Polar Codes Theory And Applications

Many good performance channel codes have been proposed since the significant theory by Shannon in the past60years for achieving channel capacity. The polar code made by Arikan is the first code that can reach channel capacity of a general discrete memory-less symmetric channel, which is fully discussed by scholar after publish. In this paper, mainly discuss the principle of channel polarization, polar code encoder and decoder, the rate of polar code, arbitrary polar matrices and the applications. To understand polar code, parallel channel interpretation is proposed, and a Fano-like decoder is mentioned based on the likelihood ratio various after the wrong historical information bit. For multilevel parallel channel, an equal distribution principle is discussed.Firstly, we discuss the channel cutoff rate improvement aspect which is the early work of Arikan before finding polar code. After proving the combining and slipping the channel into correlation subchannel can lead to channel polarization which means the channel become either noiseless or noisy with the probability of I(W) noiseless channel. We propose a parallel interpretation to verify the polarization phenomenon and give a simplify prove of the relationship between polarizing matrix and polar rate based on the study of Arikan and Korada. Polar code is depend on the channel condition which is not like general code, we use factor graph and density evolution on the construction of polar code to build a code under Gaussian channel. For AWGN case, we can use arbitrary input polar code to approach capacity. To improve the performance of SC decoder, list or stack decoder is mentioned to avoid bit error, based on this, we discuss the influence of error information bit and likelihood ratio on parallel channel interpretation and a Fano-like decoder can be used to avoid error at high SNR. At last, we talk something about the application of polar code in source coding and multilevel channel polarization, giving a equal distribution principle under asymptotic analysis, and a little discussion on polar code retransmission.Finally, comparing with FFT, we give a code field existing and looking at the channel as a low pass filter guess and talk about some open problem and future study on polar code.