Comparison Of Several Types Of Recursive Algorithms On Polarization Codes

Since Shannon proposed the modern error-correcting coding theory,searching for the error-correcting code that approximates or even reaches channel capacity has been the goal of this field.From Reed-Solomon codes and Bose-Chaudhuri-Hocquengherm codes,which were first discovered,to Turbo codes and low density parity check codes,which are widely used now,these codes are gradually approaching the channel capacity in terms of performance.However,polar codes,proposed by Turkish Professor Arikan in 2009,are the first ones that can be strictly proven to achieve the channel capacity,which is a major breakthrough in channel coding field.Polar codes have drawn the attention of scholars since they were proposed,and they have been successfully adopted in the coding scheme of 5G communication.This paper mainly deals with the research on polar codes and their decoding algorithms.Some theories involved in polar codes are briefly reviewed,such as channel polarization,channel combining,channel splitting,polar coding and so on.On the decoding of polar codes,we studied three basic algorithms,including successive cancellation(SC)decoding algorithm,successive cancellation list(SCL)decoding algorithm and belief propagation(BP)decoding algorithm.Firstly,the complexities of SC algorithm,SCL algorithm and BP algorithm are compared in this paper theoretically.The simulation analysis of the error correction performance of polar codes is also executed with the help of Matlab.In additive white Guassian noise channels with binary phase shift keying,we find that SCL algorithm performs better than SC algorithm at the same block length for different signal-to-noise ratios(SNRs).Simultaneously,the error correction performance of SCL algorithm improves,as the number of its lists increases.In the case of low SNRs,BP algorithm shows poor error correction performance.So we increase the SNR,and the performance of BP algorithm improves a lot.At the same time,we compare the differences among variant numbers of maximum iteration of BP algorithm.It is found that the performance of BP algorithm approaches when the number of iterations reaches a certain value.Comprehensively,SCL algorithm has the best performance.In addition,most studies on polar codes are developed based on the generator matrix of 2×2 kernel matrix at present.As a comparison,we also studied the polar codes with a generator matrix which is G3427.According to the analysis of the channel polarization process,the detailed algorithms of the polar codes corresponding to the 3×3 kernel matrix in the coding and decoding process are given.In the binary erasure channel,SC algorithm is used to compare the error correction performance of polar codes generated by different kernel matrices.Simulation results show that when the code rate is less than 0.15,the bit error rate(BER)of both codes increases rapidly with the increase of the code rate,and the polar codes generated by the 2x2 kernel matrix obviously perform better than those generated by the 3×3 kernel matrix.When the code rate is greater than 0.15,the increase of their BERs decreases with the increase of the code rate,and the performance gap betAveen the two types of codes gradually narrows and tends to be consistent.Lastly,the code length will also affect the error correction performance of these two kinds of codes.When the code length is close to each other,the polar codes generated by the 2×2 kernel matrix are better than those generated by the 3×3 kernel matrix.Although the polar codes generated by the 2×2 kernel matrix perform better than those generated by the 3x3 kernel matrix in most cases,there is no significant difference in error correction performance between the two types of codes when the code rate reaches 0.4.