An Ergodic Theory of Binary Operations—Part II: Applications to Polarization
| Autor: | Rajai Nasser |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Discrete mathematics
Physics Sequence Open problem 020206 networking & telecommunications 02 engineering and technology Library and Information Sciences 01 natural sciences 010305 fluids & plasmas Computer Science Applications Binary operation 0103 physical sciences 0202 electrical engineering electronic engineering information engineering Exponent Ergodic theory Random variable Quasigroup Decoding methods Information Systems |
| Zdroj: | IEEE Transactions on Information Theory. 63:1063-1083 |
| ISSN: | 1557-9654 0018-9448 |
| DOI: | 10.1109/tit.2016.2617958 |
| Popis: | An open problem in polarization theory is to determine the binary operations that always lead to polarization (in the general multilevel sense) when they are used in Arikan style constructions. This paper, which is presented in two parts, solves this problem by providing a necessary and sufficient condition for a binary operation to be polarizing. This (second) part provides a foundation of polarization theory based on the ergodic theory of binary operations which we developed in the first part. We show that a binary operation is polarizing if and only if it is uniformity preserving and its right-inverse is strongly ergodic. The rate of polarization of single user channels is studied. It is shown that the exponent of any polarizing operation cannot exceed $\frac {1}{2}$ , which is the exponent of quasi-group operations. We also study the polarization of multiple access channels (MAC). In particular, we show that a sequence of binary operations is MAC-polarizing if and only if each binary operation in the sequence is polarizing. It is shown that the exponent of any MAC-polarizing sequence cannot exceed $\frac {1}{2}$ , which is the exponent of sequences of quasi-group operations. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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