The availability of location estimates of mobile devices (MDs) is vital for several important applications such as law enforcement, disaster management, battlefield operations, vehicular communication, traffic safety, emergency response, and preemption. While global positioning system (GPS) is usually sufficient in outdoor clear sky conditions, its functionality is limited in urban canyons and indoor locations due to the absence of clear line-of-sight between the MD to be localized and a sufficient number of navigation satellites. In such scenarios, the ubiquitous nature of cellular networks makes them a natural choice for localization of MDs. Traditionally, localization in cellular networks has been studied using system level simulations by fixing base station (BS) geometries. However, with the increasing irregularity of the BS locations (especially due to capacity-driven small cell deployments), the system insights obtained by considering simple BS geometries may not carry over to real-world deployments. This necessitates the need to study localization performance under statistical (random) spatial models, which is the main theme of this work. In this thesis, we use powerful tools from stochastic geometry and point process theory to develop a tractable analytical model to study the localizability (ability to get a location fix) of an MD in single-tier and heterogeneous cellular networks (HetNets). More importantly, we study how availability of information about the location of proximate BSs at the MD impacts localizability. To this end, we derive tractable expressions, bounds, and approximations for the localizability probability of an MD. These expressions depend on several key system parameters, and can be used to infer valuable system insights. Using these expressions, we quantify the gains achieved in localizability of an MD when information about the location of proximate BSs is incorporated in the model. As expected, our results demonstrate that localizability improves with the increase in density of BS deployments. / Master of Science / Location based services form an integral part of vital day-to-day applications such as traffic control, emergency response, and navigation. Traditionally, users have relied on the global positioning system system (GPS) for localizing a device. GPS systems rely on the availability of clear line-of-sight between the devices to be localized and a sufficient number of navigation satellites. Since it is not possible to have these line-of-sight links, especially in urban canyons and indoor locations, the ubiquity of cellular networks makes them a natural choice for localization. Typically, localization using cellular networks is studied using simulations, which are carried out by fixing the network configuration including the geometry of the base stations (BSs) as well as the number of BSs that participate in localization. This limits the scope of the results obtained since a change in the network configuration would mean that one must do another set of time consuming simulations with the new network parameters. This motivates the need to develop an analytical model to study the impact of fundamental system-design factors such as BS geometries, number of participating BSs, propagation effects, and channel conditions on localization in cellular networks. Such analysis would make it convenient to infer how changing these system parameters affects localization. In this thesis, we develop a general analytical model to study the localizability (ability of get a location fix) of a device in a cellular network. In particular, we study how information about the location of BSs in the proximity of the device to be localized affects localizability. We derive expressions for metrics such as the localizability probability of a device. Our results help quantify the gains achieved in localizability performance when information about the location of BSs in the vicinity of the device to be localized is available at the device. Our results concretely demonstrate that including this additional information significantly improves the localizability performance, especially in regions with dense BS deployments.

The past two decades have seen a surge in the number of applications requiring precise positioning data. Modern cellular networks offer many services based on the user's location, such as emergency services (e.g., E911), and emerging wireless sensor networks are being used in applications spanning environmental monitoring, precision agriculture, warehouse and manufacturing logistics, and traffic monitoring, just to name a few. In these sensor networks in particular, obtaining precise positioning data of the sensors gives vital context to the measurements being reported. While the Global Positioning System (GPS) has traditionally been used to obtain this positioning data, the deployment locations of these cellular and sensor networks in GPS-constrained environments (e.g., cities, indoors, etc.), along with the need for reliable positioning, requires a localization scheme that does not rely solely on GPS. This has lead to localization being performed entirely by the network infrastructure itself, or by the network infrastructure aided, in part, by GPS. In the literature, benchmarking localization performance in these networks has traditionally been done in a deterministic manner. That is, for a fixed setup of anchors (nodes with known location) and a target (a node with unknown location) a commonly used benchmark for localization error, such as the Cramer-Rao lower bound (CRLB), can be calculated for a given localization strategy, e.g., time-of-arrival (TOA), angle-of-arrival (AOA), etc. While this CRLB calculation provides excellent insight into expected localization performance, its traditional treatment as a deterministic value for a specific setup is limited. Rather than trying to gain insight into a specific setup, network designers are more often interested in aggregate localization error statistics within the network as a whole. Questions such as: "What percentage of the time is localization error less than x meters in the network?" are commonplace. In order to answer these types of questions, network designers often turn to simulations; however, these come with many drawbacks, such as lengthy execution times and the inability to provide fundamental insights due to their inherent ``block box'' nature. Thus, this dissertation presents the first analytical solution with which to answer these questions. By leveraging tools from stochastic geometry, anchor positions and potential target positions can be modeled by Poisson point processes (PPPs). This allows for the CRLB of position error to be characterized over all setups of anchor positions and potential target positions realizable within the network. This leads to a distribution of the CRLB, which can completely characterize localization error experienced by a target within the network, and can consequently be used to answer questions regarding network-wide localization performance. The particular CRLB distribution derived in this dissertation is for fourth-generation (4G) and fifth-generation (5G) sub-6GHz networks employing a TOA localization strategy. Recognizing the tremendous potential that stochastic geometry has in gaining new insight into localization, this dissertation continues by further exploring the union of these two fields. First, the concept of localizability, which is the probability that a mobile is able to obtain an unambiguous position estimate, is explored in a 5G, millimeter wave (mm-wave) framework. In this framework, unambiguous single-anchor localization is possible with either a line-of-sight (LOS) path between the anchor and mobile or, if blocked, then via at least two NLOS paths. Thus, for a single anchor-mobile pair in a 5G, mm-wave network, this dissertation derives the mobile's localizability over all environmental realizations this anchor-mobile pair is likely to experience in the network. This is done by: (1) utilizing the Boolean model from stochastic geometry, which statistically characterizes the random positions, sizes, and orientations of reflectors (e.g., buildings) in the environment, (2) considering the availability of first-order (i.e., single-bounce) reflections as well as the LOS path, and (3) considering the possibility that reflectors can either facilitate or block reflections. In addition to the derivation of the mobile's localizability, this analysis also reveals that unambiguous localization, via reflected NLOS signals exclusively, is a relatively small contributor to the mobile's overall localizability. Lastly, using this first-order reflection framework developed under the Boolean model, this dissertation then statistically characterizes the NLOS bias present on range measurements. This NLOS bias is a common phenomenon that arises when trying to measure the distance between two nodes via the time delay of a transmitted signal. If the LOS path is blocked, then the extra distance that the signal must travel to the receiver, in excess of the LOS path, is termed the NLOS bias. Due to the random nature of the propagation environment, the NLOS bias is a random variable, and as such, its distribution is sought. As before, assuming NLOS propagation is due to first-order reflections, and that reflectors can either facilitate or block reflections, the distribution of the path length (i.e., absolute time delay) of the first-arriving multipath component (MPC) is derived. This result is then used to obtain the first NLOS bias distribution in the localization literature that is based on the absolute delay of the first-arriving MPC for outdoor time-of-flight (TOF) range measurements. This distribution is shown to match exceptionally well with commonly assumed gamma and exponential NLOS bias models in the literature, which were only attained previously through heuristic or indirect methods. Finally, the flexibility of this analytical framework is utilized by further deriving the angle-of-arrival (AOA) distribution of the first-arriving MPC at the mobile. This distribution gives novel insight into how environmental obstacles affect the AOA and also represents the first AOA distribution, of any kind, derived under the Boolean model. In summary, this dissertation uses the analytical tools offered by stochastic geometry to gain new insights into localization metrics by performing analyses over the entire ensemble of infrastructure or environmental realizations that a target is likely to experience in a network. / Doctor of Philosophy / The past two decades have seen a surge in the number of applications requiring precise positioning data. Modern cellular networks offer many services based on the user's location, such as emergency services (e.g., E911), and emerging wireless sensor networks are being used in applications spanning environmental monitoring, precision agriculture, warehouse and manufacturing logistics, and traffic monitoring, just to name a few. In these sensor networks in particular, obtaining precise positioning data of the sensors gives vital context to the measurements being reported. While the Global Positioning System (GPS) has traditionally been used to obtain this positioning data, the deployment locations of these cellular and sensor networks in GPS-constrained environments (e.g., cities, indoors, etc.), along with the need for reliable positioning, requires a localization scheme that does not rely solely on GPS. This has lead to localization being performed entirely by the network infrastructure itself, or by the network infrastructure aided, in part, by GPS. When speaking in terms of localization, the network infrastructure consists of what are called anchors, which are simply nodes (points) with a known location. These can be base stations, WiFi access points, or designated sensor nodes, depending on the network. In trying to determine the position of a target (i.e., a user, or a mobile), various measurements can be made between this target and the anchor nodes in close proximity. These measurements are typically distance (range) measurements or angle (bearing) measurements. Localization algorithms then process these measurements to obtain an estimate of the target position. The performance of a given localization algorithm (i.e., estimator) is typically evaluated by examining the distance, in meters, between the position estimates it produces vs. the actual (true) target position. This is called the positioning error of the estimator. There are various benchmarks that bound the best (lowest) error that these algorithms can hope to achieve; however, these benchmarks depend on the particular setup of anchors and the target. The benchmark of localization error considered in this dissertation is the Cramer-Rao lower bound (CRLB). To determine how this benchmark of localization error behaves over the entire network, all of the various setups of anchors and the target that would arise in the network must be considered. Thus, this dissertation uses a field of statistics called stochastic geometry} to model all of these random placements of anchors and the target, which represent all the setups that can be experienced in the network. Under this model, the probability distribution of this localization error benchmark across the entirety of the network is then derived. This distribution allows network designers to examine localization performance in the network as a whole, rather than just for a specific setup, and allows one to obtain answers to questions such as: "What percentage of the time is localization error less than x meters in the network?" Next, this dissertation examines a concept called localizability, which is the probability that a target can obtain a unique position estimate. Oftentimes localization algorithms can produce position estimates that congregate around different potential target positions, and thus, it is important to know when algorithms will produce estimates that congregate around a unique (single) potential target position; hence the importance of localizability. In fifth generation (5G), millimeter wave (mm-wave) networks, only one anchor is needed to produce a unique target position estimate if the line-of-sight (LOS) path between the anchor and the target is unimpeded. If the LOS path is impeded, then a unique target position can still be obtained if two or more non-line-of-sight (NLOS) paths are available. Thus, over all possible environmental realizations likely to be experienced in the network by this single anchor-mobile pair, this dissertation derives the mobile's localizability, or in this case, the probability the LOS path or at least two NLOS paths are available. This is done by utilizing another analytical tool from stochastic geometry known as the Boolean model, which statistically characterizes the random positions, sizes, and orientations of reflectors (e.g., buildings) in the environment. Under this model, considering the availability of first-order (i.e., single-bounce) reflections as well as the LOS path, and considering the possibility that reflectors can either facilitate or block reflections, the mobile's localizability is derived. This result reveals the roles that the LOS path and the NLOS paths play in obtaining a unique position estimate of the target. Using this first-order reflection framework developed under the Boolean model, this dissertation then statistically characterizes the NLOS bias present on range measurements. This NLOS bias is a common phenomenon that arises when trying to measure the distance between two nodes via the time-of-flight (TOF) of a transmitted signal. If the LOS path is blocked, then the extra distance that the signal must travel to the receiver, in excess of the LOS path, is termed the NLOS bias. As before, assuming NLOS propagation is due to first-order reflections and that reflectors can either facilitate or block reflections, the distribution of the path length (i.e., absolute time delay) of the first-arriving multipath component (MPC) (or first-arriving ``reflection path'') is derived. This result is then used to obtain the first NLOS bias distribution in the localization literature that is based on the absolute delay of the first-arriving MPC for outdoor TOF range measurements. This distribution is shown to match exceptionally well with commonly assumed NLOS bias distributions in the literature, which were only attained previously through heuristic or indirect methods. Finally, the flexibility of this analytical framework is utilized by further deriving angle-of-arrival (AOA) distribution of the first-arriving MPC at the mobile. This distribution yields the probability that, for a specific angle, the first-arriving reflection path arrives at the mobile at this angle. This distribution gives novel insight into how environmental obstacles affect the AOA and also represents the first AOA distribution, of any kind, derived under the Boolean model. In summary, this dissertation uses the analytical tools offered by stochastic geometry to gain new insights into localization metrics by performing analyses over all of the possible infrastructure or environmental realizations that a target is likely to experience in a network.

Polychalcogenides REX₂₋δ (X=S, Se, Te; 0≤δ≤0.2) of trivalent rare earth metals RE have been investigated in recent years to shed light on the structural diversity as a function of compositional, metric, thermodynamic, and electronic situation. Whereas the former aspects have comparable influence on the structures of all polychalcogenides REX₂₋δ, the bonding situation was assumed different for tellurides due to tellurium's higher tendency to delocalize electrons. The crystal structures generally contain puckered [REX] double slabs and planar [X] layers, the latter hosting different distortions from a square-like arrangement. The distortion patterns of sulfides and selenides can be understood by a Zintl-type approach; they are dominated by localization of valence electrons in mono- (X²⁻) or dinuclear (X₂²⁻) anions only. This review discusses crystal structures of some rare-earth metal polytellurides RETe₂₋δ (0≤δ≤0.2) and bonding features in the chalcogenide layers and relates them to their sulfide and selenide counterparts.