291 research outputs found
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
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