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The focus of this dissertation is the interpolation and approximation of multivariate bandlimited functions via sampled (function) values. The first set of results investigates polynomial interpolation in connection with multivariate bandlimited functions. To this end, the concept of a uniformly invertible Riesz basis is developed (with examples), and is used to construct Lagrangian polynomial interpolants for particular classes of sampled square-summable data. These interpolants are used to derive two asymptotic recovery and approximation formulas. The first recovery formula is theoretically straightforward, with global convergence in the appropriate metrics; however, it becomes computationally complicated in the limit. This complexity is sidestepped in the second recovery formula, at the cost of requiring a more local form of convergence. The second set of results uses oversampling of data to establish a multivariate recovery formula. Under additional restrictions on the sampling sites and the frequency band, this formula demonstrates a certain stability with respect to sampling errors. Computational simplifications of this formula are also given.

It has been well reported that the use of multidimensional constellation signals can help to reduce the bit error rate in Additive Gaussian channels by using the hyperspace geometry more efficiently. Similarly, in fading channels, dimensionality provides an inherent signal space diversity (distinct components between two constellations points), so the amplitude degradation of the signal are combated significantly better. Moreover, the set of n-dimensional signals also provides great compatibility with various Trellis Coded modulation schemes: N-dimensional signaling joined with a convolutional encoder uses fewer redundant bits for each 2D signaling interval, and increases intra-subset minimum squared Euclidean distance (MSED) to approach the ultimate capacity limit predicted by Shannon's theory. The multidimensional signals perform better for the same complexity than two-dimensional schemes. The inherent constellation expansion penalty factor paid for using classical mapping structures can be decreased by enlarging the constellation's dimension. In this thesis, a multidimensional signal set construction paradigm that completely avoids the constellation expansion penalty is used in Band-limited channels and in fading channels. As such, theoretical work on performance analysis and computer simulations for Quadrature-Quadrature Phase Shift Keying (Q2PSK), Constant Envelope (CE) Q2PSK, and trellis-coded 16D CEQ2PSK in ideal band-limited channels of various bandwidths is presented along with a novel discussion on visualization techniques for 4D Quadrature-Quadrature Phase Shift Keying (Q2PSK), Saha's Constant Envelope (CE) Q2PSK, and Cartwright's CEQ2PSK in ideal band-limited channels. Furthermore, a metric designed to be used in fading channels, with Hamming Distance (HD) as a primary concern and Euclidean distance (ED) as secondary is also introduced. Simulation results show that the 16D TCM CEQ2PSK system performs well in channels with AWGN and fading, even with the simplest convolutional encoder tested; achievable coding gains using 16-D CEQ2PSK Expanded TCM schemes under various conditions are finally reported.

Analytic models for the performance of the forward link of 3G CDMA systems using different maximum ratio combining (MRC) RAKE finger weight assignments are presented. The spreading modulations under investigation are the balanced QPSK and the complex-spreading QPSK. The models are computationally efficient, accurate, and applicable to Root Raised Cosine (RRC) pulse shaping with any roll-off factor, variable processing gain, chip rate, and data rate for orthogonal codes, random codes, quasi-orthogonal codes, Gaussian noise, and realistic channel models. The expressions derived are then used to obtain the so-called orthogonality factor, which is commonly used in system-level simulations. Next, closed-form mathematical expressions for the variance due to infinite chips interfering in systems using arbitrary Nyquist pulses are derived. These expressions are applicable to both the forward link and the reverse link. For the latter, the existing knowledge on the accurate and efficient estimation of the performance of CDMA systems is extended by presenting closed mathematical expressions for bandlimited systems using arbitrary Nyquist pulses for both BPSK and Offset-QPSK (OQPSK) modulation. The impact of adjacent channel interference in bandlimited systems is subsequently considered. Finally, mathematical expressions for the accurate and efficient estimation of a CDMA system using RRC pulse-shaping and a RAKE receiver in the presence of multipath interference are presented.

No description available.

Master of Science / Department of Mathematics / Nathan Albin / A new methodology is introduced for use in discrete periodic extension of non-periodic functions. The methodology is based on a band-limited step function, and utilizes the computational efficiency of FC-Gram (Fourier Continuation based on orthonormal Gram polynomial basis on the extension stage) extension database. The discrete periodic extension is a technique for augmenting a set of uniformly-spaced samples of a smooth function with auxiliary values in an extension region. If a suitable extension is constructed, the interpolating trigonometric polynomial found via an FFT(Fast Fourier Transform) will accurately approximate the original function in its original interval. The discrete periodic extension is a key construction in the FC-Gram algorithm which is successfully implemented in several recent efficient and high-order PDEs solvers. This thesis focuses on a new flexible discrete periodic extension procedure that performs at least as well as the FC-Gram method, but with somewhat simpler implementation and significantly decreased setup time.

Nas últimas décadas, diversos sistemas de comunicação baseados em caos foram propostos. Dentre eles, vários utilizam uma função para codificar uma mensagem em um sinal caótico, que é caracterizado como um sinal de banda larga. Dado que o canal de transmissão é limitado em banda por natureza, é necessário determinar e controlar o espectro do sinal caótico transmitido por esse sistema. Nesse sentido, um sistema de comunicação em banda limitada, baseado em sincronismo caótico, foi proposto recentemente utilizando-se filtros digitais para controlar a largura de banda dos sinais transmitidos. Esses filtros, inseridos no sistema de comunicação, modificam o sistema original gerador do sinal caótico, tornando-se necessário analisar como essa inserção afeta o sincronismo caótico. Nessa tese, apresenta-se uma análise desse sistema de comunicação digital de tempo discreto, baseado em sincronismo caótico, considerando-se um canal com ruído aditivo branco gaussiano. As condições necessárias para a sincronização desse sistema são obtidas analiticamente, por meio de um teorema, para um mapa gerador de caos qualquer. O desempenho desse sistema é avaliado em termos da taxa de erro de bit, e, para melhorar seu desempenho, propõe-se filtrar o ruído fora da banda do sinal na entrada do receptor. Apesar das condições de sincronismo terem sido determinadas, a inserção dos filtros também pode modificar a natureza caótica dos sinais, e não há garantia que os sinais transmitidos sejam caóticos. Para analisar a natureza caótica dos sinais transmitidos pelo sistema de comunicação, o maior expoente de Lyapunov é obtido numericamente em função dos coeficientes dos filtros, dos parâmetros do mapa e da função de codificação da mensagem. / In recent decades, several chaos-based communication systems have been proposed. Many of them use a function to encode a message into a chaotic signal, which is characterized as wideband. Since every transmission channel is bandlimited in nature, it is necessary to determine and to control the spectrum of the chaotic signal transmitted by this system. This way, a bandlimited chaos-based communication system was recently proposed using digital filters and chaotic synchronization. These filters, inserted in the communication system, modify the original chaotic generator system, becoming necessary to study how their insertion affect chaotic synchronization. In this work, we present an analysis of this discrete-time chaos-based digital communication system considering an additive white Gaussian noise channel. The synchronization conditions of this system is analytically obtained, through a theorem, for a generic chaos generator map. The system performance is evaluated in terms of bit error rate, and, to obtain a performance improvement, it is also proposed to filter the out-of-band noise in the receiver. Although the conditions for chaotic synchronization have been determined, the filters insertion can also modify the chaotic nature of the signals, and there is no guarantee that the transmitted signals remain chaotic. To analyze the chaotic nature of the communication system transmitted signals, the largest Lyapunov exponent is numerically accessed as a function of the filters coefficients, the parameters of the map and the message coding function.

Nas últimas décadas, diversos sistemas de comunicação baseados em caos foram propostos. Dentre eles, vários utilizam uma função para codificar uma mensagem em um sinal caótico, que é caracterizado como um sinal de banda larga. Dado que o canal de transmissão é limitado em banda por natureza, é necessário determinar e controlar o espectro do sinal caótico transmitido por esse sistema. Nesse sentido, um sistema de comunicação em banda limitada, baseado em sincronismo caótico, foi proposto recentemente utilizando-se filtros digitais para controlar a largura de banda dos sinais transmitidos. Esses filtros, inseridos no sistema de comunicação, modificam o sistema original gerador do sinal caótico, tornando-se necessário analisar como essa inserção afeta o sincronismo caótico. Nessa tese, apresenta-se uma análise desse sistema de comunicação digital de tempo discreto, baseado em sincronismo caótico, considerando-se um canal com ruído aditivo branco gaussiano. As condições necessárias para a sincronização desse sistema são obtidas analiticamente, por meio de um teorema, para um mapa gerador de caos qualquer. O desempenho desse sistema é avaliado em termos da taxa de erro de bit, e, para melhorar seu desempenho, propõe-se filtrar o ruído fora da banda do sinal na entrada do receptor. Apesar das condições de sincronismo terem sido determinadas, a inserção dos filtros também pode modificar a natureza caótica dos sinais, e não há garantia que os sinais transmitidos sejam caóticos. Para analisar a natureza caótica dos sinais transmitidos pelo sistema de comunicação, o maior expoente de Lyapunov é obtido numericamente em função dos coeficientes dos filtros, dos parâmetros do mapa e da função de codificação da mensagem. / In recent decades, several chaos-based communication systems have been proposed. Many of them use a function to encode a message into a chaotic signal, which is characterized as wideband. Since every transmission channel is bandlimited in nature, it is necessary to determine and to control the spectrum of the chaotic signal transmitted by this system. This way, a bandlimited chaos-based communication system was recently proposed using digital filters and chaotic synchronization. These filters, inserted in the communication system, modify the original chaotic generator system, becoming necessary to study how their insertion affect chaotic synchronization. In this work, we present an analysis of this discrete-time chaos-based digital communication system considering an additive white Gaussian noise channel. The synchronization conditions of this system is analytically obtained, through a theorem, for a generic chaos generator map. The system performance is evaluated in terms of bit error rate, and, to obtain a performance improvement, it is also proposed to filter the out-of-band noise in the receiver. Although the conditions for chaotic synchronization have been determined, the filters insertion can also modify the chaotic nature of the signals, and there is no guarantee that the transmitted signals remain chaotic. To analyze the chaotic nature of the communication system transmitted signals, the largest Lyapunov exponent is numerically accessed as a function of the filters coefficients, the parameters of the map and the message coding function.

Η κωδικοποίηση στο χρόνο αποτελεί έναν πραγματικού-χρόνου, ασύγχρονο μηχανισμό κωδικοποίησης της πληροφορίας ενός αναλογικού σήματος πεπερασμένου εύρους ζώνης σε μία χρονική ακολουθία (χρονικά δείγματα), βάση της οποίας το σήμα μπορεί να ανακατασκευαστεί. Τα κυκλώματα τα οποία παράγουν αυτά τα χρονικά δείγματα και οι αλγόριθμοι οι οποίοι πραγματοποιούν την ανακατασκευή αναφέρονται ως Μηχανές Κωδικοποίησης στο Χρόνο (ΤΕΜs) και Μηχανές Αποκωδικοποίησης στο χρόνο (TDMs), αντίστοιχα. Αυτή η διαδικασία μπορεί να αντιμετωπιστεί ως ένας μηχανισμός δειγμaτοληψίας εξαρτώμενος από το σήμα και από τις παραμέτρους των παραπάνω κυκλωμάτων. Σε αυτήν τη διπλωματική εργασία, παρουσιάζουμε το θεωρητικό και μαθηματικό υπόβαθρο αυτής της καινοτόμας μεθόδου αναπαράστασης της πληροφορίας. / Time encoding is a real-time, asynchronous mechanism for encoding the information of an analog bandlimited signal into a time sequence (time samples) based on which the signal can be reconstructed. The circuits generating these time samples and the algorithms carrying out the reconstruction are referred to as Time Encoding Machines (TEMs) and Time Decoding Machines (TDMs), respectively. This procedure can be addressed as a sampling scheme which depends on the signal and the parameters of the above circuits. In this diploma thesis, we present the theoretical and mathematical framework of this innovative information representation procedure.

Bandlimited optical intensity channels, arising in applications such as indoor infrared communications and visible light communications (VLC), require that all signals satisfy a bandwidth constraint as well as average, peak and non-negative amplitude constraints. However, the signaling designed for conventional radio frequency (RF) electrical channels cannot be applied directly, since they take energy constraints into consideration instead of amplitude constraints. In addition, conventional transmission techniques optimized for broad-band optical channels such as fiber optics, terrestrial/satellite-to-satellite free-space optical (FSO) communications are typically not bandwidth efficient. In this thesis, a two-dimensional signal space for bandlimited optical intensity channels is presented. A novel feature of this model is that the non-negativity and peak amplitude constraints are relaxed. The signal space parameterizes the likelihood of a negative or peak amplitude excursions in the output. Although the intensity channel only supports non-negative amplitudes, the impact of clipping on system performance is shown to be negligible if the likelihood of negative amplitude excursion is small enough. For a given signal space, a tractable approximation approach using a finite series is applied to accurately compute the likelihood of clipping under average and peak optical power constraints. The uncoded asymptotic optical power and spectral efficiencies using two-dimensional lattice constellations are computed. The Monte-Carlo (MC) simulation results show that for a given average or peak optical power, schemes designed in the presented signal space haver higher spectral efficiency than M-ary pulse amplitude modulation (PAM) using previously established techniques. / Thesis / Master of Applied Science (MASc)