2010/01/15 by S. Hamed Hassani, Hassani, S. Hamed, Rudiger Urbanke +1
Computer Science · Mathematics · #FOS: Computer and information sciences #Information Theory (cs.IT) #cs.IT #math.IT
paper · pdf · doi:10.48550/arxiv.1001.2766
Submitted to ISIT 2010
arxiv created 2010/01/28 · arxiv updated 2010/02/26
We consider the asymptotic behavior of the polarization process for polar codes when the blocklength tends to infinity. In particular, we study the problem of asymptotic analysis of the cumulative distribution ℙ(Zn ≤ z), where Zn=Z(Wn) is the Bhattacharyya process, and its dependence to the rate of transmission R. We show that for a BMS channel W, for R < I(W) we have limn → ∞ ℙ (Zn ≤ 2^-2^(n)/(2)+√(n) \fracQ-1((R)/(I(W)))2 +o(√(n))) = R and for R<1- I(W) we have limn → ∞ ℙ (Zn ≥ 1-2^-2^(n)/(2)+ √(n) \fracQ-1((R)/(1-I(W)))2 +o(√(n))) = R, where Q(x) is the probability that a standard normal random variable will obtain a value larger than x. As a result, if we denote by ℙe SC(n,R) the probability of error using polar codes of block-length N=2n and rate R<I(W) under successive cancellation decoding, then log(-log(ℙe SC(n,R))) scales as (n)/(2)+√(n)\fracQ-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(ℙe MAP(n,R))).