Research On Construction Theory Of High-Performance Polar Codes Based On The Polar Spectrum

Polar code,as the first class of constructive code that can theoretically achieve channel capacity,is another breakthrough achievement in the field of channel coding after the invention of Turbo code and LDPC code.With the application of polar codes in 5G eMBB scenarios,other typical scenarios of 5G/B5G also use polar code as a candidate for channel coding.The high-performance construction method of polar codes is one of the key prerequisites to ensure their wide applications,and the theoretical performance analysis of polar codes can provide an effective basis for the construction.However,at present,the construction and optimization of polar-coded transmission systems guided by theoretical performance analysis still lack an effective analysis tool and a systematic framework.The researches considering factors such as fading channel characteristics and signal mapping in non-orthogonal transmission systems are far from enough.On this account,this dissertation studies the construction theory of polar codes and its application in fading channel scenarios and non-orthogonal transmission systems.Starting from the algebraic coding structure of polar codes,we propose a tool named polar spectrum and establish a systematic framework to guide the construction of polar-coded transmission systems.Specifically,this dissertation includes the following four aspects of innovative work.1)This dissertation researches the theoretical performance analysis framework and construction methods of polar codes.First,we propose a new analysis tool entitled polar spectrum and design an efficient iterative enumeration algorithm to calculate the polar spectrum based on the algebraic structure of polar codes.Then,based on the polar spectrum,we establish a complete framework to analyze the theoretical performance of polar codes.Under the guidance of this framework,the union bound and union-Bhattacharyya(UB)bound on the error probability of polarized channels under successive cancellation(SC)decoding are deduced,revealing the influence of weight distribution of polar codes and channel parameters on the reliability of polarized channels.Based on the UB bound,we further design two explicit construction metrics:UB bound weight(UBW)and simplified UBW(SUBW).Finally,by extending the above theoretical analysis and construction method of polar codes to concatenated polar codes,this dissertation proposes the concept of polar coset spectrum and analyzes the error probability of polarization-adjusted convolutional codes under SC decoding to design a generalized construction metric named polar coset weight(PCW).Simulation results indicate that the proposed UBW,SUBW,and PCW constructions can outperform the traditional construction methods under the successive cancellation list(SCL)decoding.2)Based on the above work,this dissertation studies the theoretical performance and construction methods of polar codes under fast fading channels.First,according to the probability distribution characteristics of fading coefficients in fast fading channels,combined with the polar spectrum,the upper bound of the error probability of polarized channels and that of the block error rate(BLER)of polar codes under SC decoding are deduced,so that a systematic theoretical analysis framework under fast fading channels is established.Under the guidance of the framework,for three typical fast fading channels,i.e.,Rayleigh,Rician,and Nakagami,we establish the relationship between the fading parameters and the upper bound of error probability of polarized channels through simple analytical expressions.Then,we design two explicit construction metrics for the fast fading channels,namely the logarithmic upper-bound weight(LUW)and the minimum-weight LUW(MLUW).Since the polar spectrum can be calculated offline,these two constructions have a linear complexity.Simulation results show that the proposed LUW and MLUW metrics can achieve better performance than the traditional construction methods under SCL decoding.3)By extending the above innovative work to the block fading channel scenario,this dissertation analyzes and designs polar codes under the block fading channel.First,in the case of block mapping,we propose the concept of split polar spectrum by enumerating the weight distribution of codewords on each fading sub-block.A systematic framework is established to analyze and design polar codes under block fading channel based on the split polar spectrum.For the special case L=2 in the block mapping,we design an enumeration algorithm to exactly calculate the split polar spectrum.Then,for arbitrary diversity order,we approximate the split polar spectrum under random mapping and deduce the upper bound of error probability of polarized channels.Finally,we propose the design criteria to construct polar codes over the block fading channel,including the ful1 diversity criterion and the product distance criterion.Guided by these design criteria,the construction metrics,respectively named polarized diversity weight(PDW)and simplified PDW(SPDW)are proposed to design polar codes in both mappings.Simulation results indicate that the proposed PDW and SPDW metrics can achieve both diversity gain and coding gain compared to the GA method or the one based on the Bhattacharyya parameter.4)The dissertation researches the theoretical performance and construction methods of polar-coded non-orthogonal transmission systems.First,in the aspect of non-orthogonal multiple access,by decomposing the signal set in the multiuser codebook mapping space of the pattern division multiple access(PDMA)system,we analyze the upper bound of the error probability of each user’s polarized channels based on the polar spectrum,and construct a practical polar-coded PDMA(PC-PDMA)scheme.Compared with the PC-PDMA system designed through Monte-Carlo simulations,the proposed PC-PDMA system can achieve similar or better performance,while significantly reducing the construction complexity and processing delay.Then,in the aspect of non-orthogonal multicarrier modulation,we propose the "generalized modulation→bit" two-stage channel polarization transform framework.For the first stage,we propose two types of polar-coded generalized frequency division multiplexing(PC-GFDM)schemes,i.e.,MLC-PC-GFDM and BICM-PC-GFDM.The former is designed to optimize the system performance based on the multi-level coding(MLC)scheme.While the latter is proposed to reduce the processing latency and complexity by using the bit-interleaved coded modulation(BICM).Simulation results show that the proposed PC-GFDM systems can outperform Turbo-coded GFDM(TC-GFDM)systems.Overall,the proposed polar spectrum,as a potential theoretical tool,can be used to study the theoretical performance and construction methods of polar codes,concatenated polar codes,and polar-coded non-orthogonal transmission systems under different channel conditions,which builds a bridge between the weight distribution of polar codes and theoretical performance analysis.The corresponding construction based on polar spectrum reveals the essential factors of the reliability of polarized channels,provides a new design perspective for the construction of high-performance polar codes and promotes the related research directions.