An elementary description of polarization process

TL;DR

This paper provides an elementary proof of the polarization process in successive cancellation decoding by analyzing the moments of likelihood-related functions, offering insights into channel transformations and decoding complexity.

Contribution

It introduces a simple, elementary approach to understanding polarization in SC decoding using random functions and their moments, simplifying prior complex proofs.

Findings

Moments of likelihood functions are squared during channel transformations.

Product of these moments tends to zero as code length increases.

Decoding channels can be ordered with complexity of order n log n.

Abstract

We analyze successive cancellation (SC) decoder by using two random functions. The first function is related to the likelihoods of 0 and 1 in each code position, while the second gives the difference between their posterior probabilities. We then study the second power moments of both functions. We show that these moments are being squared in channel transformations, while their product tends to 0 for growing lengths $n$. This gives an elementary proof of polarization properties of SC decoding. We also derive a simple ordering of decoding channels with construction complexity of order $n\log n$.