The advances in modern communication systems have brought about devices with more functionality, better performance, smaller size, lighter weight and lower cost. Meanwhile, the requirement for newer devices has become more demanding than ever. Tunability and flexibility are both long-desired features. Tunable devices are ‘smart’ in the sense that they can adapt to the dynamic environment or varying user demand as well as correct the minor deviations due to manufacturing fluctuations, therefore making it possible to reduce system complexity and overall cost. It is also desired that electronics be flexible to provide conformability and portability. Previously, tunable devices on flexible substrates have been realized mainly by dicing and assembling. This approach is straightforward and easy to carry out. However, it will become a “mission impossible” when it comes to assembling a large amount of rigid devices on a flexible substrate. Moreover, the operating frequency is often limited by the parasitic effect of the interconnection between the diced device and the rest of the circuit on the flexible substrate. A recent effort utilized a strain-sharing Si/SiGe/Si nanomembrane to transfer a device onto a flexible substrate. This approach works very well for silicon based devices with small dimensions, such as transistors and varactor diodes. Large-scale fabrication capability is still under investigation. A new transfer technique is proposed and studied in this research. Tunable BST (Barium Strontium Titanate) IDCs (inter-digital capacitors) are first fabricated on a silicon substrate. The devices are then transferred onto a flexible LCP (liquid crystalline polymer) substrate using iv wafer bonding of the silicon substrate to the LCP substrate, followed by silicon etching. This approach allows for monolithic fabrication so that the transferred devices can operate in millimeter wave frequency. The tunability, capacitance, Q factor and equivalent circuit are studied. The simulated and measured performances are compared. BST capacitors on LCP substrates are also compared with those on sapphire substrates to prove that this transfer process does not impair the performance. A primary study of a reflectarray antenna unit cell is also conducted for loss and phase swing performance. The BST thin film layout and bias line positions are studied in order to reduce the total loss. Transferring a full-size BST-based reflectarray antenna onto an LCP substrate is the ultimate goal, and this work is ongoing at the University of Central Florida (UCF). HFSS is used to simulate the devices and to prove the concept. All of the devices are fabricated in the clean room at UCF. Probe station measurements and waveguide measurements are performed on the capacitors and reflectarray antenna unit cells respectively. This work is the first comprehensive demonstration of this novel transfer technique.

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A..=2b, where A 2 Rnn is a large, sparse symmetric positive definite matrix and b 2 Rn is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LLT is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L..T z, with x = A..1=2z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form n = A..=2b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t..=2 and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A..=2b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.