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

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

2010-01-15

arxiv.org/abs/1001.2766v3

Computer Science

Information Theory

Submitted to ISIT 2010

Scientific paper

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 $\mathbb{P}(Z_n \leq z)$, where $Z_n=Z(W_n)$ 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 $\lim_{n \to \infty} \mathbb{P} (Z_n \leq 2^{-2^{\frac{n}{2}+\sqrt{n} \frac{Q^{-1}(\frac{R}{I(W)})}{2} +o(\sqrt{n})}}) = R$ and for $R<1- I(W)$ we have $\lim_{n \to \infty} \mathbb{P} (Z_n \geq 1-2^{-2^{\frac{n}{2}+ \sqrt{n} \frac{Q^{-1}(\frac{R}{1-I(W)})}{2} +o(\sqrt{n})}}) = R$, 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 $\mathbb{P}_e ^{\text{SC}}(n,R)$ the probability of error using polar codes of block-length $N=2^n$ and rate $R

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