Last-Pair Swapping Polar Codes: A Structure to Improve Polarization under Finite-State Modulation

TL;DR

This paper introduces a last-pair swapping structure for polar codes to enhance polarization efficiency under finite-state modulation, achieving performance comparable to memoryless channels and surpassing conventional codes under certain conditions.

Contribution

It proposes a novel last-pair swapping polar code structure that improves polarization under finite-state modulation, with proven theoretical advantages and validated through simulations.

Findings

Achieves polarization efficiency equal to conventional codes on memoryless channels.

Surpasses conventional polar codes under bijective finite-state modulation.

Demonstrates SNR gain in CPM with AWGN channels.

Abstract

A novel structure of polar codes is proposed for finite-state modulation (FSM), in order to improve polarization under it, and approach the polarization efficiency that conventional polar codes achieve under memoryless channels. We choose a particular class of FSM for research, termed bijective FSM, and observe an explicit polarization loss under bijective FSM. To eliminate the loss, we propose a novel polar coding structure by substituting the last kernel of each layer in polar coding structure with a swapping matrix, thereby termed last-pair swapping structure. We prove that under bijective FSM the proposed structure achieves identical polarization efficiency with that of conventional one on memoryless channels, and exceeds that of conventional one under bijective FSM. Furthermore, we give a plausible generalization of last-pair swapping polar code: on a broader class termed sub-injective FSM. Simulation corroborates that under sub-injective FSM polarization efficiency of the proposed polar code exceeds that of conventional one. And simulation results of error rate are given on continuous phase modulation (CPM) with additional white Gaussian noise (AWGN) channels, showing a considerable signal-to-noise power ratio (snr) gain of last-pair swapping polar code over conventional one, and identical performances between the proposed polar code under bijective FSM and conventional one on memoryless channels.