Practical polar code construction over parallel channels

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 channel-mapping 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.