In this thesis, we propose different CORDIC architectures which solve the problems of long-latency in traditional pipeline CORDIC and the large-area cost in table-based CORDIC. The original table-based CORDIC can be divided into two stages, coarse stage and fine stage. We also propose the three-stage architectures, composed of traditional pipeline CORDIC, Rom/Multiplier architecture and linear approximation. Detailed analysis and estimation in area and latency of these different two-stage and three-stage architectures with different bit accuracy are given in order to determine the best architecture design for a particular precision. Finally, we choose one of the architectures to implement, compare the results, and show its applications.

In this thesis, an efficiency method for reducing the rotation ROM size in table-based architecture is proposed. The original rotation can be divided into two stages, coarse stage and fine stage. Our approach modifies the previous two-stage rotation method and proposes a multi-stage architecture and discuses three-stage phase calculation. The effect of table reduction is more apparently for higher accuracy requirement in the three-stage architecture. The total area of the previous two-stage architecture is larger than the proposed table-reduced three-stage architecture because the table size takes a significant ratio of the total area especially when the required bit accuracy is large. In the proposed three-stage design, there are two different types of architectures, depending on the rotation angles in the first and second rotation stages. Comparison of different types of architecture with the previous method shows that our designs indeed reduce the table size and the total area significantly.

Background: Measures of central tendency are one of the foundational concepts of statistics, with the most commonly used measures being mean, median, and mode. While these are all very simple to calculate when data conform to a unimodal symmetric distribution, either discrete or continuous, measures of central tendency are more challenging to calculate for data distributed asymmetrically. There is a gap in the current statistical literature on computing median and mode for most skewed unimodal continuous distributions. For example, for a standardized normal distribution, mean, median, and mode are all equal to 0. The mean, median, and mode are all equal to each other. For a more general normal distribution, the mode and median are still equal to the mean. Unfortunately, the mean is highly affected by extreme values. If the distribution is skewed either positively or negatively, the mean is pulled in the direction of the skew; however, the median and mode are more robust statistics and are not pulled as far as the mean. The traditional response is to provide an estimate of the median and mode as current methodological approaches are limited in determining their exact value once the mean is pulled away. Methods: The purpose of this study is to test a new statistical method, utilizing the first order and second order partial derivatives in Taylor series expansion, for approximating the median and mode of skewed unimodal continuous distributions. Specifically, to compute the approximated mode, the first order derivatives of the sum of the first three terms in the Taylor series expansion is set to zero and then the equation is solved to find the unknown. To compute the approximated median, the integration from negative infinity to the median is set to be one half and then the equation is solved for the median. Finally, to evaluate the accuracy of our derived formulae for computing the mode and median of the skewed unimodal continuous distributions, simulation study will be conducted with respect to skew normal distributions, skew t-distributions, skew exponential distributions, and others, with various parameters. Conclusions: The potential of this study may have a great impact on the advancement of current central tendency measurement, the gold standard used in public health and social science research. The study may answer an important question concerning the precision of median and mode estimates for skewed unimodal continuous distributions of data. If this method proves to be an accurate approximation of the median and mode, then it should become the method of choice when measures of central tendency are required.

In this thesis we present results from quantum Monte Carlo for the fully-frustrated honeycomb lattice. The XXZ model is of interest in the classical limit, as there is a mapping between the classical fully-frustrated honeycomb Ising model groundstates and the classical hard-core dimer model groundstate. The aim of this work is to explore the effect of quantum fluctuations on the fully-frustrated honeycomb model to see what sort of interesting physics arises. One might expect unusual physics due to the quantum hard-core dimer model, where interesting physics are known to exist. This is because there is a duality mapping between the classical dimer model and the classical fully-frustrated honeycomb Ising model. Indeed, by studying the fully-frustrated honeycomb XXZ model we find that in some cases the system orders into crystal-like structures, a case of order-by-disorder. The most interesting case, when the frustrating bonds are chosen randomly, reveals to us a novel state without any discernible order while at the same time avoiding the freezing one would expect of a glass. This state is a featureless system lacking low temperature magnetic susceptibility---a candidate ``quantum spin liquid''. Future work that might more easily measure quantum spin liquid criteria is suggested.

In this thesis we present results from quantum Monte Carlo for the fully-frustrated honeycomb lattice. The XXZ model is of interest in the classical limit, as there is a mapping between the classical fully-frustrated honeycomb Ising model groundstates and the classical hard-core dimer model groundstate. The aim of this work is to explore the effect of quantum fluctuations on the fully-frustrated honeycomb model to see what sort of interesting physics arises. One might expect unusual physics due to the quantum hard-core dimer model, where interesting physics are known to exist. This is because there is a duality mapping between the classical dimer model and the classical fully-frustrated honeycomb Ising model. Indeed, by studying the fully-frustrated honeycomb XXZ model we find that in some cases the system orders into crystal-like structures, a case of order-by-disorder. The most interesting case, when the frustrating bonds are chosen randomly, reveals to us a novel state without any discernible order while at the same time avoiding the freezing one would expect of a glass. This state is a featureless system lacking low temperature magnetic susceptibility---a candidate ``quantum spin liquid''. Future work that might more easily measure quantum spin liquid criteria is suggested.

The Riemann zeta function ζ(s) is one of the most fundamental functions in number theory. Euler demonstrated that ζ(s) is closely connected to the prime numbers and Riemann gave proofs of the basic analytic properties of the zeta function. Values of the zeta function and its derivatives have been studied by several mathematicians. Apostol in particular gave a computable formula for the values of the derivatives of ζ(s) at s = 0. The Hurwitz zeta function ζ(s,q) is a generalization of ζ(s). We modify Apostolʼs methods to find values of the derivatives of ζ(s,q) with respect to s at s = 0. As a consequence, we obtain relations among certain important constants, the generalized Stieltjes constants. We also give numerical estimates of several values of the derivatives of ζ(s,q).

Surface electromyography records the motor unit action potential signals in the vicinity of the electrode to reveal information on muscle activation. Decomposition of sEMG signals for characterization of constituent motor unit action potentials in terms of amplitude and firing times is useful for clinical research as well as diagnosis of neurological disorders. Successful decomposition of sEMG signals would allow for pertinent motor unit action potential information to be acquired without discomfort to the subject or the need for a well-trained operator (compared with intramuscular EMG). To determine amplitudes and firing times for motor unit action potentials in an sEMG recording, Szlavik's perturbation based decomposition may be applied. The decomposition was initially applied to synthetic sEMG signals and then to experimental data collected from the biceps brachii. Szlavik's decomposition estimator yields satisfactory results for synthetic and experimental sEMG signals with reasonable complexity.

This thesis systematically derives an innovative approach, called FOMFE, for fingerprint ridge orientation modeling based on 2D Fourier expansions, and explores possible applications of FOMFE to various aspects of a fingerprint identification system. Compared with existing proposals, FOMFE does not require prior knowledge of the landmark singular points (SP) at any stage of the modeling process. This salient feature makes it immune from false SP detections and robust in terms of modeling ridge topology patterns from different typological classes. The thesis provides the motivation of this work, thoroughly reviews the relevant literature, and carefully lays out the theoretical basis of the proposed modeling approach. This is followed by a detailed exposition of how FOMFE can benefit fingerprint feature analysis including ridge orientation estimation, singularity analysis, global feature characterization for a wide variety of fingerprint categories, and partial fin gerprint identification. The proposed methods are based on the insightful use of theory from areas such as Fourier analysis of nonlinear dynamic systems, analytical operators from differential calculus in vector fields, and fluid dynamics. The thesis has conducted extensive experimental evaluation of the proposed methods on benchmark data sets, and drawn conclusions about strengths and limitations of these new techniques in comparison with state-of-the-art approaches. FOMFE and the resulting model-based methods can significantly improve the computational efficiency and reliability of fingerprint identification systems, which is important for indexing and matching fingerprints at a large scale.

Uncertainty analysis is an important part of system design. The formula for error propagation through a system model that is most-often cited in literature is based on a first-order Taylor series. This formula makes several important assumptions and has several important limitations that are often ignored. This thesis explores these assumptions and addresses two of the major limitations. First, the results obtained from propagating error through nonlinear systems can be wrong by one or more orders of magnitude, due to the linearization inherent in a first-order Taylor series. This thesis presents a method for overcoming that inaccuracy that is capable of achieving fourth-order accuracy without significant additional computational cost. Second, system designers using a Taylor series to propagate error typically only propagate a mean and variance and ignore all higher-order statistics. Consequently, a Gaussian output distribution must be assumed, which often does not reflect reality. This thesis presents a proof that nonlinear systems do not produce Gaussian output distributions, even when inputs are Gaussian. A second-order Taylor series is then used to propagate both skewness and kurtosis through a system model. This allows the system designer to obtain a fully-described non-Gaussian output distribution. The benefits of having a fully-described output distribution are demonstrated using the examples of both a flat rolling metalworking process and the propeller component of a solar-powered unmanned aerial vehicle.

This thesis studies the use of nonparametric econometric methods to reconcile the empirical behaviour of financial asset prices with theoretical valuation models. The confrontation of economic theory with asset price data requires various functional form assumptions about the preferences and beliefs of investors. Nonparametric methods provide a flexible class of models that can prevent misspecification of agents’ utility functions or the distribution of asset returns. Evidence for potential nonlinearity is seen in the presence of non-Gaussian distributions and excessive volatility of stock returns, or non-monotonic stochastic discount factors in option prices. More robust model specifications are therefore likely to contribute to risk management and return predictability, and lend credibility to economists’ assertions. Each of the chapters in this thesis relaxes certain functional form assumptions that seem most important for understanding certain asset price data. Chapter 1 focuses on the state-price density in option prices, which confounds the nonlinearity in both the preferences and the beliefs of investors. To understand both sources of nonlinearity in equity prices, Chapter 2 introduces a semiparametric generalization of the standard representative agent consumption-based asset pricing model. Chapter 3 returns to option prices to understand the relative importance of changes in the distribution of returns and in the shape of the pricing kernel. More specifically, Chapter 1 studies the use of noisy high-frequency data to estimate the time-varying state-price density implicit in European option prices. A dynamic kernel estimator of the conditional pricing function and its derivatives is proposed that can be used for model-free risk measurement. Infill asymptotic theory is derived that applies when the pricing function is either smoothly varying or driven by diffusive state variables. Trading times and moneyness levels are modelled by marked point processes to capture intraday trading patterns. A simulation study investigates the performance of the estimator using an iterated plug-in bandwidth in various scenarios. Empirical results using S&P 500 E-mini European option quotes finds significant time-variation at intraday frequencies. An application towards delta- and minimum variance-hedging further illustrates the use of the estimator. Chapter 2 proposes a semiparametric asset pricing model to measure how consumption and dividend policies depend on unobserved state variables, such as economic uncertainty and risk aversion. Under a flexible specification of the stochastic discount factor, the state variables are recovered from cross-sections of asset prices and volatility proxies, and the shape of the policy functions is identified from the pricing functions. The model leads to closed-form price-dividend ratios under polynomial approximations of the unknown functions and affine state variable dynamics. In the empirical application uncertainty and risk aversion are separately identified from size-sorted stock portfolios exploiting the heterogeneous impact of uncertainty on dividend policy across small and large firms. I find an asymmetric and convex response in consumption (-) and dividend growth (+) towards uncertainty shocks, which together with moderate uncertainty aversion, can generate large leverage effects and divergence between macroeconomic and stock market volatility. Chapter 3 studies the nonparametric identification and estimation of projected pricing kernels implicit in the pricing of options, the underlying asset, and a riskfree bond. The sieve minimum-distance estimator based on conditional moment restrictions avoids the need to compute ratios of estimated risk-neutral and physical densities, and leads to stable estimates even in regions with low probability mass. The conditional empirical likelihood (CEL) variant of the estimator is used to extract implied densities that satisfy the pricing restrictions while incorporating the forwardlooking information from option prices. Moreover, I introduce density combinations in the CEL framework to measure the relative importance of changes in the physical return distribution and in the pricing kernel. The nonlinear dynamic pricing kernels can be used to understand return predictability, and provide model-free quantities that can be compared against those implied by structural asset pricing models.