Transmission lines, quantum graphs and fluctuations on complex networks

Abstract

High-frequency devices are commonplace and at their foundations often lie cable networks forming fundamental sub-systems with the primary role of transferring energy and information. With increasing demand for ”more electric” systems, the emerging trends in Internet of Things (IoT), as well as the surge in global mobile data traffic, the complexities of the underlying networks become more challenging to model deterministically. In such scenarios, statistical approaches are best suited because it becomes daunting to accurately model details of such networks. In this thesis, I present a quantum graph (QG) approach of modelling the transfer of energy and information through complex networks. In its own right, the graph model is used to predict the scattering spectrum in wired communications, as well as statistical predictions in wireless communication systems. I derive a more generalised vertex scattering matrix that takes into account cables of different characteristics connected at a common node. This is significant in real applications where different kinds of cables are connected. For example, in digital subscriber line (DSL) networks, the final loop drop may consist of cables with different characteristics. The proposed graph model is successfully validated both with a Transmission Line (TL) approach, and with laboratory experiments. The model agrees well with the universal predictions of Random Matrix Theory (RMT). In particular, the statistics of resonance is compared with the predictions of Weyl's law, while the level-spacing distribution is compared with the Wigner's surmise, which is a telltale signature of chaotic mixing of the underlying waves. In addition, I propose an analogue of the so-called random coupling model (RCM), which is important in the study of electromagnetic waves propagating in chaotic cavities. To achieve this, I present a procedure for symmetrising the transfer operator, which we use to modify the QG model in order for it to be comparable to RCM. Unlike the RCM which depends on Gaussian random variables, the graph model works for both Gaussian and non-Gaussian statistics. We use the analogue model to investigate the impact of different boundary conditions on the distribution of energy in networks with different topologies and connectivities. I further present a novel technique of using quantum graphs to predict the statistics of multi-antenna wireless communication systems. In this context, I present three different ways of calculating the probability density function of Shannon channel capacity. The analytical calculations compare favourably with numerical simulations of networks of varying complexities. In the area of wired communications, the graph model is used to model the distribution of data in G.fast networks (the fourth-generation Digital Subscriber Line (DSL) networks), using realistic cable parameters from the so-called TNO-Ericsson model. In particular, we show that quantum graph formalism can be used to simulate the in-premises distribution network at G.fast frequencies. The parameters of CAD$55$ (or B$05$a) cables and the in-house distribution network reported in the International Telecommunication Union documentation were used in the simulations