Comparison between continuous-time asynchronous and discrete-time synchronous iterative decoding

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.