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.
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.
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.
In this paper, we show that 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 and degrade the decoding performance. When the decoder is nonsingular, the corresponding dynamic equations can be solved analytically to derive outputs of the decoder. The proposed approach paves the way for further analytical analysis on the dynamics of analog min-sum decoders.
The goal of error control coding is to encode information in such a way, that in the event that errors occur during transmission over a noisy communication channel or during storage in an unreliable memory, the receiver can correct the errors and recover the original transmitted information. Low-density parity-check (LDPC) codes are a kind of high performance linear block code, which are already used in many recent communication systems. Information rates guaranteed by these codes approach the theoretical Shannon capacity limit. In addition, LDPC codes are now widely used in practice and included in many communication standards. Therefore study of these codes is very important and practical approaches to their design and decoding techniques are of great interest. There exist very good decoding algorithms for LDPC codes with respect to different channel models. However, LDPC codes suffer from the infamous error floor problem at high signal-to-noise ratios. This problem is attributed to the trapping sets by Richardson. It is shown that the dominant trapping sets of regular LDPC codes, so called absorption sets, undergo a two-phase dynamic behavior in the iterative message-passing decoding algorithm. We present a linear dynamic model for the iteration behavior of these sets. It can be seen that they undergo an initial geometric growth phase which stabilizes in a final bit-flipping behavior where the algorithm reaches a fixed point. Our analysis is shown to lead to very accurate numerical calculations of the error floor bit error rates down to error rates that are inaccessible by simulation. The topologies of the dominant absorption sets of two example codes, the IEEE 802.3an [2048,1723] regular (6,32) LDPC code and the Tanner [155,64,20] regular (3,5) LDPC code, are identified and tabulated by using topological features in combination with search algorithms, respectively. To make our analysis more complete, we provide more solid evidence showing that trapping sets are equivalent to absorption sets. In other words, we argue that absorption sets characterize all failure mechanisms. Some insights from our linear analysis can be borrowed to approach this problem. Another insight that this formula provides to us is the means to reduce the error floor, which is another important part of what we hoped to achieve by developing the formula in the first place. By allowing the log-likelihood ratios (LLRs) utilized by the message-passing decoding to grow bigger in precision length, the absorption sets as a trouble-causing structure will be successfully corrected. Therefore, the absorption sets, generically born with the code design, will no longer threaten the error correcting performance as long as the messages have enough room and time to grow. This translates to greater LLR clipping thresholds and more iterations, both of which are preset at the decoder. However, the actual settings are dependent on how much the subgraph of the dominant absorption set resembles that of a non-zero minimum-weight codeword, therefore the settings are empirical. In general more likeness in their topologies, the more effort it costs to correct the absorption set.
It is well-known that the performance of low-density parity-check codes is compromised by the emergence of an error floor, which is caused by so-called trapping sets. By identifying the dominant trapping sets, this error floor can be estimated analytically using a linear model. In this paper, this linear approach is improved to make the analysis more accurate. Then, guided by the error probability formula, a simple but effective method to improve the code performance in the error floor region is proposed, by introducing a boosting factor to the log-likelihood ratios returned from unsatisfied check nodes during the early decoding stages. It is shown that the effect of the dominant trapping sets can thus be reduced. The dominance of trapping sets can be further reduced by extending the computational range of the LLRs at the decoder. To illustrate these ideas, a short regular LDPC code constructed by Tanner, as well as the IEEE 802.3 LDPC code, are studied.
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.
The error floor phenomenon in many low-density parity-check (LDPC) codes is caused by combinatorial objects in their Tanner graph, known as absorbing sets. In this paper, we highlight a threshold behavior for the min-sum decoding algorithm in the graph of an absorbing set with fixed-point representation of messages. For an absorbing set of interest in a binary LDPC code we can compute the threshold, a novel real-valued parameter that is closely related to its harmfulness. We show that absorbing sets with negative thresholds cannot trap the decoder if the dynamic range of the extrinsic messages is large enough. We also prove that, in regular LDPC codes, absorbing sets with negative thresholds exist if the variable node degree is odd. The examples presented in this paper show that odd-column-weight LDPC codes can have many absorbing sets with negative thresholds, but that these absorbing sets do not trap a well-designed decoder. Simulations show a good agreement between the results of the analysis presented in this paper and the performance of practical decoders with fixed-point messages.