Channel Polarization and Polar Codes; Capacity Achieving

A new proposed method for constructing codes that achieves the symmetric capacity (the capacity of the channel with the same probabilities for the inputs), ( ) I W , of any Binary Discrete Memoryless Channel W (B-DMC) will be introduced in this tutorial. This method is based on channel polarization. Channel polarization is the fact that we can synthesize N channels, ( ) { } :1 i N W i N ≤ ≤ , out of N independent copies of DMC channels W, such that if N goes to infinity, the fraction of indices for which ( ) ( ) i N I W is around 1 approaches ( ) I W and the fraction of indices for which ( ) ( ) i N I W is around 0 approaches ( ) 1 I W − . We can clearly see some sort of polarization that can help us to construct such codes that improve the performance in terms of probability of block error; we send data through the channels with ( ) ( ) i N I W around 1 with rate 1 and with rate 0 through the rest of channels. The whole idea of channel polarization and polar codes can be stated in one precise theorem: Given any B-DMC W with ( ) 0 I W > and any target rate ( ) R I W < , there exists a sequence of polar codes { } ; 1 n n ≥ C such that n C has block-length 2 N = , rate R ≥ , the probability of block error under successive cancellation decoding bounded as ( ) 1 4 , e P N R O N −   ≤     independently of the code rate. This performance is achievable encoder and decoder complexity of ( ) log O N N for each.