KESİRLİ FOURIER DÖNÜŞÜMÜNÜN MAXWELL DENKLEMLERİNE UYGULANMASI
Özet
Fractional Fourier Transform (FrFT) is the generalization of the classical Fourier Transform (FT). FrFT can be applied to various fields that have already made use of the FT. FrFT defines the fractional Fourier domains by generalizing the time and frequency domains. In the application fields of FrFT, the system performance can be optimized over the fractional order of the FrFT.
In the diffraction theory, under Fresnel approximations, fields radiated from an aperture can be given in the form of Fresnel Integral (FRI). FRI can be written in terms of FrFT. Radiated fields from the aperture can be computed by employing the fast and accurate computation methods of the FrFT. In this thesis, numerical computation methods of the FrFT are studied and their relation to continuous FrFT is given. FrFT computation methods can be grouped into two categories as fast FrFT (fFrFT) and discrete FrFT (dFrFT). In fFrFT, computation of the convolution operator in the FrFT is evaluated by the Fast Fourier Transform (FFT). In dFrFT method, signal vector to be transformed is multiplied by the fractional power of the Discrete Fourier Transform (DFT) matrix. fFrFT and dFrFT methods assume constant sampling intervals for the signals in their inputs and outputs. If the sampling intervals of the signals are chosen as the assumed intervals, the output samples match the samples of the continuous FrFT. However, if the sampling intervals are not equal to the assumed intervals, angle and phase corrections have to be made in order to get the continuous FrFT samples at the output. In this study, angle and phase corrections needed for the fFrFT method are derived. The performances of the fFrFT and dFrFT methods are compared over their similarities to the continuous FrFT samples. The samples of the continuous FrFT are obtained by the adaptive Gauss-Kronrod (GK) numerical integral computation method. It is observed that the performance of these two methods in the computation of the continuous FrFT samples are close to each other.
In the scope of this thesis, for the first time in the literature, fFrFT method is applied to the evaluation of FRI, by employing the angle and phase corrections. The performance of the FrFT methods in evaluation of the diffraction integrals are compared with their difference to the Rayleigh-Sommerfeld Integral (RSI), which is valid in a broader region than the FRI. RSI, is computed with the GK method. It is observed that the diffraction patterns of fFrFt and dFrFT are similar and they are close to the values of the GK method in the Fresnel region. Through the outside of the Fresnel region, it is observed that the methods compute different values than that of the GK method. The computation speeds of the methods are compared by employing the computation times of the diffraction patterns. It is observed that fFrFT method is ten times faster than the dFrFT method; and a thousand times faster than the GK method. In the scope of this thesis, with the derived angle and phase corrections for the fFrFT, a fast and accurate method for evaluation of FRI for different sampling intervals is developed.
In electromagnetics, radiation integrals have to be solved in order to obtain the fields radiated from a current source. In this study, electric field vector components radiated from a current source, under Fresnel approximation, are given in terms of the FrFT. The electromagnetic fields are given in terms of vector potentials, and derivatives with respect to observation coordinates are evaluated. The obtained integral forms are written in terms of FrFT. Employing the numerical FrFT methods, fast and efficient computation of the electromagnetic field vector components are performed. In the scope of this thesis, first time in the literature, the computation of the electromagnetic fields radiated from a current source in the vector form is performed by employing the FrFT.
In this thesis, the developed method is applied to the computation of radiated fields for the dipole, cross-dipole and aperture antennas. It is observed that, employing the developed method in the Fresnel region, vector components of the radiated electromagnetic fields from the current source can be obtained rapidly and efficiently.
