Complex Analysis of Channel Polarization on discrete BMS Channels

2026-05-05

AI Summary

This work addresses the lack of precise analytical characterization of polarization for discrete binary-input memoryless symmetric (BMS) channels at finite block lengths. The authors propose a Component Evolution (CE) framework that, for the first time, adopts a complex-analytic perspective by treating the Bhattacharyya parameter as a real-valued instance of a complex-valued channel functional. Within this framework, they systematically derive closed-form expressions for the Bhattacharyya parameters of bit-channels at arbitrary polarization levels. The approach reveals the extremal roles of the binary erasure channel (BEC) and binary symmetric channel (BSC) in the polarization process and uncovers a novel recursive structure specific to BSC-derived bit-channels. This provides a precise and unified analytical tool for understanding polarization across general BMS channels.

Abstract

We develop component evolution (CE), a complex-analytic framework 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) through the lens of complex analysis, and revealing new channel-dependent recursions for a class of BSC bit-channels.

Problem

• channel polarization
• BMS channels
• Bhattacharyya parameter
• finite-blocklength
• bit-channels

Innovation

• component evolution
• complex analysis
• channel polarization
• Bhattacharyya parameter
• BMS channels