Transactions Papers On the Dynamics of Analog Min-Sum Iterative Decoders: An Analytical Approach
Saied Hemati, Abbas Yongaçoğlu · 2010
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.