On the scaling of polar codes: I. The behavior of polarized channels
S. Hamed Hassani, Rudiger L. Urbanke · 2010
We consider the asymptotic behavior of the polarization process for polar codes when the blocklength tends to infinity. In particular, we study the asymptotics of the cumulative distribution P(Zn≤ z), where Zn= Z(Wn) is the Bhattacharyya process, and its dependence on the rate of transmission R. We show that for a BMS channel W, for Rn→8P (Zn≤ 2-2n/2+√n(Q-1(R/I(W)/2)+o(√n))) = R and for Rn→8P (Zn≤ 2-2n/2+√n(Q-1(R/I(W)/2)+o(√n))) = R, where Q(x) is the probability that a standard normal random variable exceeds x. As a result, if we denote by PeSC(n,R) the probability of error using polar codes of block-length N = 2nand rate ReSC(n,R))) scales as n/2+√n(Q-1(R/I(W)/2)+o(√n)). We also prove that the same result holds for the block error probability using the MAP decoder, i.e., for log(-log(PeMAP(n,R))).