Scaling Law of Turbo Codes over the Binary Erasure Channel

Since turbo codes with iterative decoding were introduced [1], a huge amount of research activities have been conducted on the evaluation or prediction of the performance of turbo codes. The iterative turbo decoding algorithm can be analyzed by studying two open-loop constituent decoders separately [2]–[6]. Given the statistic of communication channel output, the behavior of each constituent decoder is represented by a transfer function between the statistics of extrinsic information. Note that in this paper, the extrinsic information will be called the extrinsic. The widely used statistic of extrinsic is the mutual information with the corresponding information bit [2]. The asymptotic extrinsic information transfer (EXIT) function of the constituent decoder over the binary erasure channel (BEC) can be analytically derived by using asymptotic analysis [3], [7] under the assumption that turbo decoding is a stationary process. By using asymptotic analysis, we can find the threshold erasure probability of the communication channel, at which the waterfall occurs in the bit erasure probability curve. The slope of the waterfall, however, depends on the information blocklength, so the relation between the information blocklength and the bit erasure probability in the waterfall region needs to be discovered. In this paper, we find the scaling law of the bit erasure probability of turbo codes over BEC in terms of the information blocklength, and show that the proposed result can be used to predict efficiently the bit erasure probability in the waterfall region.