Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> À1 <s> The Radon transform of a time-frequency distribution produces local areas of signal concentration that facilitate interpretation of multicomponent signals. The Radon transform can be efficiently implemented with dechirping in the time domains; however, only half of the possible projections through the time-frequency plane can be realized because of aliasing. It is shown that the frequency dual to dechirping exists, so that all of the time-frequency plane projections can be calculated efficiently. Some Radon transforms of Wigner distributions are demonstrated. > <s> BIB001 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> À1 <s> The authors present a kernel design technique based on using the Radon transform of the modulus of the ambiguity function of the signal for determination of angles and distances of radially distributed contents of the autoterms in the ambiguity domain. The proposed kernel effectively reduces the cross-terms and noise for linear FM signals. The result is a tool for high-resolution time-frequency representation of nonstationary, primarily linear FM signals. > <s> BIB002 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> À1 <s> In this study the degree p = 1 is assigned to the ordinary Fourier transform. The fractional Fourier transform, for example with degree P = 1/2, performs an ordinary Fourier transform if applied twice in a row. Ozaktas and Mendlovic [ “ Fourier transforms of fractional order and their optical implementation,” Opt. Commun. (to be published)] introduced the fractional Fourier transform into optics on the basis of the fact that a piece of graded-index (GRIN) fiber of proper length will perform a Fourier transform. Cutting that piece of GRIN fiber into shorter pieces corresponds to splitting the ordinary Fourier transform into fractional transforms. I approach the subject of fractional Fourier transforms in two other ways. First, I point out the algorithmic isomorphism among image rotation, rotation of the Wigner distribution function, and fractional Fourier transforming. Second, I propose two optical setups that are able to perform a fractional Fourier transform. <s> BIB003 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> À1 <s> Two recently described transforms are shown to be related. The Radon–Wigner transform is the squared modulus of the fractional Fourier transform. This new theorem may serve to translate signal and image processing results between different signal representations. Some consequences regarding moments are presented, including a new fractional-Fourier-transform uncertainty relation. Implications for processing are suggested. <s> BIB004 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> À1 <s> The fractional Fourier transform is a mathematical operation that generalizes the well-known Fourier transform. This operation has been shown to have physical and optical fundamental meanings, and it has been experimentally implemented by relatively simple optical setups. Based on the fractional Fourier-transform operation, a new space-frequency chart definition is introduced. By the application of various geometric operations on this new chart, such as radial and angular shearing and rotation, optical systems may be designed or analyzed. The field distribution, as well as full information about the spectrum and the space–bandwidth product, can be easily obtained in all the stages of the optical system. <s> BIB005 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> À1 <s> We consider the Cohen (1989) class of time-frequency distributions, which can be obtained from the Wigner distribution by convolving it with a kernel characterizing that distribution. We show that the time-frequency distribution of the fractional Fourier transform of a function is a rotated version of the distribution of the original function, if the kernel is rotationally symmetric. Thus, the fractional Fourier transform corresponds to rotation of a relatively large class of time-frequency representations (phase-space representations), confirming the important role this transform plays in the study of such representations. <s> BIB006 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> À1 <s> The Wigner distribution and many other members of the Cohen class of generalized phase-space distributions of a signal all share certain translation properties and the property that their two marginal distributions of energy density along the time and along the frequency axes equal the signal power and the spectral energy density. A natural generalization of this last property is shown to be a certain relationship through the Radon transform between the distribution and the signal's fractional Fourier transform. It is shown that the Wigner distribution is now distinguished by being the only member of the Cohen class that has this generalized property as well as a generalized translation property. The inversion theorem for the Wigner distribution is then extended to yield the fractional Fourier transforms. <s> BIB007 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> À1 <s> The fractional Fourier transform (FRFT) is a useful tool for signal processing. It is the generalization of the Fourier transform. Many fractional operations, such as fractional convolution, fractional correlation, and the fractional Hilbert transform, are defined from it. In fact, the FRFT can be further generalized into the linear canonical transform (LCT), and we can also use the LCT to define several canonical operations. In this paper, we discuss the relations between the operations described above and some important time-frequency distributions (TFDs), such as the Wigner distribution function (WDF), the ambiguity function (AF), the signal correlation function, and the spectrum correlation function. First, we systematically review the previous works in brief. Then, some new relations are derived and listed in tables. Then, we use these relations to analyze the applications of the FRPT/LCT to fractional/canonical filter design, fractional/canonical Hilbert transform, beam shaping, and then we analyze the phase-amplitude problems of the FRFT/LCT. For phase-amplitude problems, we find, as with the original Fourier transform, that in most cases, the phase is more important than the amplitude for the FRFT/LCT. We also use the WDF to explain why fractional/canonical convolution can be used for space-variant pattern recognition. <s> BIB008 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> À1 <s> The principal axes of the time-frequency representation of a signal are defined as those mutually orthogonal directions in the time-frequency plane for which the width of the signal's fractional power spectrum is minimum or maximum. The time-frequency kernels used in the Cohen class of time-frequency representations are then rotated in the time-frequency plane, in order to align the kernels' preferred axes to the signal's principal axes. It is shown that the resulting time-frequency representations show a better reduction of cross-terms without too severely degrading the autoterms than the corresponding original time-frequency representations. <s> BIB009 </s> Fractional Fourier transform as a signal processing tool: An overview of recent developments <s> À1 <s> A new signal-adaptive joint time-frequency distribution for the analysis of nonstationary signals is proposed. It is based on a fractional-Fourier-domain realization of the weighted Wigner distribution producing auto-terms close to the ones in the Wigner distribution itself, but with reduced cross-terms. Improvement over the standard time-frequency representations is achieved when the principal axes of a signal (defined as mutually orthogonal directions in the time-frequency plane for which the width of the signal's fractional power spectrum is minimum or maximum) do not correspond to time and frequency. The computational cost of this fractional-domain realization is the same as the computational cost of the realizations in the time or the frequency domain, since the windowed Fourier transform of the fractional Fourier transform of a signal corresponds to the short-time Fourier transform of the signal itself, with the window being the fractional Fourier transform of the initial one. The appropriate fractional domain is found from the knowledge of three second-order fractional Fourier transform moments. Numerical simulations confirm a qualitative advantage in the time-frequency representation, when the calculation is done in the optimal fractional domain. The approach can be generalized to the time-frequency distributions from the Cohen class. <s> BIB010
y-t,t-f ffðy,tÞg is rotational symmetric BIB002 BIB006 BIB009 BIB010 , where fðy,tÞ is a two-dimensional kernel function. For example, the FRFT corresponds to the rotation of the Wigner distribution (WD) in the time-frequency plane BIB009 BIB003 BIB005 BIB007 BIB008 : WD FaðxðtÞÞ ðt,f Þ ¼ WD x ðtcosaÀf sina,usina þf cosaÞ ð 14Þ The relationship between FRFT and Radon-Wigner distribution (RWD) (e.g., BIB001 ) is studied in BIB005 BIB004 , and it was shown that the RWD is the squared modulus of the FRFT: Additionally, we should point out that the FRFT could be potentially useful for obtaining time-frequency distributions without cross-terms. For example, WD and AF exhibit different behaviors when considering signals with multiple components. Namely, the WD has signal components (i.e., autoterms) concentrated around their instantaneous frequencies and group delays, while interferences (cross-terms) are located between these components. On the other hand, the AF has the auto-terms concentrated around the origin while the interferences are dislocated from the origin. The connection between these two time-frequency representations is the 2D Fourier transform: with t 2 ½À1,1. The AF of this signal is depicted in Fig. 2 (a). All four auto-terms are located in the middle of the graph (i.e., the origin of the ambiguity plane). Each of four horizontal and vertical objects with respect to the origin represents two crossterms, while the corner objects represent single cross-terms. In the case of the WD (Fig. 2(b) ), the corner objects are autoterms, while the central object represents the four cross-terms. The remaining objects contain two cross terms each. However, all 16 components (4 auto terms and 12 cross-terms) are clearly separated as shown in Fig. 2 (c) when we used the FRFT operator with a ¼ 0:5 in (16).