Calculus of variations on time scales and discrete fractional calculus

Estudamos problemas do cálculo das variações e controlo óptimo no contexto das escalas temporais. Especificamente, obtemos condições necessárias de optimalidade do tipo de Euler–Lagrange tanto para lagrangianos dependendo de derivadas delta de ordem superior como para problemas isoperimétricos. Desenvolvemos também alguns métodos directos que permitem resolver determinadas classes de problemas variacionais através de desigualdades em escalas temporais. No último capítulo apresentamos operadores de diferença fraccionários e propomos um novo cálculo das variações fraccionário em tempo discreto. Obtemos as correspondentes condições necessárias de Euler– Lagrange e Legendre, ilustrando depois a teoria com alguns exemplos.

Comparison of extrasystolic ECG signal classifiers using discrete wavelet transforms

This work compares and contrasts results of classifying time-domain ECG signals with pathological conditions taken from the MITBIH arrhythmia database. Linear discriminant analysis and a multi-layer perceptron were used as classifiers. The neural network was trained by two different methods, namely back-propagation and a genetic algorithm. Converting the time-domain signal into the wavelet domain reduced the dimensionality of the problem at least 10-fold. This was achieved using wavelets from the db6 family as well as using adaptive wavelets generated using two different strategies. The wavelet transforms used in this study were limited to two decomposition levels. A neural network with evolved weights proved to be the best classifier with a maximum of 99.6% accuracy when optimised wavelet-transform ECG data wits presented to its input and 95.9% accuracy when the signals presented to its input were decomposed using db6 wavelets. The linear discriminant analysis achieved a maximum classification accuracy of 95.7% when presented with optimised and 95.5% with db6 wavelet coefficients. It is shown that the much simpler signal representation of a few wavelet coefficients obtained through an optimised discrete wavelet transform facilitates the classification of non-stationary time-variant signals task considerably. In addition, the results indicate that wavelet optimisation may improve the classification ability of a neural network. (c) 2005 Elsevier B.V. All rights reserved.

Use of the statistical properties of the wavelet-transform coefficients for optimization of integration time in Fourier transform spectrometry

We show that an analysis of the mean and variance of discrete wavelet coefficients of coaveraged time-domain interferograms can be used as a specification for determining when to stop coaveraging. We also show that, if a prediction model built in the wavelet domain is used to determine the composition of unknown samples, a stopping criterion for the coaveraging process can be developed with respect to the uncertainty tolerated in the prediction.

Billiards in a General Domain with Random Reflections

We study stochastic billiards on general tables: a particle moves according to its constant velocity inside some domain D R(d) until it hits the boundary and bounces randomly inside, according to some reflection law. We assume that the boundary of the domain is locally Lipschitz and almost everywhere continuously differentiable. The angle of the outgoing velocity with the inner normal vector has a specified, absolutely continuous density. We construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location/velocity. We mainly focus on the case of bounded domains. Then, we prove exponential ergodicity of these two Markov processes, we study their invariant distribution and their normal (Gaussian) fluctuations. Of particular interest is the case of the cosine reflection law: the stationary distributions for the two processes are uniform in this case, the discrete time chain is reversible though the continuous time process is quasi-reversible. Also in this case, we give a natural construction of a chord ""picked at random"" in D, and we study the angle of intersection of the process with a (d - 1) -dimensional manifold contained in D.

Equivalent representation of mutual coupling in transmission lines by discrete element

This paper describes a computational model based on lumped elements for the mutual coupling between phases in transmission lines without the explicit use of modal transformation matrices. The self and mutual parameters and the coupling between phases are modeled using modal transformation techniques. The modal representation is developed from the intrinsic consideration of the modal transformation matrix and the resulting system of time-domain differential equations is described as state equations. Thus, a detailed profile ofthe currents and the voltages through the line can be easily calculated using numerical or analytical integration methods. However, the original contribution of the article is the proposal of a time-domain model without the successive phase/mode transformations and a practical implementation based on conventional electrical circuits, without the use of electromagnetic theory to model the coupling between phases. © 2003-2012 IEEE.

Representation of transmission lines: comparison of models and parameters distributed discrete parameters

A transmission line is characterized by the fact that its parameters are distributed along its length. This fact makes the voltages and currents along the line to behave like waves and these are described by differential equations. In general, the differential equations mentioned are difficult to solve in the time domain, due to the convolution integral, but in the frequency domain these equations become simpler and their solutions are known. The transmission line can be represented by a cascade of π circuits. This model has the advantage of being developed directly in the time domain, but there is a need to apply numerical integration methods. In this work a comparison of the model that considers the fact that the parameters are distributed (Universal Line Model) and the fact that the parameters considered concentrated along the line (π circuit model) using the trapezoidal integration method, and Simpson's rule Runge-Kutta in a single-phase transmission line length of 100 km subjected to an operation power. © 2003-2012 IEEE.

Nerve fiber layer in glaucomatous hemifield loss: a case-control study with time- and spectral-domain optical coherence tomography

Purpose: To evaluate the retinal nerve fiber layer measurements with time-domain (TD) and spectral-domain (SD) optical coherence tomography (OCT), and to test the diagnostic ability of both technologies in glaucomatous patients with asymmetric visual hemifield loss. Methods: 36 patients with primary open-angle glaucoma with visual field loss in one hemifield (affected) and absent loss in the other (non-affected), and 36 age-matched healthy controls had the study eye imaged with Stratus-OCT (Carl Zeiss Meditec Inc., Dublin, California, USA) and 3 D OCT-1000 (Topcon, Tokyo, Japan). Peripapillary retinal nerve fiber layer measurements and normative classification were recorded. Total deviation values were averaged in each hemifield (hemifield mean deviation) for each subject. Visual field and retinal nerve fiber layer "asymmetry indexes" were calculated as the ratio between affected versus non-affected hemifields and corresponding hemiretinas. Results: Retinal nerve fiber layer measurements in non-affected hemifields (mean [SD] 87.0 [17.1] mu m and 84.3 [20.2] mu m, for TD and SD-OCT, respectively) were thinner than in controls (119.0 [12.2] mu m and 117.0 [17.7] mu m, P<0.001). The optical coherence tomography normative database classified 42% and 67% of hemiretinas corresponding to non-affected hemifields as abnormal in TD and SD-OCT, respectively (P=0.01). Retinal nerve fiber layer measurements were consistently thicker with TD compared to SD-OCT. Retinal nerve fiber layer thickness asymmetry index was similar in TD (0.76 [0.17]) and SD-OCT (0.79 [0.12]) and significantly greater than the visual field asymmetry index (0.36 [0.20], P<0.001). Conclusions: Normal hemifields of glaucoma patients had thinner retinal nerve fiber layer than healthy eyes, as measured by TD and SD-OCT. Retinal nerve fiber layer measurements were thicker with TD than SD-OCT. SD-OCT detected abnormal retinal nerve fiber layer thickness more often than TD-OCT.

Native EEG and treatment effects in neuroleptic-naïve schizophrenic patients: time and frequency domain approaches

Time domain analysis of electroencephalography (EEG) can identify subsecond periods of quasi-stable brain states. These so-called microstates assumingly correspond to basic units of cognition and emotion. On the other hand, Global Field Synchronization (GFS) is a frequency domain measure to estimate functional synchronization of brain processes on a global level for each EEG frequency band [Koenig, T., Lehmann, D., Saito, N., Kuginuki, T., Kinoshita, T., Koukkou, M., 2001. Decreased functional connectivity of EEG theta-frequency activity in first-episode, neuroleptic-naive patients with schizophrenia: preliminary results. Schizophr Res. 50, 55-60.]. Using these time and frequency domain analyzes, several previous studies reported shortened microstate duration in specific microstate classes and decreased GFS in theta band in drug naïve schizophrenia compared to controls. The purpose of this study was to investigate changes of these EEG parameters after drug treatment in drug naïve schizophrenia. EEG analysis was performed in 21 drug-naive patients and 21 healthy controls. 14 patients were reevaluated 2-8 weeks (mean 4.3) after the initiation of drug administration. The results extended findings of treatment effect on brain functions in schizophrenia, and imply that shortened duration of specific microstate classes seems a state marker especially in patients with later neuroleptic responsive, while lower theta GFS seems a state-related phenomenon and that higher gamma GFS is a trait like phenomenon.

On the use of discrete cosine transforms for multicarrier communications

In this correspondence, the conditions to use any kind of discrete cosine transform (DCT) for multicarrier data transmission are derived. The symmetric convolution-multiplication property of each DCT implies that when symmetric convolution is performed in the time domain, an element-by-element multiplication is performed in the corresponding discrete trigonometric domain. Therefore, appending symmetric redun-dancy (as prefix and suffix) into each data symbol to be transmitted, and also enforcing symmetry for the equivalent channel impulse response, the linear convolution performed in the transmission channel becomes a symmetric convolution in those samples of interest. Furthermore, the channel equalization can be carried out by means of a bank of scalars in the corresponding discrete cosine transform domain. The expressions for obtaining the value of each scalar corresponding to these one-tap per subcarrier equalizers are presented. This study is completed with several computer simulations in mobile broadband wireless communication scenarios, considering the presence of carrier frequency offset (CFO). The obtained results indicate that the proposed systems outperform the standardized ones based on the DFT.

Full-field Fourier-domain optical coherence tomography

Full-field Fourier-domain optical coherence tomography (3F-OCT) is a full-field version of spectraldomain/swept-source optical coherence tomography. A set of two-dimensional Fourier holograms is recorded at discrete wavenumbers spanning the swept-source tuning range. The resultant three-dimensional data cube contains comprehensive information on the three-dimensional morphological layout of the sample that can be reconstructed in software via three-dimensional discrete Fourier-transform. This method of recording of the OCT signal confers signal-to-noise ratio improvement in comparison with "flying-spot" time-domain OCT. The spatial resolution of the 3F-OCT reconstructed image, however, is degraded due to the presence of a phase cross-term, whose origin and effects are addressed in this paper. We present theoretical and experimental study of imaging performance of 3F-OCT, with particular emphasis on elimination of the deleterious effects of the phase cross-term.

Numerical modelling of industrial microwave heating

The numerical modelling of electromagnetic waves has been the focus of many research areas in the past. Some specific applications of electromagnetic wave scattering are in the fields of Microwave Heating and Radar Communication Systems. The equations that govern the fundamental behaviour of electromagnetic wave propagation in waveguides and cavities are the Maxwell's equations. In the literature, a number of methods have been employed to solve these equations. Of these methods, the classical Finite-Difference Time-Domain scheme, which uses a staggered time and space discretisation, is the most well known and widely used. However, it is complicated to implement this method on an irregular computational domain using an unstructured mesh. In this work, a coupled method is introduced for the solution of Maxwell's equations. It is proposed that the free-space component of the solution is computed in the time domain, whilst the load is resolved using the frequency dependent electric field Helmholtz equation. This methodology results in a timefrequency domain hybrid scheme. For the Helmholtz equation, boundary conditions are generated from the time dependent free-space solutions. The boundary information is mapped into the frequency domain using the Discrete Fourier Transform. The solution for the electric field components is obtained by solving a sparse-complex system of linear equations. The hybrid method has been tested for both waveguide and cavity configurations. Numerical tests performed on waveguides and cavities for inhomogeneous lossy materials highlight the accuracy and computational efficiency of the newly proposed hybrid computational electromagnetic strategy.

A hybrid approach for resolving the electromagnetic fields inside a waveguide loaded with a lossy medium

In this work a novel hybrid approach is presented that uses a combination of both time domain and frequency domain solution strategies to predict the power distribution within a lossy medium loaded within a waveguide. The problem of determining the electromagnetic fields evolving within the waveguide and the lossy medium is decoupled into two components, one for computing the fields in the waveguide including a coarse representation of the medium (the exterior problem) and one for a detailed resolution of the lossy medium (the interior problem). A previously documented cell-centred Maxwell’s equations numerical solver can be used to resolve the exterior problem accurately in the time domain. Thereafter the discrete Fourier transform can be applied to the computed field data around the interface of the medium to estimate the frequency domain boundary condition in-formation that is needed for closure of the interior problem. Since only the electric fields are required to compute the power distribution generated within the lossy medium, the interior problem can be resolved efficiently using the Helmholtz equation. A consistent cell-centred finite-volume method is then used to discretise this equation on a fine mesh and the underlying large, sparse, complex matrix system is solved for the required electric field using the iterative Krylov subspace based GMRES iterative solver. It will be shown that the hybrid solution methodology works well when a single frequency is considered in the evaluation of the Helmholtz equation in a single mode waveguide. A restriction of the scheme is that the material needs to be sufficiently lossy, so that any penetrating waves in the material are absorbed.

Digital signal processing

Signal Processing (SP) is a subject of central importance in engineering and the applied sciences. Signals are information-bearing functions, and SP deals with the analysis and processing of signals (by dedicated systems) to extract or modify information. Signal processing is necessary because signals normally contain information that is not readily usable or understandable, or which might be disturbed by unwanted sources such as noise. Although many signals are non-electrical, it is common to convert them into electrical signals for processing. Most natural signals (such as acoustic and biomedical signals) are continuous functions of time, with these signals being referred to as analog signals. Prior to the onset of digital computers, Analog Signal Processing (ASP) and analog systems were the only tool to deal with analog signals. Although ASP and analog systems are still widely used, Digital Signal Processing (DSP) and digital systems are attracting more attention, due in large part to the significant advantages of digital systems over the analog counterparts. These advantages include superiority in performance,s peed, reliability, efficiency of storage, size and cost. In addition, DSP can solve problems that cannot be solved using ASP, like the spectral analysis of multicomonent signals, adaptive filtering, and operations at very low frequencies. Following the recent developments in engineering which occurred in the 1980's and 1990's, DSP became one of the world's fastest growing industries. Since that time DSP has not only impacted on traditional areas of electrical engineering, but has had far reaching effects on other domains that deal with information such as economics, meteorology, seismology, bioengineering, oceanology, communications, astronomy, radar engineering, control engineering and various other applications. This book is based on the Lecture Notes of Associate Professor Zahir M. Hussain at RMIT University (Melbourne, 2001-2009), the research of Dr. Amin Z. Sadik (at QUT & RMIT, 2005-2008), and the Note of Professor Peter O'Shea at Queensland University of Technology. Part I of the book addresses the representation of analog and digital signals and systems in the time domain and in the frequency domain. The core topics covered are convolution, transforms (Fourier, Laplace, Z. Discrete-time Fourier, and Discrete Fourier), filters, and random signal analysis. There is also a treatment of some important applications of DSP, including signal detection in noise, radar range estimation, banking and financial applications, and audio effects production. Design and implementation of digital systems (such as integrators, differentiators, resonators and oscillators are also considered, along with the design of conventional digital filters. Part I is suitable for an elementary course in DSP. Part II (which is suitable for an advanced signal processing course), considers selected signal processing systems and techniques. Core topics covered are the Hilbert transformer, binary signal transmission, phase-locked loops, sigma-delta modulation, noise shaping, quantization, adaptive filters, and non-stationary signal analysis. Part III presents some selected advanced DSP topics. We hope that this book will contribute to the advancement of engineering education and that it will serve as a general reference book on digital signal processing.

Mean and variance of estimates of the bispectrum of a harmonic random process-an analysis including leakage effects

Analytical expressions are derived for the mean and variance, of estimates of the bispectrum of a real-time series assuming a cosinusoidal model. The effects of spectral leakage, inherent in discrete Fourier transform operation when the modes present in the signal have a nonintegral number of wavelengths in the record, are included in the analysis. A single phase-coupled triad of modes can cause the bispectrum to have a nonzero mean value over the entire region of computation owing to leakage. The variance of bispectral estimates in the presence of leakage has contributions from individual modes and from triads of phase-coupled modes. Time-domain windowing reduces the leakage. The theoretical expressions for the mean and variance of bispectral estimates are derived in terms of a function dependent on an arbitrary symmetric time-domain window applied to the record. the number of data, and the statistics of the phase coupling among triads of modes. The theoretical results are verified by numerical simulations for simple test cases and applied to laboratory data to examine phase coupling in a hypothesis testing framework

Estimation of vibration amplitude in Fourier domain optical coherence tomography interferometric signals from Doppler spectrum

In presented method combination of Fourier and Time domain detection enables to broaden the effective bandwidth for time dependent Doppler Signal that allows for using higher-order Bessel functions to calculate unambiguously the vibration amplitudes.