Complex Analysis of Channel Polarization on Discrete BMS Channels

TL;DR

This paper introduces a complex function theory framework called component evolution (CE) to analyze finite-blocklength channel polarization on discrete BMS channels, providing new analytic tools and structural insights.

Contribution

The paper develops CE, a novel complex analysis framework that derives explicit expressions and structural properties for channel polarization on BMS channels.

Findings

CE provides analytic expressions for Bhattacharyya parameters at any polarization level.

CE offers new evidence of extremality for BEC and BSC channels.

Reveals new channel-dependent recursions for BSC bit-channels.

Abstract

We develop component evolution (CE), a framework based on complex function theory for finite-blocklength channel polarization on discrete binary-input memoryless output-symmetric (BMS) channels. In this view, the Bhattacharyya parameter is treated as a real-valued instance of a broader class of complex-valued channel functionals. CE systematically derives analytic expressions for the Bhattacharyya parameters of the bit-channels of a given discrete BMS channel at arbitrary polarization levels. CE also enables structural analysis, providing new evidence of extremality of the binary erasure channel (BEC) and binary symmetric channel (BSC), and revealing new channel-dependent recursions for a class of BSC bit-channels.