On the scaling of polar codes: I. The behavior of polarized channels

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(Z
n
≤ z), where Z
n
= Z(W
n
) is the Bhat-tacharyya process, and its dependence on the rate of transmission R. We show that for a BMS channel W, for R < I(W) we have lim
n→8
ℙ equations R and for R < 1 − I(W) we have
n→8
ℙ equations R, where Q(x) is the probability that a standard normal random variable exceeds x. As a result, if we denote by ℙ
^SC
e
(n,R) the probability of error using polar codes of block-length N = 2
ⁿ
and rate R < I(W) under successive cancellation decoding, then log(−log(ℙ
^SC
e
(n,R))) scales as equations. We also prove that the same result holds for the block error probability using the MAP decoder, i.e., for log(−log(ℙ
^MAP
e
(n,R))).