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

TL;DR

This paper analyzes the asymptotic behavior of polarized channels in polar codes, revealing how error probabilities scale with blocklength and rate, and establishing results for both successive cancellation and MAP decoding.

Contribution

It provides a detailed asymptotic analysis of the polarization process for polar codes, including error probability scaling laws for large blocklengths and different decoding methods.

Findings

Error probabilities scale as a function of blocklength and rate.

Results apply to both successive cancellation and MAP decoding.

Provides precise asymptotic formulas involving normal distribution quantiles.

Abstract

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 $R<I(W)$ under successive cancellation decoding, then $\log(-\log(\mathbb{P}_e ^{\text{SC}}(n,R)))$ scales as $\frac{n}{2}+\sqrt{n}\frac{Q^{-1}(\frac{R}{I(W)})}{2}+ o(\sqrt{n})$. We also prove that the same result holds for the block error probability using the MAP decoder, i.e., for $\log(-\log(\mathbb{P}_e ^{\text{MAP}}(n,R)))$.