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We consider the problem of numerical integration, where one aims to approximate an integral of a given continuous function from the function values at given sampling points, also known as quadrature points. A useful framework for such an approximation process is provided by the theory of reproducing kernel Hilbert spaces and the concept of the worst case quadrature error. However, the computation of optimal quadrature points, which minimize the worst case quadrature error, is in general a challenging task and requires efficient algorithms, in particular for large numbers of points. The focus of this thesis is on the efficient computation of optimal quadrature points on the torus T^d, the sphere S^d, and the rotation group SO(3). For that reason we present a general framework for the minimization of the worst case quadrature error on Riemannian manifolds, in order to construct numerically such quadrature points. Therefore, we consider, for N quadrature points on a manifold M, the worst case quadrature error as a function defined on the product manifold M^N. For the optimization on such high dimensional manifolds we make use of the method of steepest descent, the Newton method, and the conjugate gradient method, where we propose two efficient evaluation approaches for the worst case quadrature error and its derivatives. The first evaluation approach follows ideas from computational physics, where we interpret the quadrature error as a pairwise potential energy. These ideas allow us to reduce for certain instances the complexity of the evaluations from O(M^2) to O(M log(M)). For the second evaluation approach we express the worst case quadrature error in Fourier domain. This enables us to utilize the nonequispaced fast Fourier transforms for the torus T^d, the sphere S^2, and the rotation group SO(3), which reduce the computational complexity of the worst case quadrature error for polynomial spaces with degree N from O(N^k M) to O(N^k log^2(N) + M), where k is the dimension of the corresponding manifold. For the usual choice N^k ~ M we achieve the complexity O(M log^2(M)) instead of O(M^2). In conjunction with the proposed conjugate gradient method on Riemannian manifolds we arrive at a particular efficient optimization approach for the computation of optimal quadrature points on the torus T^d, the sphere S^d, and the rotation group SO(3). Finally, with the proposed optimization methods we are able to provide new lists with quadrature formulas for high polynomial degrees N on the sphere S^2, and the rotation group SO(3). Further applications of the proposed optimization framework are found due to the interesting connections between worst case quadrature errors, discrepancies and potential energies. Especially, discrepancies provide us with an intuitive notion for describing the uniformity of point distributions and are of particular importance for high dimensional integration in quasi-Monte Carlo methods. A generalized form of uniform point distributions arises in applications of image processing and computer graphics, where one is concerned with the problem of distributing points in an optimal way accordingly to a prescribed density function. We will show that such problems can be naturally described by the notion of discrepancy, and thus fit perfectly into the proposed framework. A typical application is halftoning of images, where nonuniform distributions of black dots create the illusion of gray toned images. We will see that the proposed optimization methods compete with state-of-the-art halftoning methods.

We consider the problem of numerical integration, where one aims to approximate an integral of a given continuous function from the function values at given sampling points, also known as quadrature points. A useful framework for such an approximation process is provided by the theory of reproducing kernel Hilbert spaces and the concept of the worst case quadrature error. However, the computation of optimal quadrature points, which minimize the worst case quadrature error, is in general a challenging task and requires efficient algorithms, in particular for large numbers of points. The focus of this thesis is on the efficient computation of optimal quadrature points on the torus T^d, the sphere S^d, and the rotation group SO(3). For that reason we present a general framework for the minimization of the worst case quadrature error on Riemannian manifolds, in order to construct numerically such quadrature points. Therefore, we consider, for N quadrature points on a manifold M, the worst case quadrature error as a function defined on the product manifold M^N. For the optimization on such high dimensional manifolds we make use of the method of steepest descent, the Newton method, and the conjugate gradient method, where we propose two efficient evaluation approaches for the worst case quadrature error and its derivatives. The first evaluation approach follows ideas from computational physics, where we interpret the quadrature error as a pairwise potential energy. These ideas allow us to reduce for certain instances the complexity of the evaluations from O(M^2) to O(M log(M)). For the second evaluation approach we express the worst case quadrature error in Fourier domain. This enables us to utilize the nonequispaced fast Fourier transforms for the torus T^d, the sphere S^2, and the rotation group SO(3), which reduce the computational complexity of the worst case quadrature error for polynomial spaces with degree N from O(N^k M) to O(N^k log^2(N) + M), where k is the dimension of the corresponding manifold. For the usual choice N^k ~ M we achieve the complexity O(M log^2(M)) instead of O(M^2). In conjunction with the proposed conjugate gradient method on Riemannian manifolds we arrive at a particular efficient optimization approach for the computation of optimal quadrature points on the torus T^d, the sphere S^d, and the rotation group SO(3). Finally, with the proposed optimization methods we are able to provide new lists with quadrature formulas for high polynomial degrees N on the sphere S^2, and the rotation group SO(3). Further applications of the proposed optimization framework are found due to the interesting connections between worst case quadrature errors, discrepancies and potential energies. Especially, discrepancies provide us with an intuitive notion for describing the uniformity of point distributions and are of particular importance for high dimensional integration in quasi-Monte Carlo methods. A generalized form of uniform point distributions arises in applications of image processing and computer graphics, where one is concerned with the problem of distributing points in an optimal way accordingly to a prescribed density function. We will show that such problems can be naturally described by the notion of discrepancy, and thus fit perfectly into the proposed framework. A typical application is halftoning of images, where nonuniform distributions of black dots create the illusion of gray toned images. We will see that the proposed optimization methods compete with state-of-the-art halftoning methods.

International Telemetering Conference Proceedings / October 23-26, 2000 / Town & Country Hotel and Conference Center, San Diego, California / Bit errors often occur in a wireless communications link when impairments alter the transmitted signal. It is advantageous to be able to predict how well a system will tolerate transmission problems. This paper details laboratory performance measurements and comparisons in terms of evaluating configurations of a digital receiver for Feher patented Quadrature Phase Shift Keying (FQPSK-B) demodulation. The transmitted signal is subjected to calibrated levels of impairments while the receiver performance is monitored in real-time.

This work addresses the question of the efficient numerical solution of time-domain boundary integral equations with retarded potentials arising in the problems of acoustic and electromagnetic scattering. The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature. This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known. Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.g. high convergence order, stability, low dissipation and dispersion. As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations. Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm. The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems: the need to construct many dense matrices and to evaluate many singular and near-singular integrals. The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method (HF FMM) and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times. The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested. The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In this work we prove that kernels of the boundary integral operators $w_n^h(d)$ ($h$ is the time step and $t_n=nh$ is the time) exhibit exponential decay outside of the neighborhood of $d=nh$ (this is the consequence of the Huygens principle). The size of the support of these kernels for fixed $h$ increases with $n$ as $n^a,a<1$, where $a$ depends on the order of the Runge-Kutta method and is (typically) smaller for Runge-Kutta methods of higher order. Numerical experiments demonstrate that theoretically predicted values of $a$ are quite close to optimal. In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled. The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity. The efficiency of the approach is confirmed by extensive numerical experiments.

Today’s communication systems are in need of spectrally efficient and high throughput techniques more than ever because of high data rate applications and the scarcity and expense of bandwidth. To cope with increased data rate demands, more base stations are needed which is not cost and energy efficient in cellular networks. It has been shown that wireless relay networks can provide higher network throughput and increase power efficiency with low complexity and cost. Furthermore, network resources can be utilized more efficiently by using network coding in relay networks. A wireless relay network in which multiple nodes exchange information with the help of relay node(s) is called a multi-way relay channel (MWRC). MWRCs are expected to be an integral part of next generation wireless standards. The main focus of this dissertation is the investigation of transmission schemes in an MWRC to improve the throughput and error performance. An MWRC with full data exchange is assumed in which a half-duplex relay station (RS) is the enabler of communication. One of the challenges with signal demodulation in MWRCs is the existence of ambiguous points in the received constellation. The first part of this dissertation investigates a transmission scheme for full data exchange in MWRC that benefits from these points and improves its throughput by 33% compared to traditional relaying. Then an MWRC is considered where a RS assists multiple nodes to exchange messages. A different approach is taken to avoid ambiguous points in the superposition of user symbols at the relay. This can be achieved by employing complex field network coding (CFNC) which results in full data exchange in two communication phases. CFNC may lead to small Euclidean distances between constellation points, resulting in poor error performance. To improve this performance, the optimal user precoding values are derived such that the power efficiency of the relay constellation is highest when channel state information is available at the users. The error performance of each user is then analyzed and compared with other relaying schemes. Finally, focusing on the uplink of multi-way relay systems, the performance of an MWRC is studied in which users can employ arbitrary modulation schemes and the links between the users and the relay have different gains, e.g. Rayleigh fading. Analytical expressions for the exact average pairwise error probability of these MWRCs are derived. The probability density function (PDF) and the mean of the minimum Euclidean distance of the relay constellation are closely approximated, and a tight upper bound on the symbol error probability is developed. / Graduate

L'objectif de cette thèse est l'analyse d'erreur a posteriori et la proposition de stratégies d'adaptivité basées sur des critères d'arrêt et de raffinement local de maillage. Nous traitons une classe d'équations paraboliques dégénér ées multidimensionnelles modélisant des problèmes importants pour l'industrie. Au chapitre 1 nous considérons le problème de Stefan instationaire a deux phases qui modélise un processus de changement de phase régi par la loi de Fourier. Nous régularisons la relation entre l'enthalpie et la température et nous discrétisons le problème par la méthode d'Euler implicite en temps et un schéma numérique conforme en espace tel que les élément finis conformes, ou les volumes finis centrés aux sommets du maillage. Nous démontrons une borne supérieure de la norme duale du résidu, de l'erreur sur l'enthalpie dans L2(0; T;H-1) et de l'erreur sur la température dans L2(0; T;L2), par des estimateurs d'erreur entièrement calculables. Ces estimateurs comprennent : un estimateur associé à l'erreur de régularisation, un estimateur associé à l'erreur d'une méthode de linéarisation (par exemple, la méthode de Newton), un estimateur associé à l'erreur en temps et un estimateur associé à l'erreur du schéma en espace. Par conséquent, ces estimateurs permettent de formuler un algorithme adaptatif de résolution où les erreurs associées peuvent être équilibrées. Nous proposons également une stratégie de raffinement local de maillages. En fin, nous prouvons l'efficacité de nos estimations d'erreur a posteriori. Un test numérique illustre l'efficacité de nos estimateurs et la performance de l'algorithme adaptatif. En particulier, des indices d'efficacité proches de la valeur optimale de 1 sont obtenus. Au chapitre 2 nous développons des estimations d'erreur a posteriori pour l'écoulement de Darcy polyphasique et isothermique, décrit par un système couplé d'équations aux dérivées partielles non linéaires et d'équations algébriques non linéaires. Ce système est discrétisé en espace par une méthode de volume finis centrés par maille et la méthode d'Euler implicite en temps. Nous etablissons une borne supérieure d'une norme duale du résidu augmentée d'un terme qui tiens compte de la non-conformité des volumes finis par des estimateurs d'erreur a posteriori entièrement calculables. Dans ce chapitre, nous nous concentrons sur la formulation d'un critère d'arrêt de l'algorithme de linéarisation du problème discrète (tel que la méthode de Newton) avec un critère d'arrêt du solveur algébrique de résolution du système linéarité (par exemple la méthode GMRes), de sort que les contributions des estimateurs d'erreur correspondant n'affectent plus la somme globale des estimateurs d'erreur de manière significative. Nous appliquons notre analyse sur des exemples réalistes d'ingénierie de réservoir pour confirmer qu'en général notre ajustement des critères d'arrêt apporte une économie significative (jusqu'au un ordre de magnitude en termes du nombre total des itérations du solveur algébrique), déjà sur des maillages fixes, et ceci sans perte notable de précision. Au chapitre 3 nous complétons le modèle décrit au chapitre 2 en considérant une condition non-isothermique pour l'écoulement a fin de traiter le modèle général d'écoulement polyphasique thermique dans les milieux poreux. Pour ce problème, nous développons des estimateurs d'erreur analogues a ceux du chapitre 2 pour lesquels nous établissons une borne supérieure d'erreur entièrement calculable, pour une norme duale du résidu complétée par un terme d'évaluation de la non-conformité. Nous montrons ensuite comment estimer séparément chaque composante d'erreur, ce qui nous permet d'ajuster les critères d'arrêt et d'équilibrer les contributions des différents estimateurs d'erreur : erreur d'approximation en temps, erreur d'approximation en espace, erreur de linéarisation et erreur du solveur algébrique. Ce chapitre se termine par une application des estimateurs au modèle d'huile morte. La preuve de l'efficacité de notre estimation a postiriori est egalement fournie. Finalement, au chapitre 4 nous considérons les procédés de récupération assistée d'huile. Plus précisément, nous étudions une technique de récupération thermique d'huile de type huile morte par injection de vapeur destinée a augmenter la mobilité des hydrocarbures. Dans ce chapitre, nous appliquons l'analyse a posteriori des chapitres 2 et 3, nous proposons une formule de quadrature pour simplifier l'évaluation des estimateurs, nous proposons un algorithme adaptatif de raffinement de maillages en espace et en temps basé sur les estimateurs et nous illustrons pas des essais numériques sur des exemples réalistes la performance de cette stratégie de raffinement. Notamment, des gains significatifs sont réalisés en terme du nombre de mailles nécessaires pour la simulation sur des exemples en dimension trois.

The scope of this dissertation is to investigate the merits and implications of using multilevel modulation formats in optical communications systems. Following the trend in academia and industry, special focus is placed on quadrature phase-shift keying (QPSK), and specifically on polarization division multiplexed (PDM) QPSK. A special kind of receiver is investigated thoroughly, the digital coherent receiver, the equivalent of the coherent quadrature demodulator in classical communications nomenclature. A large number of digital signal processing (DSP) algorithms are implemented, some of them novel, and their performance is examined, analyzed, and compared in a number of practical system scenarios. The impact of transmitter / receiver imperfections and a number of optical fiber impairments on system performance is studied. Experimental results taken from proof-of-concept experiments are also analyzed. / Η διατριβή αυτή έχει ως σκοπό τη μελέτη οπτικών τηλεπικοινωνιακών συστημάτων που χρησιμοποιούν τετραδικές διαμορφώσεις φάσης, πολυπλεξία κατά πόλωση και σύμφωνους ψηφιακούς δέκτες διαφοροποίησης φάσης και πόλωσης. Μελετήθηκαν αλγόριθμοι επεξεργασίας σήματος κατάλληλοι για εξάλειψη της επίδρασης των φαινομένων διάδοσης και των μη ιδανικοτήτων οπτικών τηλεπικοινωνιακών συστημάτων. Η μελέτη έγινε με προσομοίωση Monte-Carlo, με χρήση ημιαναλυτικής μεθόδου προσδιορισμού της πιθανότητας σφάλματος τηλεπικοινωνιακού συστήματος και με ανάλυση πειραματικών δεδομένων. Τα πειραματικά δεδομένα ελήφθησαν από οπτικό τηλεπικοινωνιακό σύστημα με τετραδική διαμόρφωση φάσης και πολυπλεξία κατά πόλωση με ρυθμούς συμβόλων 0.1-10 GBd (0.4-40 Gb/s). Μελετήθηκαν αλγόριθμοι επανένωσης των πολώσεων, αλγόριθμοι αποπολύπλεξης των πολώσεων, αλγόριθμοι διόρθωσης της ανισοσταθμίας ορθογωνιότητας, αλγόριθμοι εκτίμησης και αφαίρεσης της ενδιάμεσης συχνότητας και αλγόριθμοι εκτίμησης και αφαίρεσης του θορύβου φάσης των laser.

The subject matter of this dissertation is the design, analysis and practical implementation of a new numerical method to approximate the eigenvalues and eigenfunctions of regular Sturm–Liouville problems, given in Liouville’s normal form, defined on compact intervals, with self-adjoint separated boundary conditions. These are classical problems in computational mathematics which lie on the interface between numerical analysis and spectral theory, with important applications in physics and chemistry, not least in the approximation of energy levels and wave functions of quantum systems. Because of their great importance, many numerical algorithms have been proposed over the years which span a vast and diverse repertoire of techniques. When compared with previous approaches, the principal advantage of the numerical method proposed in this dissertation is that it is accompanied by error bounds which: (i) hold uniformly over the entire eigenvalue range, and, (ii) can attain arbitrary high-order. This dissertation is composed of two parts, aggregated according to the regularity of the potential function. First, in the main part of this thesis, this work considers the truncation, discretization, practical implementation and MATLAB software, of the new approach for the classical setting with continuous and piecewise analytic potentials (Ramos and Iserles, 2015; Ramos, 2015a,b,c). Later, towards the end, this work touches upon an extension of the new ideas that enabled the truncation of the new approach, but instead for the general setting with absolutely integrable potentials (Ramos, 2014).

This thesis deals with optical synthesis of unmodulated and modulated microwave signals. Generation of microwave signals based on optical heterodyning is discussed in detail. The effect of phase noise of laser on heterodyned output has been studied for different phase noise profiles. Towards this, we propose a generic algorithm to numerically model the linewidth broadening of a laser due to phase noise. Generation of microwave signals is demonstrated practically by conducting an optical heterodyning experiment. Signals ranging in frequency from 12.5 MHz to 27 GHz have been generated. Limitations of optical heterodyning based approach in terms of phase noise performance and frequency stability are discussed and practically demonstrated. A hardware-efficient Optical Phase Locked Loop (OPLL) is proposed to overcome these issues. Phase noise tracking performance of the proposed OPLL has been experimentally demonstrated. Phase noise values as low as -105 dBc/Hz at 10 KHz offset have been achieved. Optical modulators, owing to their extremely low electro-optic response time, can support high frequency modulating signals. This makes them highly attractive in comparison to their microwave counterparts. In this thesis, we propose techniques to generate microwave signals modulated at very high bit rates by down-converting the corresponding modulated optical signals to microwave domain. Down-conversion required for this process is achieved by optical heterodyning. The proposed concept has been theoretically analyzed, simulated and experimentally validated. Amplitude Modulated and ASK modulated microwave signals have been generated as Proof-of-Concept. Limitations posed by OPLL in generation of angle modulated microwave signals by optical heterodyning have been brought out. Schemes overcoming these limitations have been proposed towards generation of BPSK and QPSK modulated microwave signals. Integrated Optics (IO) technology has been studied as a means of implementation of the proposed concepts. IO components like Sinusoidal bends, Y-branch splitters and Electro-Optic-Modulators (EOMs) have been designed towards optical synthesis of modulated microwave signals. Propagation of modulated optical signal through these IO components has also been studied. An all-optic scheme based on Optical Beam Forming is proposed for transmission of QPSK modulated signal. Limitation of phase-shifting based approach, in terms of beam-squint, has been brought out. True-Time-Delay based approach has been proposed for applications demanding wide instantaneous bandwidth to avoid beam-squint. Algorithms / numerical methods required for analyses and simulations associated with the above-mentioned tasks have been evolved. This study is envisaged to provide useful insight into the realization of high-speed, compact, light-weight data transmitting systems based on Integrated Optics for future onboard spacecraft applications. This work, we believe, is a step towards realization of an Integrated Optic System-on-Chip solution for specific microwave data transmission applications.