10 research outputs found
Study of the operational SNR while constructing polar codes
Channel coding is commonly based on protecting information to be communicated across an unreliable medium, by adding patterns of redundancy into the transmission path. Also referred to as forward error control coding (FECC), the technique is widely used to enable correcting or at least detecting bit errors in digital communication systems. In this paper we study an original FECC known as polar coding which has proven to meet the typical use cases of the next generation mobile standard. This work is motivated by the suitability of polar codes for the new coming wireless era. Hence, we investigate the performance of polar codes in terms of bit error rate (BER) for several codeword lengths and code rates. We first perform a discrete search to find the best operating signal-to-noise ratio (SNR) at two different code rates, while varying the blocklength. We find in our extensive simulations that the BER becomes more sensitive to operating SNR (OSNR) as long as we increase the blocklength and code rate. Finally, we note that increasing blocklength achieves an SNR gain, while increasing code rate changes the OSNR domain. This trade-off sorted out must be taken into consideration while designing polar codes for high-throughput application
Polar Codes for Arbitrary Classical-Quantum Channels and Arbitrary cq-MACs
We prove polarization theorems for arbitrary classical-quantum (cq) channels.
The input alphabet is endowed with an arbitrary Abelian group operation and an
Ar{\i}kan-style transformation is applied using this operation. It is shown
that as the number of polarization steps becomes large, the synthetic
cq-channels polarize to deterministic homomorphism channels which project their
input to a quotient group of the input alphabet. This result is used to
construct polar codes for arbitrary cq-channels and arbitrary classical-quantum
multiple access channels (cq-MAC). The encoder can be implemented in operations, where is the blocklength of the code. A quantum successive
cancellation decoder for the constructed codes is proposed. It is shown that
the probability of error of this decoder decays faster than
for any .Comment: 30 pages. Submitted to IEEE Trans. Inform. Theory and in part to
ISIT201
Ergodic Theory Meets Polarization. I: An Ergodic Theory for Binary Operations
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 Ar{\i}kan 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 (first) part of the
paper introduces the mathematical framework that we will use in the second part
to characterize the polarizing operations. We define uniformity preserving,
irreducible, ergodic and strongly ergodic operations and we study their
properties. The concepts of a stable partition and the residue of a stable
partition are introduced. We show that an ergodic operation is strongly ergodic
if and only if all its stable partitions are their own residues. We also study
the products of binary operations and the structure of their stable partitions.
We show that the product of a sequence of binary operations is strongly ergodic
if and only if all the operations in the sequence are strongly ergodic. In the
second part of the paper, we provide a foundation of polarization theory based
on the ergodic theory of binary operations that we develop in this part.Comment: 34 pages, 1 figure. Accepted to IEEE Trans. Inform. Theory and
presented in part at ISIT'1
Fourier Analysis of MAC Polarization
A problem of the polar code construction for multiple access channels (MACs) is that they do not always achieve the whole capacity region. Although polar codes achieve the sum-capacity of symmetric MACs, polarization may induce a loss in the capacity region which prevents polar codes from achieving the whole capacity region. This paper provides a single letter necessary and sufficient condition which characterizes all the MACs that do not lose any part of their capacity region by polarization
Achieving the Fundamental Limit of Lossless Analog Compression via Polarization
In this paper, we study the lossless analog compression for i.i.d.
nonsingular signals via the polarization-based framework. We prove that for
nonsingular source, the error probability of maximum a posteriori (MAP)
estimation polarizes under the Hadamard transform, which extends the
polarization phenomenon to analog domain. Building on this insight, we propose
partial Hadamard compression and develop the corresponding analog successive
cancellation (SC) decoder. The proposed scheme consists of deterministic
measurement matrices and non-iterative reconstruction algorithm, providing
benefits in both space and computational complexity. Using the polarization of
error probability, we prove that our approach achieves the
information-theoretical limit for lossless analog compression developed by Wu
and Verdu.Comment: 48 pages, 5 figures. This work was presented in part at the 2023 IEEE
Global Communications Conferenc
Polarization and Channel Ordering: Characterizations and Topological Structures
Information theory is the field in which we study the fundamental limitations of communication. Shannon proved in 1948 that there exists a maximum rate, called capacity, at which we can reliably communicate information through a given channel. However, Shannon did not provide an explicit construction of a practical coding scheme that achieves the capacity. Polar coding, invented by Arikan, is the first low-complexity coding technique that achieves the capacity of binary-input memoryless symmetric channels. The construction of these codes is based on a phenomenon called polarization. The study of polar codes and their generalization to arbitrary channels is the subject of polarization theory, a subfield of information and coding theories. This thesis consists of two parts. In the first part, we provide solutions to several open problems in polarization theory. The first open problem that we consider is to determine the binary operations that always lead to polarization when they are used in Arikan-style constructions. In order to solve this problem, we develop an ergodic theory for binary operations. This theory is used to provide a necessary and sufficient condition that characterizes the polarizing binary operations, both in the single-user and the multiple-access settings. We prove that the exponent of a polarizing binary operation cannot exceed 1/2. Furthermore, we show that the exponent of an arbitrary quasigroup operation is exactly 1/2. This implies that quasigroup operations are among the best polarizing binary operations. One drawback of polarization in the multiple-access setting is that it sometimes induces a loss in the symmetric capacity region of a given multiple-access channel (MAC). An open problem in MAC polarization theory is to determine all the MACs that do not lose any part of their capacity region by polarization. Using Fourier analysis, we solve this problem by providing a single-letter necessary and sufficient condition that characterizes all these MACs in the general setting where we have an arbitrary number of users, and each user uses an arbitrary Abelian group operation on his input alphabet. We also study the polarization of classical-quantum (cq) channels. The input alphabet is endowed with an arbitrary Abelian group operation, and an Arikan-style transformation is applied using this operation. We show that as the number of polarization steps becomes large, the synthetic cq-channels polarize to deterministic homomorphism channels that project their input to a quotient group of the input alphabet. This result is used to construct polar codes for arbitrary cq-channels and arbitrary classical-quantum multiple-access channels (cq-MAC). In the second part of this thesis, we investigate several problems that are related to three orderings of communication channels: degradedness, input-degradedness, and the Shannon ordering. We provide several characterizations for the input-degradedness and the Shannon ordering. Two channels are said to be equivalent if they are degraded from each other. Input-equivalence and Shannon-equivalence between channels are similarly defined. We construct and study several topologies on the quotients of the spaces of discrete memoryless channels (DMC) by the equivalence, the input-equivalence and the Shannon-equivalence relations. Finally, we prove the continuity of several channel parameters and operations under various DMC topologies