Polarization Decomposition and Its Applications

TL;DR

This paper introduces a polarization decomposition framework for binary-input memoryless channels, defining a polarization factor that simplifies capacity analysis and enables efficient computation and visualization of subchannel capacities.

Contribution

The work presents a novel polarization factor and an associated algorithm, providing the first efficient method to compute symmetric capacities for all subchannels in arbitrary BMCs.

Findings

Explicit formulation of the polarization factor as a function of block length and subchannel index.

Development of an efficient algorithm for polarization factor computation.

Establishment of a one-to-one correspondence between the polarization factor and n-ary trees.

Abstract

The polarization decomposition of arbitrary binary-input memoryless channels (BMCs) is studied in this work. By introducing the polarization factor (PF), defined in terms of the conditional entropy of the channel output under various input configurations, we demonstrate that the symmetric capacities of the polarized subchannels can be uniformly expressed as functions of the PF. The explicit formulation of the PF as a function of the block length and subchannel index is derived. Furthermore, an efficient algorithm is proposed for the computation of the PF. Notably, we establish a one-to-one correspondence between each PF and an $n$-ary tree. Leveraging this tree structure, we develop a pruning method to determine the conditional entropy associated with different input relationships. The proposed polarization framework offers both theoretical insights and practical advantages, including intuitive visualization of polarization behavior and efficient polar code construction. To the best of our knowledge, this is the first approach that enables the efficient computation of symmetric capacities for all subchannels in arbitrary BMCs.