Polar Codes for Fast Fading Channel: Design Based on Polar Spectrum

As the first class of constructive channel codes achieving the symmetric capacity of the binary-input discrete memoryless channels, polar codes have been extended to the fading channel. But most constructions of polar codes in fading channel are only suitable for the Rayleigh fading and involve complex iterative-calculation. In this paper, we establish a systematic framework in term of the polar spectrum to analyze and construct polar codes in various fast fading channels, such as Rician, Rayleigh and Nakagami channels. In these three fading channels, by using the polar spectrum, we derive the upper bound of the error probability of the polarized channel and that of block error rate of polar codes. The analysis based on the polar spectrum explicitly reveals the relationship between the diversity order and the codeword weight. Furthermore, we propose two construction metrics, named logarithmic upper-bound weight (LUW) and minimum-weight LUW (MLUW) respectively, to design the polar codes in fast fading channels. These two constructions are simple with a linear complexity and explicit for the practical implementation. In addition, for various fading channels, they can construct polar codes with similar or better performance over those based on traditional methods.