Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> FRFT and linear canonical transform <s> The fractional Fourier transform (FRFT) has been used for many years, and it is useful in many applications. Most applications of the FRFT are based on the design of fractional filters (such as removal of chirp noise and the fractional Hilbert transform) or on fractional correlation (such as scaled space-variant pattern recognition). In this study we introduce several types of simplified fractional Fourier transform (SFRFT). Such transforms are all special cases of a linear canonical transform (an affine Fourier transform or an ABCD transform). They have the same capabilities as the original FRFT for design of fractional filters or for fractional correlation. But they are simpler than the original FRFT in terms of digital computation, optical implementation, implementation of gradient-index media, and implementation of radar systems. Our goal is to search for the simplest transform that has the same capabilities as the original FRFT. Thus we discuss not only the formulas and properties of the SFRFT's but also their implementation. Although these SFRFT's usually have no additivity properties, they are useful for the practical applications. They have great potential for replacing the original FRFT's in many applications. <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> FRFT and linear canonical 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 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> FRFT and linear canonical transform <s> The linear canonical transform (the LCT) is a useful tool for optical system analysis and signal processing. It is parameterized by a 2/spl times/2 matrix {a, b, c, d}. Many operations, such as the Fourier transform (FT), fractional Fourier transform (FRFT), Fresnel transform, and scaling operations are all the special cases of the LCT. We discuss the eigenfunctions of the LCT. The eigenfunctions of the FT, FRFT, Fresnel transform, and scaling operations have been known, and we derive the eigenfunctions of the LCT based on the eigenfunctions of these operations. We find, for different cases, that the eigenfunctions of the LCT also have different forms. When |a+d| 2, the eigenfunctions become the chirp multiplication and chirp convolution of self-similar functions (fractals). Besides, since many optical systems can be represented by the LCT, we can thus use the eigenfunctions of the LCT derived in this paper to discuss the self-imaging phenomena in optics. We show that there are usually varieties of input functions that can cause the self-imaging phenomena for an optical system. <s> BIB003
The linear canonical transform (LCT) is a multiparameter integral transform and it represents a generalization of many mathematical transformations (e.g., the Fourier transform, FRFT, Fresnel transform) BIB002 BIB003 . It should be mentioned that the LCT is also called the affine Fourier transform, the generalized Fresnel transform, the Collins formula, the ABCD transform, or the almost Fourier and almost Fresnel transformation BIB002 BIB003 . The transformation is useful in many practical applications such as optics, radar system analysis, filter design, phase retrieval, and pattern recognition BIB003 . The LCT is defined by BIB002 BIB003 BIB001 LCT L ðxðtÞÞ ¼ where ad Àbc =1, L ¼ fa,b,c,dg. Using the above equation, it is straightforward to show that the FRFT is the special case of LCT where fa,b,c,dg ¼ fcosa,sina,Àsina,cosag with some phase correction BIB002 BIB003 BIB001 . In other words,