Fourier analysis, Walsh-Fourier analysis, and wavelet analysis have often been used in time series analysis. Fourier analysis can be used to detect periodic components that have sinusoidal shape; however, it might be misleading when the periodic components are not sinusoidal. Walsh-Fourier analysis is suitable for revealing the rectangular trends of time series. The flaw of the Walsh-Fourier analysis is that Walsh functions are not periodic. The resulting Walsh-Fourier analysis is more difficult to interpret than classical Fourier analysis. Wavelet analysis is very useful in analyzing and describing time series with gradual frequency changes. Wavelet analysis also has a shortcoming by giving no exact meaning to the concept of frequency because wavelets are not periodic functions. In addition, all three analysis methods above require equally-spaced time series observations. In this dissertation, by using a sequence of periodic step functions, a new analysis method, adaptive Fourier analysis, and its theory are developed. These can be applied to time series data where patterns may take general periodic shapes that include sinusoids as special cases. Most importantly, the resulting adaptive Fourier analysis does not require equally-spaced time series observations. / Ph. D.

Univ., Diss.--Tübingen, 1998. / A rev. version of the author's dissertation (doctoral--Tübingen). Includes bibliographical references. Literaturverz. S. [163] - 170.

This thesis is chiefly concerned with a classical conjecture of Littlewood's regarding the L¹-norm of the Fourier transform, and the closely related idem-potent theorem. The vast majority of the results regarding these problems are, in some sense, qualitative or at the very least infinitary and it has become increasingly apparent that a quantitative state of affairs is desirable. Broadly speaking, the first part of the thesis develops three new tools for tackling the problems above: We prove a new structural theorem for the spectrum of functions in A(G); we extend the notion of local Fourier analysis, pioneered by Bourgain, to a much more general structure, and localize Chang's classic structure theorem as well as our own spectral structure theorem; and we refine some aspects of Freiman's celebrated theorem regarding the structure of sets with small doubling. These tools lead to improvements in a number of existing additive results which we indicate, but for us the main purpose is in application to the analytic problems mentioned above. The second part of the thesis discusses a natural version of Littlewood's problem for finite abelian groups. Here the situation varies wildly with the underlying group and we pay special attention first to the finite field case (where we use Chang's Theorem) and then to the case of residues modulo a prime where we require our new local structure theorem for A(G). We complete the consideration of Littlewood's problem for finite abelian groups by using the local version of Chang's Theorem we have developed. Finally we deploy the Freiman tools along with the extended Fourier analytic techniques to yield a fully quantitative version of the idempotent theorem.

Thesis (MSc)--University of Stellenbosch, 2007. / ENGLISH ABSTRACT: The dielectric-barrier discharge (DBD) is a source of non-equilibrium plasma that has seen widespread industrial application in recent years. A high-voltage arbitrary-waveform generator has been designed, built and characterised for the purpose of investigating the influence that the applied voltage waveform has on the operation of a DBD. The developed arbitrary-waveform generator is based on the principle of Fourier synthesis. Up to twenty Fourier components are generated by means of a digital circuit board, and then separately amplified by Class-AB amplifiers. Twenty step-up transformers are subsequently used to transform the Fourier components to higher voltages; the summation of the Fourier components are realised by the series connection of the transformer secondary sides. It was found that the digital generation of the Fourier components is very accurate and provides for the easy configuration of arbitrary waveforms. Furthermore, the amplification of the Fourier components by the Class-AB amplifiers introduces very little distortion. The principle of adding the Fourier components via the step-up transformers has been demonstrated; however, the large distributed capacitances of the transformers adversely affect the operation of the Class-AB amplifiers, leading to the introduction of distortion into the generated waveform. Furthermore, it was found that care had to be taken to limit the introduction of EMI through the system’s large ground plane. / AFRIKAANSE OPSOMMING: Die di¨elektriese versperringsontlading (DVO) is ’n bron van nie-ekwilibrium plasma wat in die afgelope jare wye toepassing in die nywerheid gevind het. ’n Arbitrˆere-golfvorm hoogspanningskragbron is ontwerp, gebou en gekarakteriseer, met die doel om die invloed wat die aangewende spanningsgolfvorm het op die werking van die DVO, te ondersoek. Die ontwikkelde arbitrˆere golfvormgenerator is gebaseer op die beginsels van Fourier samestelling. Tot twintig Fourier komponente word digitaal gegenereer, en dan afsonderlik versterk deur Klas-AB versterkers. Twintig transformators word dan gebruik om die Fourier komponente na ho¨er spannings te transformeer. Die sommasie van die Fourier komponente geskied deur die serieskakeling van die transformators se sekondˆere windings. Daar is bevind dat die digitale generasie van die Fourier komponente baie akkuraat is, en dat die arbitrˆere golfvorms maklik verstel kan word. Verder versterk die Klas-AB versterkers die Fourier komponente sonder enige noemenswaardige vervorming. Die gebruik van die transformators om die Fourier komponente saam te voeg, is gedemonstreer. Die groot verspreide kapasitansies van die transformators be¨ınvloed egter die funksioneering van die Klas-AB versterkers, wat lei tot ’n vervorming van die uittree. Daar is ook bevind dat die toetrede van EM versteurings deur die grondvlak van die sisteem problematies kan wees.

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This thesis is about digital control theory and presents an account of methods for enabling and analysing intentional non-uniform sampling in discrete compensators. Most conventional control algorithms cause numerical problems where data is collected at sampling rates that are substantially higher than the dynamics of the equivalent continuous-time operation that is being implemented. This is of relevant interest in applications of digital control, in which high sample rates are routinely dictated by the system stability requirements rather than the signal processing needs. Considerable recent progress in reducing the sample frequency requirements has been made through the use of non-uniform sampling schemes, so called alias-free signal processing. The approach prompts the simplification of complex systems and consequently enhances the numerical conditioning of the implementation algorithms that otherwise, would require very high uniform sample rates. Such means of signal representation and analysis presents a variety of options and thus is being researched and practiced in a number of areas in communications. However, the control communities have not yet investigated the use of intentional non-uniform sampling, and hence the ethos of this research project is to investigate the effectiveness of such sampling regimes, in the context of exploiting the benefits. Digital control systems exhibit bandwidth limitations enforced by their closed-loop frequency requirements, the calculation delays in the control algorithm and the interfacing conversion times. These limitations pave the way for additional phase lags within the control loop that demand very high sample rates. Since non-uniform sampling is propitious in reducing the sample frequency requirements of digital processing, it proffers the prospects of being utilised in achieving a higher control bandwidth without opting for very high uniform sample rates. The concept, to the author s knowledge, has not formally been studied and very few definite answers exist in control literature regarding the associated analysis techniques. The key contributions adduced in this thesis include the development and analysis of the control algorithm designed to accommodate intentional non-uniform sample frequencies. In addition, the implementation aspects are presented on an 8-bit microcontroller and an FPGA board. This work begins by establishing a brief historical perspective on the use of non-uniform sampling and its role for digital processing. The study is then applied to the problem of digital control design, and applications are further discoursed. This is followed by consideration of its implementation aspects on standard hardware.

As ligas Al-Si hipoeutéticas têm grande importância na indústria de fundição de peças devido às excelentes propriedades de fundição, como baixo ponto de fusão e alta fluidez. A análise térmica das curvas de resfriamento medidas durante a solidificação destas ligas pode ser utilizada para controlar a formação da macroestrutura de grãos. Esta análise envolve a determinação das temperaturas de início e final de solidificação, bem como a evolução da fração de sólido com o tempo a partir da chamada análise térmica de Fourier. Apesar desta técnica ter sido aplicada a diversas ligas comerciais, existem poucos dados referentes às ligas Al-Si binárias. Os dados são ainda mais escassos quando se deseja examinar o efeito do tratamento de inoculação do metal líquido para refino de grão. O objetivo do presente trabalho é investigar o efeito do tratamento de inoculação nas ligas binárias Al-3%Si, Al-7%Si e Al-11%Si através da análise térmica e metalográfica. Foram obtidos lingotes cilíndricos a partir do vazamento da liga Al-Si líquida com ou sem a adição de inoculante na forma da liga mãe Al-3%Ti-1%B, adicionada para se obter um teor nominal de 0,05%Ti. Curvas de resfriamento foram medidas a partir de termopares inseridos no interior da cavidade do molde, composto de areia de sílica ligada com resina fenólica. Os lingotes foram seccionados e as suas micro e macroestruturas examinadas por metalografia óptica. A análise térmica das curvas de taxa de resfriamento e a análise térmica de Fourier foram aplicadas para determinar os instantes de início e final de solidificação, a magnitude da recalescência, a evolução da fração de sólido com o tempo e a fração de sólido no momento da coesão dendrítica. Os resultados mostram que o inoculante, apesar de adicionado na mesma quantidade em todas as ligas, é mais efetivo em diminuir o tamanho de grão da liga Al-3%Si do que das ligas Al-7%Si e Al-11%Si. Esta perda de eficiência é chamada de envenenamento do inoculante pelo Si. A evolução da fração de sólido mostra que a fração de coesão diminui com o aumento do teor de Si, como observado por alguns autores. A adição do inoculante diminui a fração de sólido durante a solidificação, diminuindo também a fração de coesão. Este efeito contraria dados da literatura para o ponto de coesão medido através de métodos mecânicos. / The hypoeutectic Al-Si alloys are extremely important to the foundry industry owing to their excellent fluidity and castability. The thermal analysis of the cooling curves measured during solidification of these alloys can be used to control the formation of the grain macrostructure. This analysis consists of the determination of the temperatures at the beginning and end of solidification, as well as the calculation of the time evolution of the solid fraction by more elaborate techniques, such as, the Fourier thermal analysis. Although this technique has been applied to several aluminum commercial alloys, there are very few data about binary Al-Si alloys. Available data related to the effect of melt inoculation to decrease grain size are even scarcer. The objective of the present work is to study the effect of the inoculation treatment of the binary alloys Al-3%Si, Al-7%Si, and Al-11%Si by thermal analysis and metalography. Cylindrical ingots were cast by pouring Al-Si melts with or without the addition of inoculant in the form of Al-3%Ti-1%B to obtain 0,05%Ti. Cooling curves were measured by thermocouples located within the mold cavity, made of silica sand and phenolic resin. The ingots were sectioned and their micro and macrostructures examined by optical metalography. The simple thermal analysis and the Fourier analysis were used to determine the time of beginning and end of solidification, the size of the recalescence, the time evolution of solid fraction and the solid fraction at the dendrite coherency point. The results show that, although the same amount of inoculant is added to all melts, it is more effective to decrease the grain size of the Al-3%Si than that of the Al7%Si and Al-11%Si alloys. This fading of efficiency is referred to as poisoning of the inoculant by Si. The evolution of solid fraction shows that the solid fraction at the coherency point decreases with an increase in the Si content, as observed by other authors. The addition of inoculant also causes a decrease in the solid fraction at dendrite coherency, which disagrees with some published coherency fraction obtained by mechanical methods.

Thesis. 1978. Sc.D.--Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Vita. / Bibliography: p. 142-145. / Sc.D.

The study of signal processing has wide applications, such as in hi-fi audio, television, voice recognition and many other areas. Signals are rarely observed without noise, which obstruct our analysis of signals. Hence, it is of great interest to study the detection, approximation and removal of noise.  In this thesis we compare two methods for image denoising. The methods are each based on a data transformation. Specifically, Fourier Transform and Singular Value Decomposition are utilized in respective methods and compared on grayscale images. The comparison is based on the visual quality of the resulting image, the maximum peak signal-to-noise ratios attainable for the respective methods and their computational time. We find that the methods are fairly equal in visual quality. However, the method based on the Fourier transform scores higher in peak signal-to-noise ratio and demands considerably less computational time.

A dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science in Engineering. Johannesburg, 2017 / This study presents the development of a potentially new biomarker for three different movement disorders: Huntington’s Disease (HD), Parkinson’s Disease (PD) and Essential Tremors (ET). A Leap Motion® gaming device was used to record the trajectories of subjects’ forefinger as they trace simple patterns in the air. The patterns used were stepfunction, triangle and circle. The recorded signals were analysed using transform functions and Fourier analysis. Both analysis types yielded features from which differences between the four categories studied: PD, HD, ET and control subjects, were sought and displayed in both graphical and numerical forms. The X-axis and Y-axis of the signals were separately analysed and yielded different results. For the step-function pattern, no distinct differences between the four categories were found from the transfer function analysis whereas the Y-axis of the signal could distinguish between the categories. For the triangle pattern, the X-axis features provided a discrimination between the categories while the Y-axis feature did not. For the circle pattern, neither X-axis nor Y-axis features were able to distinguish between the categories. A Fourier analysis showed a better discrimination ability for both X- and Y- axis. This study is a preliminary one and all results indicate that more subjects of all categories are needed to develop a bio-marker for the diseases studied and that a higher order transfer function analysis is required. However, the methodology outlined in this work, comprising of both the experimental system and the analysis showed a potential to produce a biomarker for movement disorders. / MT2018