Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> FRFT as fractional powers of Fourier transform <s> We introduce the concept of Fourier transforms of fractional order, the ordinary Fourier transform being a transform of order 1. The integral representation of this transform can be used to construct a table of fractional order Fourier transforms. A generalized operational calculus is developed, paralleling the familiar one for the ordinary transform. Its application provides a convenient technique for solving certain classes of ordinary and partial differential equations which arise in quantum mechanics from classical quadratic hamiltonians. The method of solution is first illustrated by its application to the free and to the forced quantum mechanical harmonic oscillator. The corresponding Green's functions are obtained in closed form. The new technique is then extended to three-dimensional problems and applied to the quantum mechanical description of the motion of electrons in a constant magnetic field. The stationary states, energy levels and the evolution of an initial wave packet are obtained by a systematic application of the rules of the generalized operational calculus. Finally, the method is applied to the second order partial differential equation with time-dependent coefficients describing the quantum mechanical dynamics of electrons in a time-varying magnetic field. <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> FRFT as fractional powers of Fourier transform <s> Preface. Acknowledgments. Introduction. Signals, Systems, and Transformations. Wigner Distributions and Linear Canonical Transforms. The Fractional Fourier Transform. Time-Order and Space-Order Representations. The Discrete Fractional Fourier Transform. Optical Signals and Systems. Phase-Space Optics. The Fractional Fourier Transform in Optics. Applications of the Fractional Fourier Transform to Filtering, Estimation, and Signal Recovery. Applications of the Fractional Fourier Transform to Matched Filtering, Detection, and Pattern Recognition. Bibliography on the Fractional Fourier Transform. Other Cited Works. Credits. Index. <s> BIB002
Namias provided an elegant generalization of the Fourier transform to the FRFT BIB001 , by deriving the FRFT from the eigenfunctions of the Fourier transform. In fact, Namias used the fact that the Hermite-Gaussian functions ðf k ðtÞÞ are eigenfunctions of the Fourier transform and showed that the a-th order FRFT shares the eigenfunctions of the Fourier transform. More specifically, the eigenvalues of the FRFT are the a-th root of the eigenvalues of the Fourier transform BIB001 : Therefore, if we expand a signal in terms of these eigenfunctions, we obtained where By applying the fractional operator ðF a ðÁÞÞ to both sides of (6), we get From the above equation, it is clear that the kernel function can be defined as The above equation can be considered as the spectral decomposition of the kernel function of FRFT BIB002 . Using the Mehler formula, Namias in fact shows that we obtain the FRFT from (8) BIB001 .