Theoretical Study Of The Decoding Of Polar Codes

Polar codes is a linear block codes that can reach the channel capacity,it has very high research value.The encoding and decoding complexity of polar codes is nearly linear,for polar codes of length N,it's complexity is O(N log N),these theoretical results hold for very long code lengths in conjunction with successive cancelation(SC)decoding algorithm.Polar codes' performance is good in long code lengths,but in short lengths,the performance is not as good as low-density parity-check codes(LDPC)and Turbo codes.In order to improve the performance of polar codes in finite code lengths,the scholars have proposed many effective decoding algorithms,including some excellent decoding algorithms that have been applied to LDPC codes.In this paper,the encoding and decoding of polar codes will be improved accordingly,that is to say,to improve the performance of Belief Propagation(BP)decoding algorithm and Bit Flipping(BF)decoding algorithm by doing some operation on polar codes,BF is also a kind of BP decoding algorithm,but it is only applicable to the binary symmetric channel(BSC)channel.The specific work of this article is as follows:First,in order to be more suitable to use belief propagation decoding algorithm for polar codes in the form of LDPC codes,the paper analyzes the reason of 4-cycles in generate matrix,explores and introduces the method to eliminate 4-cycles by expansion.The study found that there are many forms of matrix expansion,this paper only gives two of them.Compared the expanded check matrix and the original check matrix,the density of 1 is greatly reduced.Second,polar codes was originally introduced as non-systematic codes.In this paper,the non-systematic polar codes is transformed into systematic polar codes according to the method proposed by Arikan.In order to match the new extended check matrix and be convenient to perform decoding operation,there a further study of the expansion of the systematic polar codes,and in the same time,gives the extended law.Furthermore,there are partial check nodes with degree 1 in the extended parity check matrix,in this case,the corresponding rows and columns of these check nodes can be peeled,new results are obtained in the process,and it reduces the complexity of decoding.Finally,we observe the decoding performance of the extended and peeled polar codes in the Additive Gauss White Noise channel(AWGN)and the BF decoding performance in BSC by simulation.The SC decoding algorithm is a high performance decoding algorithm for polar codes,in order to observe the difference in performance between decoding algorithms,we give the SC decoding simulation result in the AWGN.