the sum-product algorithm for binary low density parity check (LDPC) codes having isolated short cycles. Independence assumption among messages passed, assumed reasonable in all configurations of graphs, fails the most in graphical structures with short cycles. This research work is a step forward towards understanding the effect of short cycles on error floors of the sum-product algorithm. We propose a modified sum-product algorithm by considering the statistical dependency of the messages passed in a cycle of length 4. We also formulate a modified algorithm in the log domain which eliminates the numerical instability and precision issues associated with the probability domain. Simulation results show a signal to noise ratio (SNR) improvement for the modified sum-product algorithm compared to the original algorithm. This suggests that dependency among messages improves the decisions and successfully mitigates the effects of length-4 cycles in the Tanner graph. The improvement is significant at high SNR region, suggesting a possible cause to the error floor effects on such graphs. Using density evolution techniques, we analysed the modified decoding algorithm. The threshold computed for the modified algorithm is higher than the threshold computed for the sum-product algorithm, validating the observed simulation results. We also prove that the conditional entropy of a codeword given the estimate obtained using the modified algorithm is lower compared to using the original sum-product algorithm.
Error correcting codes are used in virtually all communication systems to ensure reliable transmission of information. In 1948, Shannon established an upper-bound on the maximum rate at which information can be transmitted reliably over a noisy channel. Reliably transmitting information with a rate close to this theoretical limit, known as the channel capacity, has been the goal of channel coding scientists for the last five decades. The rediscovery of low-density parity-check (LDPC) codes in the 1990s added much-renewed excitement in the coding community. LDPC codes are interesting because they can approach channel capacity under sub-optimum decoding algorithms whose complexity is linear in the code length. Unsurprisingly, LDPC codes quickly attained their popularity in practical applications such as magnetic storage, wireless and optical communications. One, if not the most, important and challenging problem in LDPC code research is the study and analysis of the error floor phenomenon. This phenomenon is described as an abrupt degradation in the frame error rate performance of LDPC codes in the high signal-to-noise ratio region. Error floor is harmful because its presence prevents the LDPC decoder from reaching very low probability of decoding failure, an important requirement for many applications. Not long after the rediscovery of LDPC codes, scientists established that error floor is caused by certain harmful structures, most commonly known as trapping sets, in the Tanner representation of a code. Since then, the study of error floor mostly consists of three major problems: 1) estimating error floor; 2) constructing LDPC codes with low error floor and 3) designing decoders that are less susceptible to error floor. Although some parts of this dissertation can be used as important elements in error floor estimation, our main contributions are a novel method for constructing LDPC codes with low error floor and a novel class of low complexity decoding algorithms that can collectively alleviate error floor. These contributions are summarized as follows. A method to construct LDPC codes with low error floors on the binary symmetric channel is presented. Codes are constructed so that their Tanner graphs are free of certain small trapping sets. These trapping sets are selected from the Trapping Set Ontology for the Gallager A/B decoder. They are selected based on their relative harmfulness for a given decoding algorithm. We evaluate the relative harmfulness of different trapping sets for the sum-product algorithm by using the topological relations among them and by analyzing the decoding failures on one trapping set in the presence or absence of other trapping sets. We apply this method to construct structured LDPC codes. To facilitate the discussion, we give a new description of structured LDPC codes whose parity-check matrices are arrays of permutation matrices. This description uses Latin squares to define a set of permutation matrices that have disjoint support and to derive a simple necessary and sufficient condition for the Tanner graph of a code to be free of four-cycles. A new class of bit flipping algorithms for LDPC codes over the binary symmetric channel is proposed. Compared to the regular (parallel or serial) bit flipping algorithms, the proposed algorithms employ one additional bit at a variable node to represent its "strength." The introduction of this additional bit allows an increase in the guaranteed error correction capability. An additional bit is also employed at a check node to capture information which is beneficial to decoding. A framework for failure analysis and selection of two-bit bit flipping algorithms is provided. The main component of this framework is the (re)definition of trapping sets, which are the most "compact" Tanner graphs that cause decoding failures of an algorithm. A recursive procedure to enumerate trapping sets is described. This procedure is the basis for selecting a collection of algorithms that work well together. It is demonstrated that decoders which employ a properly selected group of the proposed algorithms operating in parallel can offer high speed and low error floor decoding.
Low-Density Parity-Check (LDPC) codes have extensive applications in numerous communication systems. However, iterative LDPC decoders may suffer from error floor. In this paper, a low complexity post processing scheme (PPS) is proposed to significantly improve the error correction performance of a Min-Sum based LDPC decoder in the error floor region. Instead of using trapping set elimination algorithms, the proposed PPS introduces perturbations so that a decoder can escape from undesired local maximums. After decoding failure, for these unreliable code bits, the proposed PPS generates carefully modified soft messages, which are employed in the following decoding iterations. Numerical results demonstrate that the presented PPS significantly lowers the error floor for simulated LDPC codes.
Low Density Parity Check (LDPC) code approaches Shannon–limit performance for binary field and long code lengths. However, performance of binary LDPC code is degraded when the code word length is small. An optimized min-sum algorithm for LDPC code is proposed in this paper. In this algorithm unlike other decoding methods, an optimization factor has been introduced in both check node and bit node of the Min-sum algorithm. The optimization factor is obtained before decoding program, and the same factor is multiplied twice in one cycle. So the increased complexity is fairly low. Simulation results show that the proposed Optimized Min-Sum decoding algorithm performs very close to the Sum-Product decoding while preserving the main features of the Min-Sum decoding, that is low complexity and independence with respect to noise variance estimation errors.
Abstract ― Low Density Parity Check (LDPC) code approaches Shannon–limit performance for binary field and long code lengths. However, performance of binary LDPC code is degraded when the code word length is small. An optimized min-sum algorithm for LDPC code is proposed in this paper. In this algorithm unlike other decoding methods, an optimization factor has been introduced in both check node and bit node of the Min-sum algorithm. The optimization factor is obtained before decoding program, and the same factor is multiplied twice in one cycle. So the increased complexity is fairly low. Simulation results show that the proposed Optimized Min-Sum decoding algorithm performs very close to the Sum-Product decoding while preserving the main features of the Min-Sum decoding, that is low complexity and independence with respect to noise variance estimation errors.