Spatially controlled relay beamforming
Creators: name: Kalogerias, Dionysios
Publisher: name: No Publisher Supplied
Year: 2017

description: This thesis is about fusion of optimal stochastic motion control and physical layer communications. Distributed, networked communication systems, such as relay beamforming networks (e.g., Amplify &amp; Forward (AF)), are typically designed without explicitly considering how the positions of the respective nodes might affect the quality of the communication. Optimum placement of network nodes, which could potentially improve the quality of the communication, is not typically considered. However, in most practical settings in physical layer communications, such as relay beamforming, the Channel State Information (CSI) observed by each node, per channel use, although it might be (modeled as) random, it is both spatially and temporally correlated. It is, therefore, reasonable to ask if and how the performance of the system could be improved by (predictively) controlling the positions of the network nodes (e.g., the relays), based on causal side (CSI) information, and exploitting the spatiotemporal dependencies of the wireless medium. In this work, we address this problem in the context of AF relay beamforming networks. This novel, cyber-physical system approach to relay beamforming is termed as “Spatially Controlled Relay Beamforming”.    First, we discuss wireless channel modeling, however, in a rigorous, Bayesian framework. Experimentally accurate and, at the same time, technically precise channel modeling is absolutely essential for designing and analyzing spatially controlled communication systems. In this work, we are interested in two distinct spatiotemporal statistical models, for describing the behavior of the log-scale magnitude of the wireless channel:    1. Stationary Gaussian Fields: In this case, the channel is assumed to evolve as a stationary, Gaussian stochastic field in continuous space and discrete time (say, for instance, time slots). Under such assumptions, spatial and temporal statistical interactions are determined by a set of time and space invariant parameters, which completely determine the mean and covariance of the underlying Gaussian measure. This model is relatively simple to describe, and can be sufficiently characterized, at least for our purposes, both statistically and topologically. Additionally, the model is rather versatile and there is existing experimental evidence, supporting its practical applicability. Our contributions are summarized in properly formulating the whole spatiotemporal model in a completely rigorous mathematical setting, under a convenient measure theoretic framework. Such framework greatly facilitates formulation of meaningful stochastic control problems, where the wireless channel field (or a function of it) can be regarded as a stochastic optimization surface.     2. Conditionally Gaussian Fields, when conditioned on a Markovian channel state: This is a completely novel approach to wireless channel modeling. In this approach, the communication medium is assumed to behave as a partially observable (or hidden) system, where a hidden, global, temporally varying underlying stochastic process, called the channel state, affects the spatial interactions of the actual channel magnitude, evaluated at any set of locations in the plane. More specifically, we assume that, conditioned on the channel state, the wireless channel constitutes an observable, conditionally Gaussian stochastic process. The channel state evolves in time according to a known, possibly non stationary, non Gaussian, low dimensional Markov kernel. Recognizing the intractability of general nonlinear state estimation, we advocate the use of grid based approximate nonlinear filters as an effective and robust means for recursive tracking of the channel state. We also propose a sequential spatiotemporal predictor for tracking the channel gains at any point in time and space, providing real time sequential estimates for the respective channel gain map. In this context, our contributions are multifold. Except for the introduction of the layered channel model previously described, this line of research has resulted in a number of general, asymptotic convergence results, advancing the theory of grid-based approximate nonlinear stochastic filtering. In particular, sufficient conditions, ensuring asymptotic optimality are relaxed, and, at the same time, the mode of convergence is strengthened. Although the need for such results initiated as an attempt to theoretically characterize the performance of the proposed approximate methods for statistical inference, in regard to the proposed channel modeling approach, they turn out to be of fundamental importance in the areas of nonlinear estimation and stochastic control. The experimental validation of the proposed channel model, as well as the related parameter estimation problem, termed as “Markovian Channel Profiling (MCP)”, fundamentally important for any practical deployment, are subject of current, ongoing research.    Second, adopting the first of the two aforementioned channel modeling approaches, we consider the spatially controlled relay beamforming problem for an AF network with a single source, a single destination, and multiple, controlled at will, relay nodes.     We consider a time slotted system, where the relays update their positions before the beginning of each time slot. Under a general, rigorous and theoretically grounded framework, based on a version of the so-called Fundamental Lemma of Stochastic Control, we propose a novel, 2-stage stochastic programming formulation for specifying both beamforming weights and relay positions at each time slot. The objective is to maximize the expected (long term) Quality-of-Service (QoS) of the network, at each time slot, based on causal Channel State Information (CSI), while respecting a total transmit power budget at the relays. The resulting motion control problem is shown to be equivalent to a set of much simpler, 2-dimensional subproblems, which may be solved in a distributed fashion, one at each relay. However, these problems are all nonconvex, and their objectives are impossible to evaluate analytically. Then, two methods are proposed, one based on the Method of Statistical Differentials, and one relying on the multidimensional Gauss-Hermite Quadrature Rule (brute force). Both methods allow approximate, closed form evaluation of the aforementioned objectives, enabling the use of any preferable nonlinear solver, thus allowing the determination of approximately optimal relay controls.    Additionally, we show analytically that, although positions are optimized myopically at each time slot (based on all available CSI, though), the average network QoS is nondecreasing across time slots, as long as the temporal dependence of the communication medium is sufficiently strong. Synthetic numerical simulations are presented, confirming our theoretical predictions and corroborating the efficacy of the proposed approach.     The extension of this formulation, when the second of the channel modeling approaches presented above is adopted, in which the channel is modeled as partially observable Markovian system, is nontrivial and constitutes a subject of further research.