Conventional iterative decoding with flooding or parallel schedule can be formulated as a fixed-point problem solved iteratively by the successive substitution (SS) method. In this work, we investigate the dynamics of continuous-time asynchronous analog implementation of iterative decoding, and show that it can be approximated as the application of the well-known successive overrelaxation (SOR) method for solving the fixed-point problem. We observe that SOR with the optimal relaxation factor can considerably improve the performance of iterative decoding for short low-density parity-check (LDPC) codes compared to SS. Our simulation results for the application of SOR to belief propagation (sum-product) and min-sum algorithms demonstrate improvements of up to about 0.7 dB over the standard SS for randomly constructed LDPC codes. The improvement in performance increases with the maximum number of iterations and by accordingly reducing the relaxation factor. The asymptotic result, corresponding to infinite maximum number of iterations and infinitesimal relaxation factor represents the performance of analog continuous-time asynchronous iterative decoding. This means that under ideal circumstances continuous-time asynchronous analog decoders can outperform their discrete-time synchronous digital counterparts by a large margin. The proposed model for analog decoding, and the associated performance curves, can be used as an "ideal analog decoder" benchmark for performance evaluation of analog decoding circuits.
Conventional iterative decoding with flooding or parallel schedule can be formulated as a fixed-point problem solved iteratively by the successive substitution (SS) method. In this work, we investigate the dynamics of continuous-time asynchronous analog implementation of iterative decoding, and show that it can be approximated as the application of the well-known successive overrelaxation (SOR) method for solving the fixed-point problem. We observe that SOR with the optimal relaxation factor can considerably improve the performance of iterative decoding for short low-density parity-check (LDPC) codes compared to SS. Our simulation results for the application of SOR to belief propagation (sum-product) and min-sum algorithms demonstrate improvements of up to about 0.7 dB over the standard SS for randomly constructed LDPC codes. The improvement in performance increases with the maximum number of iterations and by accordingly reducing the relaxation factor. The asymptotic result, corresponding to infinite maximum number of iterations and infinitesimal relaxation factor represents the performance of analog continuous-time asynchronous iterative decoding. This means that under ideal circumstances continuous-time asynchronous analog decoders can outperform their discrete-time synchronous digital counterparts by a large margin. The proposed model for analog decoding, and the associated performance curves, can be used as an "ideal analog decoder" benchmark for performance evaluation of analog decoding circuits.
Conventional iterative decoding with flooding or parallel schedule can be formulated as a fixed-point problem solved iteratively by a successive substitution (SS) method. In this paper, we investigate the dynamics of a continuous-time (asynchronous) analog implementation of iterative decoding, and show that it can be approximated as the application of the well-known successive relaxation (SR) method for solving the fixed-point problem. We observe that SR with the optimal relaxation factor can considerably improve the error-rate performance of iterative decoding for short low-density parity-check (LDPC) codes, compared with SS. Our simulation results for the application of SR to belief propagation (sum-product) and min-sum algorithms demonstrate improvements of up to about 0.7 dB over the standard SS for randomly constructed LDPC codes. The improvement in performance increases with the maximum number of iterations, and by accordingly reducing the relaxation factor. The asymptotic result, corresponding to an infinite maximum number of iterations and infinitesimal relaxation factor, represents the steady-state performance of analog iterative decoding. This means that under ideal circumstances, continuous-time (asynchronous) analog decoders can outperform their discrete-time (synchronous) digital counterparts by a large margin. Our results also indicate that with the assumption of a truncated Gaussian distribution for the random delays among computational modules, the error-rate performance of the analog decoder, particularly in steady state, is rather independent of the variance of the distribution. The proposed simple model for analog decoding, and the associated performance curves, can be used as an "ideal analog decoder" benchmark for performance evaluation of analog decoding circuits.
Iterative decoding with flooding schedule can be formulated as a fixed-point problem solved iteratively by successive substitution (88) method. In this work, we model continuous-time analog (asynchronous) iterative decoding by a first-order differential equation, and show that it can be approximated as the application of the well-known successive over relaxation (SOR) method for solving the fixed-point problem. Simulation results for belief propagation (sum-product) and min-sum algorithms confirm that SOR, which is in general superior to the simpler 88 method, can considerably improve the performance of iterative decoding for short codes. The improvement in performance increases with the maximum number of iterations and by reducing the step size in SOR, and the asymptotic result, corresponding to infinite maximum number of iterations and infinitesimal step size represents the performance of continuous-time analog iterative decoding. This means that under ideal circumstances continuous-time analog decoders can outperform their discrete-time digital counterparts by a large margin. Moreover, the results obtained by the proposed model are surprisingly close to the results of circuit simulation of a min-sum analog decoder presented in [S. Hemati et al., 2003]. Our work also suggests a general framework for improving iterative decoding algorithms on graphs with cycles, even for synchronous digital implementations.
Abstract—Conventional iterative decoding with flooding or par-allel schedule can be formulated as a fixed-point problem solved iteratively by a successive substitution (SS) method. In this paper, we investigate the dynamics of a continuous-time (asynchronous) analog implementation of iterative decoding, and show that it can be approximated as the application of the well-known successive relaxation (SR) method for solving the fixed-point problem. We observe that SR with the optimal relaxation factor can consider-ably improve the error-rate performance of iterative decoding for short low-density parity-check (LDPC) codes, compared with SS. Our simulation results for the application of SR to belief prop-agation (sum-product) and min-sum algorithms demonstrate im-provements of up to about 0.7 dB over the standard SS for ran-domly constructed LDPC codes. The improvement in performance increases with the maximum number of iterations, and by accord-
In this paper, we show an ideal analog min-sum decoder, in the log-likelihood ratio domain, can be considered as a piecewise linear system. Many theoretical aspects of these decoders, thus, can be studied analytically. It is also shown that the dynamic equations can become singular for codes with cycles. When it is non-singular, the corresponding dynamic equations can be solved analytically to derive outputs of the decoder. We study the relationship between singularity and error floor and prove that absorption sets with degree two check nodes are singular graphs and under specific conditions the dynamic equations of an analog min-sum decoder can be reduced to that of an absorption set. The proposed approach paves the way for further analytical analysis on the dynamics of analog min-sum decoders and error floor in low-density parity-check codes. proof-of-concept analog decoders for short codes were im- plemented; however, measurement results show that in high signal to noise ratio region, decoding performance is degraded. Transistor mismatch, malfunction in comparators, and other implementation imperfections were held responsible for the observed early error floor. In (22) a simple model for ideal analog iterative decoding was introduced and its dynamic equations, which are nonlin- ear differential equations, were derived. By using numerical techniques, it was shown that the dynamics of analog iterative decoders is different from that of the conventional iterative decoders. While the former is based on application of succes- sive relaxation (SR) technique to the fixed point problem of decoding, the latter is based on the application of successive substitution (SS) method. It was also observed that due to higher chance of convergence for SR compared to SS, analog decoding can provide some coding gain for codes with cycles. In this paper, an analytical approach is used for studying the dynamics of ideal analog min-sum decoders. This study sheds more light on theoretical aspects of decoding in continuous time domain and provides a better understanding about time response of ideal analog min-sum decoders. We show that an ideal analog min-sum decoder in log-likelihood ratio (������ ) domain can be considered as a piecewise linear system; that is, for any time interval (�� 1 ,�� 2), this system can be partitioned into finitely many sub-intervals, such that on each such sub-interval �� , it is equal to a linear system. We also show that outputs of the decoder can be derived analytically by having the parity- check matrix of the code and the received information from the channel. We consider the model introduced for analog decoding in (22), and write the dynamic equation in a matrix form. We then study many important aspects of this system, such as bounds on eigenvalues and stability of the dynamic equations. We show the dynamic equations can become singular and under specific conditions we relate them to absorption sets that are responsible for error floors in LDPC codes. We also show dithering that models implementation imperfections such as mismatch can help in tackling singularity problem.
The application of successive relaxation (SR) to the fixed-point problem associated with the iterative decoding of low-density parity-check (LDPC) codes is studied. We consider finite-length codes decoded by belief propagation (BP) and a well-known approximation of it, referred to as min-sum (MS), over a binary input additive white Gaussian noise (BIAWGN) channel. For both algorithms, we show that the application of SR in different domains results in different error correcting performance. In particular, the performances of SR in log-likelihood ratio (LLR) and LR domains for BP and MS are compared, and it is shown that SR-MS-LLR has the best performance. For the tested codes, SR-MS-LLR outperforms standard BP by up to about 0.5dB, offering an attractive solution in terms of performance/complexity tradeoff. We also investigate the effects of quantization on SR-MS-LLR and demonstrate that at least 6–7 bits of quantization is required to capture close to floating-point performance.
The application of successive relaxation (SR) to the fixed-point problem associated with the iterative decoding of low-density parity-check (LDPC) codes was proposed by Hemati et al.. The simulation results presented by Hemati et al. for the SR version of belief propagation (BP) in the likelihood ratio (LR) domain and that of min-sum (MS) in the log-likelihood ratio (LLR) domain are based on the assumption of all-zero codeword transmission. This assumption however results in erroneous error rates when SR is applied in the LR domain. Here, we correct the simulation results reported by Hemati et al. for SR-BP in the LR domain. Furthermore, we investigate the performance of SR-BP and SR-MS in the LLR and LR domains, respectively. The results for a binary input additive white Gaussian noise (BIAWGN) channel show that for both BP and MS, the application of SR in the two domains of LR and LLR results in different error correcting performance. In particular, for the tested codes, it is shown that among the four algorithms, SR-MS-LLR has the best performance. It outperforms standard MS and BP by up to about 0.6 dB and 0.3 dB, respectively, offering an attractive solution in terms of performance/complexity tradeoff.