Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Introduction <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> Introduction <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 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> Introduction <s> Signal processing can be found in many applications and its primary goal is to provide underlying information on specific problems for the purpose of decision making. Traditional signal processing approaches assume the stationarity of signals, which in practice is not often satisfied. Hence, time or frequency descriptions alone are insufficient to provide comprehensive information about such signals. On the contrary, time-frequency analysis is more suitable for nonstationary signals. Therefore, this paper provides a status report of feature based signal processing in the time-frequency domain through an overview of recent contributions. The feature considered here is energy concentration. The paper provides an analysis of several classes of feature extractors, i.e., time-frequency representations, and feature classifiers. The results of the literature review indicate that time-frequency domain signal processing using energy concentration as a feature is a very powerful tool and has been utilized in numerous applications. The expectation is that further research and applications of these algorithms will flourish in the near future. <s> BIB003
In very simple terms, the fractional Fourier transform (FRFT) is a generalization of the ordinary Fourier transform BIB002 . Specifically, the FRFT implements the so-called order parameter a which acts on the ordinary Fourier transform operator. In other words, the a-th order fractional Fourier transform represents the a-th power of the ordinary Fourier transform operator. When a ¼ p=2, we obtain the Fourier transform, while for a ¼ 0, we obtain the signal itself. Any intermediate value of a ð0 oa op=2Þ produces a signal representation that can be considered as a rotated time-frequency representation of the signal BIB003  . Interestingly enough, the idea of the fractional powers of the Fourier operator has been ''discovered'' several times in the literature. Initially, the idea appeared in the mathematical literature between the two world wars (e.g.,   ). More publications relating to this idea appeared after the second world war, however they were sporadic (e.g.  ). The idea of fractional Fourier operator re-gains a momentum in 1980s with publications by Namias (e.g. BIB001 ). Following Namias' contributions, a large number of papers appeared in the literature during 1990s tying the concept of the fractional Fourier operators to many other fields (e.g., time-frequency analysis as described in BIB003 ). We have also witnessed a number of recent contributions attempting to understand the practical applications of the FRFT beyond optics. The main goal of this publication is to provide an overview of recent developments regarding the FRFT and its applications. Although a number of publications reviewing the FRFT has also appeared in recent years (e.g., BIB002   ), some of these publications are geared toward explaining the mathematical eloquence behind the FRFT. Our goal is to simplify the theory behind the FRFT and provide an overview suitable for a signal processing practitioner. In particular, we emphasize the practicality of the FRFT by devoting a significant part of this manuscript to its discrete realizations and applications. By attracting more practitioners to this eloquent mathematical concept, we hope to foster more applications since the FRFT offers many valuable properties, which otherwise were not available with traditional tools (e.g., the Fourier transform). The paper is organized as follows: Section 2 introduces the concept of FRFT and relates the transform to other mathematical representations. In Section 3, we discuss various implementation approaches for discrete signals. Section 4 provides an overview of some FRFT applications, while in Section 5, we provide concluding remarks along with an outline of possible future directions.