The Theoretical Analysis And Application Of Polar Code Based On Distance Spectrums

Polar code,which was presented by Arikan in 2009,has been proved to achieve the channel capacity from theoretically.It is a constructive channel coding and shows some advantages in low encoding/decoding complexity and good universality.Therefore,polar code has great research value and significance in theoretical analysis and application research.The distance spectrum of code reacts the weight distribution of the codewords.In general,the weight distribution of codewords determines the performance of code.Therefore,the distance spectrum is an effective tool of performance analysis.This paper researches the theoretical analysis of polar code based on the distance spectrum and the practical application of polar code,including the following innovation work of three aspects:(1)Establish the performance analysis scheme of polar code based on distance spectrum and design polar code transmission scheme.Firstly,a distance spectrum search algorithm which can be implemented on the memory-limited computer is proposed this paper.Secondly,relied on the distance spectrum,the theoretical performance bound of polar code is presented so as to analyzing the performance of polar code.Then from the point of performance bound,the performance of non-systematic polar code(NSPC)and systematic polar code(SPC)which is constructed by different kernel matrixes is analyzed and compared.Finally,aiming at the polar encoding transmission in inter-symbol interference(ISI)channel,the joint receiving scheme between equalization and polar decoding is designed.(2)Based on the distance spectrum and extrinsic information transfer(EXIT)chart tools,the parallel concatenated systematic polar code(PCSPC)is analyzed and designed.In light of the theory of parallel concatenated code,this paper analyzes the distance spectrum of PCSPC.Then,based on the distance spectrum of PCSPC and union bound,the performance bound of PCSPC is proposed and the performance of PCSPC is analyzed.Afterwards,the convergence of iterative decoding structure of PCSPC is analyzed relied on the EXIT chart.For the performance optimization of PCSPC,two scaling factor(SF)optimization schemes are presented,one is based on simulated EXIT(S-EXIT)chart and the other one is based on minimum weighted mean square error(MWMSE)criteria.Finally,this paper proposes a soft list successive cancellation(SSCL)decoding scheme to improve the performance of PCSPC.(3)Based on the distance spectrum and EXIT chart tools,the three-dimensional polar code(3D-PC)scheme is analyzed and designed.Firstly,in order to lower the error floor of PCSPC,a 3D-PC scheme is proposed in this paper.Secondly,the equation of the distance spectrum of 3D-PC is analyzed and derived.Then,based on th e distance spectrum of 3D-PC,the relationship between the minimum distance and the permeability of 3D-PC is analyzed.The study indicates that,the larger the permeability,the lager the minimum distance,the lower error floor and 3D-PC has larger minimum distance compared with the PCSPC.At the same time,for the iterative decoding structure of 3D-PC,this paper utilizes the EXIT chart to analyze the relationship between convergence threshold and permeability.Analysis shows that,the larger the permeability,the larger the convergence threshold and the poorer waterfall performance.Finally,considering the relationship between distance and convergence threshold with the permeability,the permeability is optimized and designed so as to reaching the balance between the performance of error floor and waterfall performance.The simulation result shows that the error floor of PCSPC can be lowered by the 3D-PC at the cost of little complexity.