An algebraic approach to physical-layer network coding

Abstract

This dissertation studies a family of the state-of-the-art PNC schemes called compute-and-forward (C&F). C&F was originally proposed and studied from an information-theoretic perspective. As such, it typically relies on several strong assumptions: very long block length, almost unbounded complexity, perfect channel state information, and no decoding errors at the relays; its benefit is often analyzed for simple network configurations. The aim of this dissertation is two-fold: first, to relax the above assumptions while preserving the performance of C&F, and second, to understand the benefit of C&F in more realistic network scenarios, such as random-access wireless networks.There are four main results in this dissertation. First, an algebraic framework is developed, which establishes a direct connection between C&F and module theory. This connection allows us to systematically design lattice codes for C&F with controlled block length and complexity. In particular, explicit design criteria are derived, concrete design examples are provided, and it is shown that nominal coding gains from 3 to 7.5 dB can be obtained with relatively short block length and reasonable decoding complexity. Second, a new C&F scheme is proposed, which, unlike conventional C&F schemes, does not require any channel state information (CSI). It is shown that this CSI-free scheme achieves, for a certain class of lattice codes, almost the same throughput as its CSI-enabled counterpart. Third, an end-to-end error control mechanism is designed, which effectively mitigates decoding errors introduced at wireless relays. In particular, the end-to-end error control problem is modeled as a finite-ring matrix channel problem, for which tight capacity bounds and capacity-approaching schemes are provided. The final part of this dissertation studies the benefit of C&F in random-access wireless networks. In particular, it is shown that C&F significantly improves the network throughput and delay performance of slotted-ALOHA-based random-access protocols.ii Acknowledgements This thesis is the result of the support, help, and advice that I have received from an amazing group of scholars and students at the University of Toronto. Without them this dissertation wouldn't have been possible. First, I feel very fortunate to have two great supervisors: Prof. Frank R. Kschischang and Prof.Baochun Li. As an advisor, Frank has consistently exceeded my expectation. He is not only a brilliant researcher and excellent teacher, but also truly a kind and caring person. He has given me a tremendous amount in the past six years, I can only thank him for a subset. In particular, I have benefited greatly from his focus on fundamental problems, his taste for beautiful mathematics, his emphasis on clarity and grace, and his dedication in developing his students to their full potential. I am very grateful to Frank for making my PhD journey so exciting and rewarding! Frank also has a great sense of humor.I will always remember how much fun we had together during our endless discus...

Performance Analysis of Csma With Multi-Packet Reception: The Inhomogeneous Case

“…Also, µ(L, P e ) is the normalized volume-to-noise ratio (which is closely related to the coding gain) for a given error probability P e in nearest neighbor decoding of the lattice L. Note that (22) and (23) apply to an arbitrary lattice code instead of asymptotically-good lattice codes, where log (2πeG(L ′ )) and log…”

“…As a new MPR technique, C&F enables a receiver to recover simultaneously transmitted packets via decoding linear equations [17][18][19][20][21][22]. Compared to other MPR techniques, C&F achieves close-to-optimal performance [23,24] under single-user decoding, making it particularly attractive for practical applications.…”

Physical Layer Network Coding Based on Integer Forcing Precoded Compute and Forward

“…In order to compare with the existent schemes, the baseline lattice network coding (LNC) scheme, which is used to practically implement CF in [3], is considered. To make a fair comparison, we have adjusted the power of the transmitters in the schemes with and without precoding, such that the total power available at the transmitters in both schemes is equal.…”

“…They have shown that the degrees of freedom (DoF) achievable by the lattice-based implementation of CF is at most two, when the channel gains are irrational. This is due to the gap between the natural channel gains and the integer coefficient of the linear combination in coherence to the observation in [3]. Further, it is proved that this limitation is not inherent to the fundamental concept of CF, but is due to the lattice-based implementation of CF.…”

“…An algebraic approach to implement CF has been introduced in [3]. In this work, the authors have related the approach introduced by Nazer and Gastpar to the fundamental theorem of finitely generated modules over the principal ideal domain (PID).…”