Developments and extensions of Roth's method of double Fourier series with applications to the solution of electromagnetic field problems

The first part of the thesis compares Roth's method with other methods, in particular the method of separation of variables and the finite cosine transform method, for solving certain elliptic partial differential equations arising in practice. In particular we consider the solution of steady state problems associated with insulated conductors in rectangular slots. Roth's method has two main disadvantages namely the slow rate of con­vergence of the double Fourier series and the restrictive form of the allowable boundary conditions. A combined Roth-separation of variables method is derived to remove the restrictions on the form of the boundary conditions and various Chebyshev approximations are used to try to improve the rate of convergence of the series. All the techniques are then applied to the Neumann problem arising from balanced rectangular windings in a transformer window. Roth's method is then extended to deal with problems other than those resulting from static fields. First we consider a rectangular insulated conductor in a rectangular slot when the current is varying sinusoidally with time. An approximate method is also developed and compared with the exact method.The approximation is then used to consider the problem of an insulated conductor in a slot facing an air gap. We also consider the exact method applied to the determination of the eddy-current loss produced in an isolated rectangular conductor by a transverse magnetic field varying sinusoidally with time. The results obtained using Roth's method are critically compared with those obtained by other authors using different methods. The final part of the thesis investigates further the application of Chebyshdev methods to the solution of elliptic partial differential equations; an area where Chebyshev approximations have rarely been used. A poisson equation with a polynomial term is treated first followed by a slot problem in cylindrical geometry.

Processing and rendering of Fourier domain optical coherence tomography images at a line rate over 524 kHz using a graphics processing unit

In Fourier domain optical coherence tomography (FD-OCT), a large amount of interference data needs to be resampled from the wavelength domain to the wavenumber domain prior to Fourier transformation. We present an approach to optimize this data processing, using a graphics processing unit (GPU) and parallel processing algorithms. We demonstrate an increased processing and rendering rate over that previously reported by using GPU paged memory to render data in the GPU rather than copying back to the CPU. This avoids unnecessary and slow data transfer, enabling a processing and display rate of well over 524,000 A-scan/s for a single frame. To the best of our knowledge this is the fastest processing demonstrated to date and the first time that FD-OCT processing and rendering has been demonstrated entirely on a GPU.

Simultaneous ultrafast optical pulse train bursts generation and shaping based on Fourier series developments using superimposed fiber Bragg gratings

We propose an all-fiber method for the generation of ultrafast shaped pulse train bursts from a single pulse based on Fourier Series Developments (FDSs). The implementation of the FSD based filter only requires the use of a very simple non apodized Superimposed Fiber Bragg Grating (S-FBG) for the generation of the Shaped Output Pulse Train Burst (SOPTB). In this approach, the shape, the period and the temporal length of the generated SOPTB have no dependency on the input pulse rate.

An Algorithm for Fresnel Diffraction Computing Based on Fractional Fourier Transform

The fractional Fourier transform (FrFT) is used for the solution of the diffraction integral in optics. A scanning approach is proposed for finding the optimal FrFT order. In this way, the process of diffraction computing is speeded up. The basic algorithm and the intermediate results at each stage are demonstrated.

Fourier Neural Networks: An Approach with Sinusoidal Activation Functions

* Supported by INTAS 2000-626, INTAS YSF 03-55-1969, INTAS INNO 182, and TIC 2003-09319-c03-03.

A Generalized Convolution with a Weight Function for the Fourier Cosine and Sine Transforms

A generalized convolution with a weight function for the Fourier cosine and sine transforms is introduced. Its properties and applications to solving a system of integral equations are considered.

On the Range of the Fourier Transform Associated with the Spherical Mean Operator

We characterize the range of some spaces of functions by the Fourier transform associated with the spherical mean operator R and we give a new description of the Schwartz spaces. Next, we prove a Paley-Wiener and a Paley-Wiener-Schawrtz theorems.

Regenerative Fourier transformation for dual-quadrature regeneration of multilevel rectangular QAM

We propose a new nonlinear optical loop mirror based configuration capable of regenerating regular rectangular quadrature amplitude modulated (QAM) signals. The scheme achieves suppression of noise distortion on both signal quadratures through the realization of two orthogonal regenerative Fourier transformations. Numerical simulations show the performance of the scheme for high constellation complexities (including 256-QAM formats).

On generalized orlicz sequence spaces of Fourier coefficients for trigonometric gap series. I.

We investigate the operator associating with a function fєLp2π, 1Fourier coefficients of ƒ with respect to a trigonometric gap system, as well as an operator from a modular space X ρs(ϕ) to the generalized Orlicz sequence space lϕ.

On the Mixed Modulus of Smoothness and a Class of Double Fourier Series

MSC 2010: 42A32; 42A20

Channel model and lower capacity bound for the transmission based on nonlinear Fourier transform

The integrability of the nonlinear Schräodinger equation (NLSE) by the inverse scattering transform shown in a seminal work [1] gave an interesting opportunity to treat the corresponding nonlinear channel similar to a linear one by using the nonlinear Fourier transform. Integrability of the NLSE is in the background of the old idea of eigenvalue communications [2] that was resurrected in recent works [3{7]. In [6, 7] the new method for the coherent optical transmission employing the continuous nonlinear spectral data | nonlinear inverse synthesis was introduced. It assumes the modulation and detection of data using directly the continuous part of nonlinear spectrum associated with an integrable transmission channel (the NLSE in the case considered). Although such a transmission method is inherently free from nonlinear impairments, the noisy signal corruptions, arising due to the ampli¯er spontaneous emission, inevitably degrade the optical system performance. We study properties of the noise-corrupted channel model in the nonlinear spectral domain attributed to NLSE. We derive the general stochastic equations governing the signal evolution inside the nonlinear spectral domain and elucidate the properties of the emerging nonlinear spectral noise using well-established methods of perturbation theory based on inverse scattering transform [8]. It is shown that in the presence of small noise the communication channel in the nonlinear domain is the additive Gaussian channel with memory and signal-dependent correlation matrix. We demonstrate that the effective spectral noise acquires colouring", its autocorrelation function becomes slow decaying and non-diagonal as a function of \frequencies", and the noise loses its circular symmetry, becoming elliptically polarized. Then we derive a low bound for the spectral effiency for such a channel. Our main result is that by using the nonlinear spectral techniques one can significantly increase the achievable spectral effiency compared to the currently available methods [9]. REFERENCES 1. Zakharov, V. E. and A. B. Shabat, Sov. Phys. JETP, Vol. 34, 62{69, 1972. 2. Hasegawa, A. and T. Nyu, J. Lightwave Technol., Vol. 11, 395{399, 1993. 3. Yousefi, M. I. and F. R. Kschischang, IEEE Trans. Inf. Theory, Vol. 60, 4312{4328, 2014. 4. Yousefi, M. I. and F. R. Kschischang, IEEE Trans. Inf. Theory, Vol. 60, 4329{4345 2014. 5. Yousefi, M. I. and F. R. Kschischang, IEEE Trans. Inf. Theory, Vol. 60, 4346{4369, 2014. 6. Prilepsky, J. E., S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, Phys. Rev. Lett., Vol. 113, 013901, 2014. 7. Le, S. T., J. E. Prilepsky, and S. K. Turitsyn, Opt. Express, Vol. 22, 26720{26741, 2014. 8. Kaup, D. J. and A. C. Newell, Proc. R. Soc. Lond. A, Vol. 361, 413{446, 1978. 9. Essiambre, R.-J., G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, J. Lightwave Technol., Vol. 28, 662{701, 2010.

Nonlinear Fourier transform for optical communications

The nonlinear Fourier transform, also known as eigenvalue communications, is a transmission and signal processing technique that makes positive use of the nonlinear properties of fibre channels. I will discuss recent progress in this field.

Optical communication based on the periodic nonlinear Fourier transform signal processing

In this work we introduce the periodic nonlinear Fourier transform (PNFT) and propose a proof-of-concept communication system based on it by using a simple waveform with known nonlinear spectrum (NS). We study the performance (addressing the bit-error-rate (BER), as a function of the propagation distance) of the transmission system based on the use of the PNFT processing method and show the benefits of the latter approach. By analysing our simulation results for the system with lumped amplification, we demonstrate the decent potential of the new processing method.

Fault diagnosis scheme for open-circuit faults in switched reluctance motor drives using fast Fourier transform algorithm with bus current detection

Reliability of power converters is of crucial importance in switched reluctance motor drives used for safety-critical applications. Open-circuit faults in power converters will cause the motor to run in unbalanced states, and if left untreated, they will lead to damage to the motor and power modules, and even cause a catastrophic failure of the whole drive system. This study is focused on using a single current sensor to detect open-circuit faults accurately. An asymmetrical half-bridge converter is considered in this study and the faults of single-phase open and two-phase open are analysed. Three different bus positions are defined. On the basis of a fast Fourier transform algorithm with Blackman window interpolation, the bus current spectrums before and after open-circuit faults are analysed in details. Their fault characteristics are extracted accurately by the normalisations of the phase fundamental frequency component and double phase fundamental frequency component, and the fault characteristics of the three bus detection schemes are also compared. The open-circuit faults can be located by finding the relationship between the bus current and rotor position. The effectiveness of the proposed diagnosis method is validated by the simulation results and experimental tests.

Proteção diferencial de transformadores de potência utilizando a transformada Wavelet

About 10% of faults involving the electrical system occurs in power transformers. Therefore, the protection applied to the power transformers is essential to ensure the continuous operation of this device and the efficiency of the electrical system. Among the protection functions applied to power transformers, the differential protection appears as one of the main schemes, presenting reliable discrimination between internal faults and external faults or inrush currents. However, when using the low frequency components of the differential currents flowing through the transformer, the main difficulty of the conventional methods of differential protection is the delay for detection of the events. However, internal faults, external faults and other disturbances related to the transformer operation present transient and can be appropriately detected by the wavelet transform. In this paper is proposed the development of a wavelet-based differential protection for detection and identification of external faults to the transformer, internal faults, and transformer energizing by using the wavelet coefficient energy of the differential currents. The obtained results reveal the advantages of using of the wavelet transform in the differential protection compared to conventional protection, since it provides reliability and speed in detection of these events.