Practical polar code construction over parallel channels

Abstract

Channel polarisation results are extended to the case of communications over parallel channels, where the channel state information is known to both the encoder and decoder. Given a set of parallel binary-input discrete memoryless channels (B-DMCs), by performing the channel polarising transformation over independent copies of these component channels, we obtain a second set of synthesised binary-input channels. Similar to the single-channel case, we prove that as the size of the transformation goes infinity, some of the resulting channels tend to completely noised, and the others tend to noise-free, where the fraction of the latter approaches the average symmetric capacity of the underlying component channels. For finite-length polar coding over parallel channels, performance is found to be relied heavily on the specific channelmapping scheme. To avoid exhaustive searching, an empirically good scheme that is called equal-capacity partition channel mapping is proposed and numerical results show that the proposed scheme significantly outperforms random mapping. Further, utilising the above results, a polar coding method for arbitrary code length is proposed, which has potential applications in practical systems.

ARUM: Polar-Coded HARQ Scheme Based on Incremental Channel Polarization

“…The selection of the repeated or punctured positions has strong impact on finite-length performance [12]. The best known universal solution is performing a bit-reversal permutation on the coded bit sequence, then repeating/puncturing |M − N | bits sequentially from the first, or in a reverse order from the last.…”

“…The channel reliabilities are changed after rate-matching, so the optimal active set A should be adjusted accordingly [12]. However, in HARQ procedure, there is no chance to change the set A once it is determined at the first transmission.…”

Practical polar code construction using generalised generator matrices

“…It is shown that using larger generator matrices allows obtaining polar codes with flexible code lengths while maintaining an error performance comparable to that of the original construction. This is an efficient way to obtain polar codes of flexible code lengths and is comparable to the contribution of [11]. In [11], the authors propose the use of puncturing together with codes with length equal to the next power of 2 to obtain the desired code length.…”

“…This is an efficient way to obtain polar codes of flexible code lengths and is comparable to the contribution of [11]. In [11], the authors propose the use of puncturing together with codes with length equal to the next power of 2 to obtain the desired code length. This approach can be used in conjunction with the proposed approach to obtain codes of desired length with minimal use of puncturing by starting with a slightly larger block length (not necessarily the next power of 2) and puncturing the unnecessary symbols/channels.…”

Polar coded HARQ scheme with Chase combining

“…The RCPP codes over AWGN channel are well studied in [7]. Given an AWGN channel and a specific RCPP code, the block error rate (BLER) can be accurately predicted under the framework of channel polarization over parallel channels [8]. The code construction and performance evaluation methods of RCPP codes are extended to the Rayleigh fading channels.…”