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EPFL

1. Nasser, Rajai. Polarization and Channel Ordering: Characterizations and Topological Structures.

Degree: 2017, EPFL

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… Advisors/Committee Members: Telatar, Emre.

Subjects/Keywords: Polar codes; ergodic theory; quasigroup; multiple-access channels; Fourier transform; classical-quantum channels; channel ordering; input degradedness; Shannon ordering; topology.

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Nasser, R. (2017). Polarization and Channel Ordering: Characterizations and Topological Structures. (Thesis). EPFL. Retrieved from http://infoscience.epfl.ch/record/231813

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Nasser, Rajai. “Polarization and Channel Ordering: Characterizations and Topological Structures.” 2017. Thesis, EPFL. Accessed December 07, 2019. http://infoscience.epfl.ch/record/231813.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Nasser, Rajai. “Polarization and Channel Ordering: Characterizations and Topological Structures.” 2017. Web. 07 Dec 2019.

Vancouver:

Nasser R. Polarization and Channel Ordering: Characterizations and Topological Structures. [Internet] [Thesis]. EPFL; 2017. [cited 2019 Dec 07]. Available from: http://infoscience.epfl.ch/record/231813.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Nasser R. Polarization and Channel Ordering: Characterizations and Topological Structures. [Thesis]. EPFL; 2017. Available from: http://infoscience.epfl.ch/record/231813

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

2. Rajan, Adithya. On the Ordering of Communication Channels.

Degree: PhD, Electrical Engineering, 2014, Arizona State University

This dissertation introduces stochastic ordering of instantaneous channel powers of fading channels as a general method to compare the performance of a communication system over two different channels, even when a closed-form expression for the metric may not be available. Such a comparison is with respect to a variety of performance metrics such as error rates, outage probability and ergodic capacity, which share common mathematical properties such as monotonicity, convexity or complete monotonicity. Complete monotonicity of a metric, such as the symbol error rate, in conjunction with the stochastic Laplace transform order between two fading channels implies the ordering of the two channels with respect to the metric. While it has been established previously that certain modulation schemes have convex symbol error rates, there is no study of the complete monotonicity of the same, which helps in establishing stronger channel ordering results. Toward this goal, the current research proves for the first time, that all 1-dimensional and 2-dimensional modulations have completely monotone symbol error rates. Furthermore, it is shown that the frequently used parametric fading distributions for modeling line of sight exhibit a monotonicity in the line of sight parameter with respect to the Laplace transform order. While the Laplace transform order can also be used to order fading distributions based on the ergodic capacity, there exist several distributions which are not Laplace transform ordered, although they have ordered ergodic capacities. To address this gap, a new stochastic order called the ergodic capacity order has been proposed herein, which can be used to compare channels based on the ergodic capacity. Using stochastic orders, average performance of systems involving multiple random variables are compared over two different channels. These systems include diversity combining schemes, relay networks, and signal detection over fading channels with non-Gaussian additive noise. This research also addresses the problem of unifying fading distributions. This unification is based on infinite divisibility, which subsumes almost all known fading distributions, and provides simplified expressions for performance metrics, in addition to enabling stochastic ordering.

Subjects/Keywords: Electrical engineering; ergodic capacity; fading; Ordering channels; stochastic order; symbol error rate; wireless communication

…for ρ ≥ 0. Recall that by (2.19), LT ordering of the channels X and Y can be… …36 3.3 SER comparison of DPSK over Pareto-type channels… …39 3.4 Ergodic capacity comparison of Pareto-type channels… …40 3.5 SER comparison of BPSK over MRC Ricean channels… …41 3.6 SER comparison of BPSK over EGC Ricean channels… 

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Rajan, A. (2014). On the Ordering of Communication Channels. (Doctoral Dissertation). Arizona State University. Retrieved from http://repository.asu.edu/items/24798

Chicago Manual of Style (16th Edition):

Rajan, Adithya. “On the Ordering of Communication Channels.” 2014. Doctoral Dissertation, Arizona State University. Accessed December 07, 2019. http://repository.asu.edu/items/24798.

MLA Handbook (7th Edition):

Rajan, Adithya. “On the Ordering of Communication Channels.” 2014. Web. 07 Dec 2019.

Vancouver:

Rajan A. On the Ordering of Communication Channels. [Internet] [Doctoral dissertation]. Arizona State University; 2014. [cited 2019 Dec 07]. Available from: http://repository.asu.edu/items/24798.

Council of Science Editors:

Rajan A. On the Ordering of Communication Channels. [Doctoral Dissertation]. Arizona State University; 2014. Available from: http://repository.asu.edu/items/24798

.