Novel Algorithms For Fourier Transform And Spectral Estimation, And Applications In Electromagnetic Engineering

As is well known, Fourier transform and spectral estimation has extremely wide applications in many areas including theoretical studies and engineering problems as well. In the area of electromagnetics, applications of Fourier transform include calculation of radar cross section, design and analysis of antenna arrays, magnetic resonance imaging, synthetic aperature imaging, fast solutions of differential or integral equations, and so on. And also, the spectral estimation has applcations in electromagnetics, such as measurement of radar target's velocity, esitimation of resonant frequencies of electromagnetic structures, and correction of ionospheric contamination for the over-the-horizon radar. However, recent applications in electromagnetics deal with more and more frequently the Fourier transform of nonuniformly spaced data or discontinuous functions, or estimation of the spectral of stochastic, time-varying signals. This motivates our effort to develope new algorithms for the Fourier transform and spectral estimation to solve the problems mentioned above.Firstly, this thesis deals with the problem of fast Fourier transform (FFT) for nonuniformly spaced data. Specifically, the least-squares error (LSE) nonuniform FFT (NUFFT) is studied in detail, and the properties of its LSE interpolation functions is derived in this thesis for the first time. In addition, the thesis generalizes the NUFFT algorithms to the arbitrary dimentional case, and applies them to two-or three-dimentional magnetic resonance imaging (MRI). Some MRI results show good trade-off between accuracy and efficiency of the NUFFT-based imaging method. The thesis also develops a NUFFT-based time-frequency convertion algorithm for the finite-difference time-domain (FDTD) simulation data. The proposed convertion algorithm is computationally and storage efficient, and is capable of performing the time-frequency convertion for multiple nonuniformly spaced frequencies of interest. Consequently, this algorithm is very robust for different FDTD application situations.Secondly, the Fourier transform of discontinuous functions is considered.In this part, the thesis at first modifies the original discontinuous FFT which is presented earlier in the literature. This modification is based on the idea of LSE-NUFFT. Then, a new fast algorithm for evaluating the Fourier transform integrals of piecewise smooth functions is proposed. The new algorithm can calculate the spectral at arbitrarily high frequencies without the limitation of finite sampling rate, and the error in the results decreases exponentially as the order of interpolation polynomials used for this algorithm. In addition, this thesis also presents an algorithm called'conformal FFT' which is based on descritization of computational domain with triangle cells. The algorithm can reduce the staircasing error of FFT for two dimentional functions with arbitrary shape contours.Thirdly, the thesis works on the estimation of harmonics and its application to the synthesis of antenna arrays. In this part, the thesis at first studies on the iterative Fourier interpolation algorithms for estimation of single tone frequency. A generalization of the Fourier interpolation algorithm is proposed, which allows us to have an additional freedom to select the interpolation Fourier coefficients for improved performance. In addition, the thesis studies the matrix pencil method which is one of harmonics estimation methods. The main contribution for this research is to apply the matrix pencil method (MPM) to the synthesis of nonuniformly spaced arrays which is usually considered as a challenging problem in the area of antenna array synthesis. The MPM-based synthesis is studied in detail for both pencil beam patterns and shaped beam patterns. The array synthesized by the MPM-based method has smaller number of elements than that with uniform spacings. This research provides a new idea of applying harmonics estimation methods to solution of electromagnetic problems.Finally, the thesis tries to solve the problem of the ionospheric phase contamination of over-the-horizon radar echo by using the idea of instantaneous spectral estimation.In this part, the thesis at first studies on a parametric decontamination method which is based on the piecewise phase polynomial model. Some improvements have made to this method by using data-driven order selection criterion which can adaptively choose the best order for the phase polynomial model according to the realistic ionospheric contamination. The thesis has also developed the decontamination methods based on time-frequency distributions such as spectrogram and pseudo Wigner-Ville distribution. The key step of such methods is to estimate the time-varying frequency of Bragg line in the sea echo. The thesis presents detailed comparison between different instantaneous estimation approaches from time-frequency distributions. In addition, a new decontamination method is proposed which utilizes a specific peak-tracking scheme to estimate the ionospheric frequency modulation from the time-frequency distribution of echo data without filtering. To further improve the performance, the thesis presents a cascaded double-correction method which is very useful in the situation that serious ionopheric contamination happens.