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There is a growing demand for the high quality TV programs such as High Definition TV (HDTV). The CATV network is often a suitable solution to address this demand using a CATV modem delivering high data rate digital signals in a cost effective manner, thereby, utilizing a complex digital modulation scheme is inevitable. Exploiting complex modulation schemes, entails a more sophisticated modulator and distribution system with much tighter tolerances. However, there are always distortions introduced to the modulated signal in the modulator degrading signal quality.<p> In this research, the effect of distortions introduced by the RF band pass filter in the modulator will be considered which cause degradations on the quality of the output Quadrature Amplitude Modulated (QAM) signal. Since the RF filter's amplitude/group delay distortions are not symmetrical in the frequency domain, once translated into the base band they have a complex effect on the QAM signal. Using Matlab, the degradation effects of these distortions on the QAM signal such as Bit Error Rate (BER) is investigated.<p> In order to compensate for the effects of the RF filter distortions, two different methods are proposed. In the first method, a complex base band compensation filter is placed after the pulse shaping filter (SRRC). The coefficients of this complex filter are determined using an optimization algorithm developed during this research. The second approach, uses a pre-equalizer in the form of a Feed Forward FIR structure placed before the pulse shaping filter (SRRC). The coefficients of this pre-equalizer are determined using the equalization algorithm employed in a test receiver, with its tap weights generating the inverse response of the RF filter. The compensation of RF filter distortions in base band, in turn, improves the QAM signal parameters such as Modulation Error Ratio (MER). Finally, the MER of the modulated QAM signal before and after the base band compensation is compared between the two methods, showing a significant enhancement in the RF modulator performance.

There is a growing demand for the high quality TV programs such as High Definition TV (HDTV). The CATV network is often a suitable solution to address this demand using a CATV modem delivering high data rate digital signals in a cost effective manner, thereby, utilizing a complex digital modulation scheme is inevitable. Exploiting complex modulation schemes, entails a more sophisticated modulator and distribution system with much tighter tolerances. However, there are always distortions introduced to the modulated signal in the modulator degrading signal quality.<p> In this research, the effect of distortions introduced by the RF band pass filter in the modulator will be considered which cause degradations on the quality of the output Quadrature Amplitude Modulated (QAM) signal. Since the RF filter's amplitude/group delay distortions are not symmetrical in the frequency domain, once translated into the base band they have a complex effect on the QAM signal. Using Matlab, the degradation effects of these distortions on the QAM signal such as Bit Error Rate (BER) is investigated.<p> In order to compensate for the effects of the RF filter distortions, two different methods are proposed. In the first method, a complex base band compensation filter is placed after the pulse shaping filter (SRRC). The coefficients of this complex filter are determined using an optimization algorithm developed during this research. The second approach, uses a pre-equalizer in the form of a Feed Forward FIR structure placed before the pulse shaping filter (SRRC). The coefficients of this pre-equalizer are determined using the equalization algorithm employed in a test receiver, with its tap weights generating the inverse response of the RF filter. The compensation of RF filter distortions in base band, in turn, improves the QAM signal parameters such as Modulation Error Ratio (MER). Finally, the MER of the modulated QAM signal before and after the base band compensation is compared between the two methods, showing a significant enhancement in the RF modulator performance.

This thesis deals with solutions of nonlinear regression problems using R programming language. The introductory theoretical part is devoted to familiarization with the principles of solving nonlinear regression models and of their applications in the program R. In both, theoretical and practical part, the most famous and used differentiator algorithms are presented, particularly the Gauss-Newton's and of the steepest descent method, for estimating the parameters of nonlinear regression. Further, in the practical part, there are some demo solutions of particular tasks using nonlinear regression methods. Overall, a large number of graphs processed by the author is used in this thesis for better comprehension.

<p dir="ltr">This thesis presents advances in numerical methods for partial differential equations (PDEs) and optimization problems, with a focus on improving efficiency, stability, and accuracy across various applications. We begin by addressing 3D Poisson-type equations, developing a GPU-accelerated spectral-element method that utilizes the tensor product structure to achieve extremely fast performance. This approach enables solving problems with over one billion degrees of freedom in less than one second on modern GPUs, with applications to Schrödinger and Cahn<i>–</i>Hilliard equations demonstrated. Next, we focus on parabolic PDEs, specifically the Cahn<i>–</i>Hilliard equation with dynamical boundary conditions. We propose an efficient energy-stable numerical scheme using a unified framework to handle both Allen<i>–</i>Cahn and Cahn<i>–</i>Hilliard type boundary conditions. The scheme employs a scalar auxiliary variable (SAV) approach to achieve linear, second-order, and unconditionally energy stable properties. Shifting to a machine learning perspective for PDEs, we introduce an unsupervised learning-based numerical method for solving elliptic PDEs. This approach uses deep neural networks to approximate PDE solutions and employs least-squares functionals as loss functions, with a focus on first-order system least-squares formulations. In the realm of optimization, we present an efficient and robust SAV based algorithm for discrete gradient systems. This method modifies the standard SAV approach and incorporates relaxation and adaptive strategies to achieve fast convergence for minimization problems while maintaining unconditional energy stability. Finally, we address optimization in the context of machine learning by developing a structure-guided Gauss<i>–</i>Newton method for shallow ReLU neural network optimization. This approach exploits both the least-squares and neural network structures to create an efficient iterative solver, demonstrating superior performance on challenging function approximation problems. Throughout the thesis, we provide theoretical analysis, efficient numerical implementations, and extensive computational experiments to validate the proposed methods. </p>

In the last decade acoustic noise has become more and more regarded as a problem. In cars, boats, trains and aircraft, low-frequency noise reduces comfort. Lightweight materials and more powerful engines are used in high-speed vehicles, resulting in a general increase in interior noise levels. Low-frequency noise is annoying and during periods of long exposure it causes fatigue and discomfort. The masking effect which low-frequency noise has on speech reduces speech intelligibility. Low-frequency noise is sought to be attenuated in a wide range of applications in order to improve comfort and speech intelligibility. The use of conventional passive methods to attenuate low-frequency noise is often impractical since considerable bulk and weight are required; in transportation large weight is associated with high fuel consumption. In order to overcome the problems of ineffective passive suppression of low-frequency noise, the technique of active noise control has become of considerable interest. The fundamental principle of active noise control is based on secondary sources producing ``anti-noise.'' Destructive interference between the generated and the primary sound fields results in noise attenuation. Active noise control systems significantly increase the capacity for attenuating low-frequency noise without major increase in volume and weight. This doctoral dissertation deals with the topic of active noise control within the passenger cabin in aircraft, and within headsets. The work focuses on methods, controller structures and adaptive algorithms for attenuating tonal low-frequency noise produced by synchronized or moderately synchronized propellers generating beating sound fields. The control algorithm is a central part of an active noise control system. A multiple-reference feedforward controller based on the novel actuator-individual normalized Filtered-X Least-Mean-Squares algorithm is introduced, yielding significant attenuation of such period noise. This algorithm is of the LMS-type, and owing to the novel normalization it can also be regarded as a Newton-type algorithm. The new algorithm combines low computational complexity with high performance. For that reason the algorithm is suitable for use in systems with a large number of control sources and control sensors in order to reduce the computional power required by the control system. The computational power of the DSP hardware is limited, and therefore algorithms with high computational complexity allow fewer control sources and sensors to be used, often with reduced noise attenuation as a result. In applications, such as controlling aircraft cabin noise, where a large multiple-channel system is needed to control the relative complex interior sound field, it is of great importance to keep down the computational complexity of the algorithm so that a large number of loudspeakers and microphones can be used. The dissertation presents theoretical work, off-line computer experiments and practical real-time experiments using the actuator-individual normalized algorithm. The computer experiments are principally based on real-life cabin noise data recorded during flight in a twin-engine propeller aircraft and in a helicopter. The practical experiments were carried out in a full-scale fuselage section from a propeller aircraft. / Buller i vår dagliga miljö kan ha en negativ inverkan på vår hälsa. I många sammanhang, i tex bilar, båtar och flygplan, förekommer lågfrekvent buller. Lågfrekvent buller är oftast inte skadligt för hörseln, men kan vara tröttande och försvåra konversationen mellan personer som vistas i en utsatt miljö. En dämpning av bullernivån medför en förbättrad taluppfattbarhet samt en komfortökning. Att dämpa lågfrekvent buller med traditionella passiva metoder, tex absorbenter och reflektorer, är oftast ineffektivt. Det krävs stora, skrymmande absorbenter för att dämpa denna typ av buller samt tunga skiljeväggar för att förhindra att bullret transmitteras vidare från ett utrymme till ett annat. Metoder som är mera lämpade vid dämpning av lågfrekvent buller är de aktiva. De aktiva metoderna baseras på att en vågrörelse som ligger i motfas med en annan överlagras och de släcker ut varandra. Bullerdämpningen erhålls genom att ett ljudfält genereras som är lika starkt som bullret men i motfas med detta. De aktiva bullerdämpningsmetoderna medför en effektiv dämpning av lågfrekvent buller samtidigt som volymen, tex hos bilkupen eller båt/flygplanskabinen ej påverkas nämnvärt. Dessutom kan fordonets/farkostens vikt reduceras vilket är tacksamt för bränsleförbrukningen. I de flesta tillämpningar varierar bullrets karaktär, dvs styrka och frekvensinnehåll. För att följa dessa variationer krävs ett adaptivt (självinställande) reglersystem som styr genereringen av motljudet. I propellerflygplan är de dominerande frekvenserna i kabinbullret relaterat till propellrarnas varvtal, man känner alltså till frekvenserna som skall dämpas. Man utnyttjar en varvtalssignal för att generera signaler, så kallade referenssignaler, med de frekvenser som skall dämpas. Dessa bearbetas av ett reglersystem som generar signaler till högtalarna som i sin tur generar motljudet. För att ställa in högtalarsignalerna så att en effektiv dämpning erhålls, används mikrofoner utplacerade i kabinen som mäter bullret. För att åstadkomma en effektiv bullerdämpning i ett rum, tex i en flygplanskabin, behövs flera högtalare och mikrofoner, vilket kräver ett avancerat reglersystem. I doktorsavhandlingen ''Active Control of Propeller-Induced Noise in Aircraft'' behandlas olika metoder för att reducera kabinbuller härrörande från propellrarna. Här presenteras olika strukturer på reglersystem samt beräkningsalgoritmer för att ställa in systemet. För stora system där många högtalare och mikrofoner används, samt flera frekvenser skall dämpas, är det viktigt att systemet inte behöver för stor beräkningskapacitet för att generera motljudet. Metoderna som behandlas ger en effektiv dämpning till låg beräkningskostnad. Delar av materialet som presenteras i avhandlingen har ingått i ett EU-projekt med inriktning mot bullerundertryckning i propellerflygplan. I projektet har flera europeiska flygplanstillverkare deltagit. Avhandlingen behandlar även aktiv bullerdämpning i headset, som används av helikopterpiloter. I denna tillämpning har aktiv bullerdämpning används för att öka taluppfattbarheten.

The work done in this thesis is to examine various methods for curve fitting. Linear least squares and non-linear least squares will be described and compared, and the Newton method, Gauss--Newton method and Levenberg--Marquardt method will be applied to example problems. / Syftet med denna uppsats är att beskriva och använda olika metoder för kurvanpassning, det vill säga att passa matematiska funktioner till data. De metoder som undersöks är Newtons metod, Gauss--Newton metoden och Levenberg--Marquardt metoden. Även skillnaden mellan linjär minsta kvadrat anpassning och olinjär minsta kvadrat anpassning. Till sist tillämpas Newton, Gauss Newton och Levenberg--Marquardt metoderna på olika exempel.

Stable and computationally efficient reconstruction methodologies are developed to solve two important medical imaging problems which use near-infrared (NIR) light as the source of interrogation, namely, diffuse optical tomography (DOT) and one of its variations, ultrasound-modulated optical tomography (UMOT). Since in both these imaging modalities the system matrices are ill-conditioned owing to insufficient and noisy data, the emphasis in this work is to develop robust stochastic filtering algorithms which can handle measurement noise and also account for inaccuracies in forward models through an appropriate assignment of a process noise. However, we start with demonstration of speeding of a Gauss-Newton (GN) algorithm for DOT so that a video-rate reconstruction from data recorded on a CCD camera is rendered feasible. Towards this, a computationally efficient linear iterative scheme is proposed to invert the normal equation of a Gauss-Newton scheme in the context of recovery of absorption coefficient distribution from DOT data, which involved the singular value decomposition (SVD) of the Jacobian matrix appearing in the update equation. This has sufficiently speeded up the inversion that a video rate recovery of time evolving absorption coefficient distribution is demonstrated from experimental data. The SVD-based algorithm has made the number of operations in image reconstruction to be rather than. 2()ONN3()ONN The rest of the algorithms are based on different forms of stochastic filtering wherein we arrive at a mean-square estimate of the parameters through computing their joint probability distributions conditioned on the measurement up to the current instant. Under this, the first algorithm developed uses a Bootstrap particle filter which also uses a quasi-Newton direction within. Since keeping track of the Newton direction necessitates repetitive computation of the Jacobian, for all particle locations and for all time steps, to make the recovery computationally feasible, we devised a faster update of the Jacobian. It is demonstrated, through analytical reasoning and numerical simulations, that the proposed scheme, not only accelerates convergence but also yields substantially reduced sample variance in the estimates vis-à-vis the conventional BS filter. Both accelerated convergence and reduced sample variance in the estimates are demonstrated in DOT optical parameter recovery using simulated and experimental data. In the next demonstration a derivative free variant of the pseudo-dynamic ensemble Kalman filter (PD-EnKF) is developed for DOT wherein the size of the unknown parameter is reduced by representing of the inhomogeneities through simple geometrical shapes. Also the optical parameter fields within the inhomogeneities are approximated via an expansion based on the circular harmonics (CH) (Fourier basis functions). The EnKF is then used to recover the coefficients in the expansion with both simulated and experimentally obtained photon fluence data on phantoms with inhomogeneous inclusions. The process and measurement equations in the Pseudo-Dynamic EnKF (PD-EnKF) presently yield a parsimonious representation of the filter variables, which consist of only the Fourier coefficients and the constant scalar parameter value within the inclusion. Using fictitious, low-intensity Wiener noise processes in suitably constructed ‘measurement’ equations, the filter variables are treated as pseudo-stochastic processes so that their recovery within a stochastic filtering framework is made possible. In our numerical simulations we have considered both elliptical inclusions (two inhomogeneities) and those with more complex shapes ( such as an annular ring and a dumbbell) in 2-D objects which are cross-sections of a cylinder with background absorption and (reduced) scattering coefficient chosen as = 0.01 mm-1 and = 1.0 mm-1respectively. We also assume=0.02 mm-1 within the inhomogeneity (for the single inhomogeneity case) and=0.02 and 0.03 mm-1 (for the two inhomogeneities case). The reconstruction results by the PD-EnKF are shown to be consistently superior to those through a deterministic and explicitly regularized Gauss-Newton algorithm. We have also estimated the unknown from experimentally gathered fluence data and verified the reconstruction by matching the experimental data with the computed one. The superiority of a modified version of the PD-EnKF, which uses an ensemble square root filter, is also demonstrated in the context of UMOT by recovering the distribution of mean-squared amplitude of vibration, related to the Young’s modulus, in the ultrasound focal volume. Since the ability of a coherent light probe to pick-up the overall optical path-length change is limited to modulo an optical wavelength, the individual displacements suffered owing to the US forcing should be very small, say within a few angstroms. The sensitivity of modulation depth to changes in these small displacements could be very small, especially when the ROI is far removed from the source and detector. The contrast recovery of the unknown distribution in such cases could be seriously impaired whilst using a quasi-Newton scheme (e.g. the GN scheme) which crucially makes use of the derivative information. The derivative-free gain-based Monte Carlo filter not only remedies this deficiency, but also provides a regularization insensitive and computationally competitive alternative to the GN scheme. The inherent ability of a stochastic filter in accommodating the model error owing to a diffusion approximation of the correlation transport may be cited as an added advantage in the context of the UMOT inverse problem. Finally to speed up forward solve of the partial differential equation (PDE) modeling photon transport in the context of UMOT for which the PDE has time as a parameter, a spectral decomposition of the PDE operator is demonstrated. This allows the computation of the time dependent forward solution in terms of the eigen functions of the PDE operator which has speeded up the forward solution, which in turn has rendered the UMOT parameter recovery computationally efficient.

The project deals with artificial neural networks. After designing and debugging the test data set and the training sample set, we created a multilayer perceptron network in the Neural NetworkToolbox (NNT) of Matlab. When creating networks, we used different training algorithms and algorithms improving the generalization of the network. When creating a radial basis network, we did not use the NNT, but a specific source code in Matlab was written. Functionality of neural networks was tested on simple training and testing patterns. Realistic training data were obtained by the simulation of twelve monoconic antennas operating in the frequency range from 2 to 6 GHz. Antennas were located inside a mathematical model of Octavia II. Using CST simulations, electromagnetic fields in a car were obtained. Trained networks are described by regressive characteristics andthe mean square error of training. Algorithms improving generalization are applied on the created and trained networks. The performance of individual networks is mutually compared.

<p>U doktorskoj diseratciji posmatrani su singularni nelinearni problemi. U prvom&nbsp;poglavlju predstavljene su oznake i osnovne definicije i teoreme koje se koriste u&nbsp;disertaciji. U drugom poglavlju prikazani su poznati postupci i njihovo pona&scaron;anje&nbsp;u slučajevima da je re&scaron;enje regularno ili singularno. Takođe su pokazane poznate&nbsp;modifikacije ovih postupaka kako bi se pobolj&scaron;ala konvergencija. Posebno su&nbsp;predstavljena četiri kvazi-Njutnova metoda i predložene njihove modifikacije u&nbsp;slučaju singularnosti re&scaron;enja. U trećem poglavlju predstavljeni su teorijski okvir&nbsp;pri definisanju graničnih sistema i neki poznati algoritmi za njihovo re&scaron;avanje i&nbsp;definisan je novi algoritam koji je podjednako efikasan ali jeftiniji za rad jer ne&nbsp;uključuje izračunavanje izvoda. Takođe, predložena je kombinacija definisanog&nbsp;algortitma sa metodom negativnog gradijenta, kao i algoritam koji predstavlja&nbsp;primenu poznatog algoritma na definisani granični sistem. U četvrtom poglavlju&nbsp;predstavljeni su numerički rezultati dobijeni primenom definisanih algoritama na&nbsp;relevantne primere i potvrđeni su teorijski dobijeni rezultati.</p> / <p>In this doctoral thesis nonlinear singular problems were observed. The first&nbsp;chapter presents basic definitions and theorems that are used in the thesis. The&nbsp;second chapter presents several methods that are commonly used and their&nbsp;behavior if the solution is regular or singular. Also, some known modifications to&nbsp;these methods are presented in order to improve convergence. In addition four&nbsp;quasi-Newton methods and their modifications in the case the singularity of the&nbsp;solution. The third chapter consists of the theoretical foundation for defining the&nbsp;bordered system, some known algorithms for solving them and new algorithm is&nbsp;defined to accelerate convergence to a singular solution. New algorithm is&nbsp;efficient but cheaper for the use since there is no derivative evaluations in it. It is&nbsp;presented synthesis of new algorithm with negative gradient method and using&nbsp;one of well known method on the bordered system as well. The fourth chapter&nbsp;presents the numerical results obtained by the defined algorithms on the relevant&nbsp;examples and theoretical results are confirmed.</p>