Multilevel Polarization of Polar Codes Over Arbitrary Discrete Memoryless Channels

It is shown that polar codes achieve the symmetric capacity of discrete memoryless channels with arbitrary input alphabet sizes. It is shown that in general, channel polarization happens in several, rather than only two levels so that the synthesized channels are either useless, perfect or "partially perfect". Any subset of the channel input alphabet which is closed under addition, induces a coset partition of the alphabet through its shifts. For any such partition of the input alphabet, there exists a corresponding partially perfect channel whose outputs uniquely determine the coset to which the channel input belongs. By a slight modification of the encoding and decoding rules, it is shown that perfect transmission of certain information symbols over partially perfect channels is possible. Our result is general regarding both the cardinality and the algebraic structure of the channel input alphabet; i.e we show that for any channel input alphabet size and any Abelian group structure on the alphabet, polar codes are optimal. It is also shown through an example that polar codes when considered as group/coset codes, do not achieve the capacity achievable using coset codes over arbitrary channels

Asymptotic Distribution of Multilevel Channel Polarization for a Certain Class of Erasure Channels

This study examines multilevel channel polarization for a certain class of erasure channels that the input alphabet size is an arbitrary composite number. We derive limiting proportions of partially noiseless channels for such a class. The results of this study are proved by an argument of convergent sequences, inspired by Alsan and Telatar's simple proof of polarization, and without martingale convergence theorems for polarization process.Comment: 31 pages; 1 figure; 1 table; a short version of this paper has been submitted to the 2018 IEEE International Symposium on Information Theory (ISIT2018

Countably Infinite Multilevel Source Polarization for Non-Stationary Erasure Distributions

Polar transforms are central operations in the study of polar codes. This paper examines polar transforms for non-stationary memoryless sources on possibly infinite source alphabets. This is the first attempt of source polarization analysis over infinite alphabets. The source alphabet is defined to be a Polish group, and we handle the Ar{\i}kan-style two-by-two polar transform based on the group. Defining erasure distributions based on the normal subgroup structure, we give recursive formulas of the polar transform for our proposed erasure distributions. As a result, the recursive formulas lead to concrete examples of multilevel source polarization with countably infinite levels when the group is locally cyclic. We derive this result via elementary techniques in lattice theory.Comment: 12 pages, 1 figure, a short version has been accepted by the 2019 IEEE International Symposium on Information Theory (ISIT2019

Performance of the coherent optical binary polarization-shift-keying heterodyne system in free space optical communications using a lognormal atmospheric turbulence model MIC-CPE 2010

In this paper, simulation results for the bit error rate (BER) performance and fading penalty of a coherent optical binary polarization shift keying (2PolSK) heterodyne system adopted for free space optical (FSO) communication links with a log-normal atmospheric turbulence model is presented. The conditional and unconditional BER expressions are derived, demonstrating the comprehensive similarity between the 2PolSK and binary frequency shift keying (2FSK) schemes with regards to the system sensitivity. The study shows that 2PolSK offers improved performance compared to the binary amplitude shift keying (2ASK

Resonant diffraction gratings with polarization-dependent efficiencies

Subwavelength-structures with different fill factors in the lateral dimensions result in unique phase shifts for the different polarization states of transmitted light.By using this additional degree of freedom for diffractive optical elements, we yield additional functionalities for compact optical systems with DOEs. As a fully operable example we present a binary subwavelength-grating which acts as a polarizing beamsplitter for TE- and TM-polarization over a wide range of incidence angles. We show our design approach, the manufacturing process with Soft-UV-Nanoimprint-Technology, as well as experimental results. We will also lay out possibilities for the design and application of multilevel polarization-functionalized gratings