RCA analysis of the polar codes and the use of feedback to aid polarization at short blocklengths

This paper uses an extension of Reciprocal Channel Approximation (RCA) to accurately and efficiently predict the frame error rate (FER) performance of polar codes by analyzing the probability density function (p.d.f) of log likelihood ratios (LLR) associated with information bits. A feedback scheme uses the RCA to predict the p.d.f of LLRs in conjunction with a repetition coding system to decrease the blocklength required for a target FER by a factor of 16. Using a rate-0.5 128-bit polar code as the initially transmitted code, the FER of the system with feedback is obtained by theoretical analysis and verified by simulation. Including the additional incremental transmissions the average blocklength for the system with feedback is 137.55 bits and the rate is 0.4653. Without feedback, a polar code with blocklength 2048 is required to achieve a comparable FER at a comparable rate. Intuitively, feedback allows the polar code to use fewer frozen bits in the initial transmission and then uses repetition codes to provide the needed reliability to resolve unreliable unfrozen bits identified by feedback.