Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> The fractional Fourier transform <s> The functional Fourier transform (FRFT), which is a generalization of the classical Fourier transform, was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community, to which it may, however, be potentially useful. The FRFT depends on a parameter /spl alpha/ and can be interpreted as a rotation by an angle /spl alpha/ in the time-frequency plane. An FRFT with /spl alpha/=/spl pi//2 corresponds to the classical Fourier transform, and an FRFT with /spl alpha/=0 corresponds to the identity operator. On the other hand, the angles of successively performed FRFTs simply add up, as do the angles of successive rotations. The FRFT of a signal can also be interpreted as a decomposition of the signal in terms of chirps. The authors briefly introduce the FRFT and a number of its properties and then present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT's relationships with time-frequency representations such as the Wigner distribution, the ambiguity function, the short-time Fourier transform and the spectrogram. These relationships have a very simple and natural form and support the FRFT's interpretation as a rotation operator. Examples of FRFTs of some simple signals are given. An example of the application of the FRFT is also given. > <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> The fractional Fourier transform <s> In recent years, the fractional Fourier transform has been the focus of many research papers. In this letter, we show that the fractional Fourier transform is nothing more than a variation of the standard Fourier transform and, as such, many of its properties, such as its inversion formula and sampling theorems, can be deduced from those of the Fourier transform by a simple change of variable. <s> BIB002 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> The fractional Fourier transform <s> The paper investigates the possibility for giving a general definition of the fractional Fourier transform (FRT) for all signal classes [one-dimensional (1-D) and multidimensional, continuous and discrete, periodic and aperiodic]. Since the definition is based on the eigenfunctions of the ordinary Fourier transform (FT), the preliminary conditions is that the signal domain/periodicity be the same as the FT domain/periodicity. Within these classes, a general FRT definition is formulated, and the FRT properties are established. In addition, the multiplicity (which is intrinsic in a fractional operator) is clearly developed. The general definition is checked in the case in which the FRT is presently available and, moreover, to establish the FRT in new classes of signals. <s> BIB003 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> The fractional 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> BIB004
The fractional Fourier transform (FRFT) is a linear operator defined as   BIB001 BIB003  and dðtÞ representing the Dirac function. Throughout the paper we use F a to denote the operator associated with the FRFT. It should be noted that we adopted notation for the FRFT found in signal processing literature. In particular, we denote the rotation angle by a. Mathematicians usually denote the rotation angle by a, where a ¼ 2a=p. It is important to point out that various a values provide transformations with distinctive properties. Hence, a can be adjusted in many applications to provide enhanced results in comparison to other existing methods. Additional computational complexity associated with such optimization efforts is often acceptable. Given the properties of the kernel, Eq. (1) is equal to x(t) when a is a multiple of 2p, and is equal to x( À t) when aþp is a multiple of 2p. Some properties associated with the FRFT are summarized as follows BIB004 BIB002 : 1. The standard Fourier transform is a special case of the FRFT with a rotation angle a ¼ p=2. 2. F 0 is the identity operator, i.e., F 0 ðxðtÞÞ ¼ xðtÞ. The same can be stated for F 2p . 3. The FRFT is a linear operator, i.e., F a ð P k c k x k ðtÞÞ ¼ P k c k F a ðx k ðtÞÞ. 4. The FRFT adheres to commutativity (i.e., F a 1 F a 2 ¼ F a 2 F a 1 ) and associativity (i.e., ðF a 3 F a 2 ÞF a 1 ¼ F a 3 ðF a 2 F a 1 ÞÞ.