Codes on Graphs and More

Modern communication systems strive to achieve reliable and efficient information transmission and storage with affordable complexity. Hence, efficient low-complexity channel codes providing low probabilities for erroneous receptions are needed. Interpreting codes as graphs and graphs as codes opens new perspectives for constructing such channel codes. Low-density parity-check (LDPC) codes are one of the most recent examples of codes defined on graphs, providing a better bit error probability than other block codes, given the same decoding complexity. After an introduction to coding theory, different graphical representations for channel codes are reviewed. Based on ideas from graph theory, new algorithms are introduced to iteratively search for LDPC block codes with large girth and to determine their minimum distance. In particular, new LDPC block codes of different rates and with girth up to 24 are presented. Woven convolutional codes are introduced as a generalization of graph-based codes and an asymptotic bound on their free distance, namely, the Costello lower bound, is proven. Moreover, promising examples of woven convolutional codes are given, including a rate 5/20 code with overall constraint length 67 and free distance 120. The remaining part of this dissertation focuses on basic properties of convolutional codes. First, a recurrent equation to determine a closed form expression of the exact decoding bit error probability for convolutional codes is presented. The obtained closed form expression is evaluated for various realizations of encoders, including rate 1/2 and 2/3 encoders, of as many as 16 states. Moreover, MacWilliams-type identities are revisited and a recursion for sequences of spectra of truncated as well as tailbitten convolutional codes and their duals is derived. Finally, the dissertation is concluded with exhaustive searches for convolutional codes of various rates with either optimum free distance or optimum distance profile, extending previously published results

Orthogonal Time Frequency Space (OTFS) Modulation for Wireless Communications

The orthogonal time frequency space (OTFS) modulation is a recently proposed multi-carrier transmission scheme, which innovatively multiplexes the information symbols in the delay-Doppler (DD) domain instead of the conventional time-frequency (TF) domain. The DD domain symbol multiplexing gives rise to a direct interaction between the DD domain information symbols and DD domain channel responses, which are usually quasi-static, compact, separable, and potentially sparse. Therefore, OTFS modulation enjoys appealing advantages over the conventional orthogonal frequency-division multiplexing (OFDM) modulation for wireless communications. In this thesis, we investigate the related subjects of OTFS modulation for wireless communications, specifically focusing on its signal detection, performance analysis, and applications. In specific, we first offer a literature review on the OTFS modulation in Chapter~1. Furthermore, a summary of wireless channels is given in Chapter 2. In particular, we discuss the characteristics of wireless channels in different domains and compare their properties. In Chapter 3, we present a detailed derivation of the OTFS concept based on the theory of Zak transform (ZT) and discrete Zak transform (DZT). We unveil the connections between OTFS modulation and DZT, where the DD domain interpretations of key components for modulation, such as pulse shaping, and matched-filtering, are highlighted. The main research contributions of this thesis appear in Chapter 4 to Chapter 7. In Chapter 4, we introduce the hybrid maximum a posteriori (MAP) and parallel interference cancellation (PIC) detection. This detection approach exploits the power discrepancy among different resolvable paths and can obtain near-optimal error performance with a reduced complexity. In Chapter 5, we propose the cross domain iterative detection for OTFS modulation by leveraging the unitary transformations among different domains. After presenting the key concepts of the cross domain iterative detection, we study its performance via state evolution. We show that the cross domain iterative detection can approach the optimal error performance theoretically. Our numerical results agree with our theoretical analysis and demonstrate a significant performance improvement compared to conventional OTFS detection methods. In Chapter 6, we investigate the error performance for coded OTFS systems based on the pairwise-error probability (PEP) analysis. We show that there exists a fundamental trade-off between the coding gain and the diversity gain for coded OTFS systems. According to this trade-off, we further provide some rule-of-thumb guidelines for code design in OTFS systems. In Chapter 7, we study the potential of OTFS modulation in integrated sensing and communication (ISAC) transmissions. We propose the concept of spatial-spreading to facilitate the ISAC design, which is able to discretize the angular domain, resulting in simple and insightful input-output relationships for both radar sensing and communication. Based on spatial-spreading, we verify the effectiveness of OTFS modulation in ISAC transmissions and demonstrate the performance improvements in comparison to the OFDM counterpart. A summary of this thesis is presented in Chapter 8, where we also discuss some potential research directions on OTFS modulation. The concept of OTFS modulation and the elegant theory of DD domain communication may have opened a new gate for the development of wireless communications, which is worthy to be further explored