Fractional Fourier transform as a signal processing tool: An overview of recent developments
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paper_title: Time-frequency feature representation using energy concentration: An overview of recent advances

paper_content:

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.
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paper_title: The Fractional Fourier Transform: With Applications in Optics and Signal Processing

paper_content:

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.
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paper_title: The Fractional Order Fourier Transform and its Application to Quantum Mechanics

paper_content:

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.
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paper_title: On the relationship between the Fourier and fractional Fourier transforms

paper_content:

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.
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paper_title: A unified framework for the fractional Fourier transform

paper_content:

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.
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paper_title: The Fractional Fourier Transform: With Applications in Optics and Signal Processing

paper_content:

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.
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paper_title: The fractional Fourier transform and time-frequency representations

paper_content:

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. >
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paper_title: The Fractional Fourier Transform: With Applications in Optics and Signal Processing

paper_content:

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.
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paper_title: The fractional Fourier transform and time-frequency representations

paper_content:

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. >
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paper_title: The Fractional Fourier Transform: With Applications in Optics and Signal Processing

paper_content:

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.
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paper_title: The Fractional Order Fourier Transform and its Application to Quantum Mechanics

paper_content:

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.
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paper_title: On fractional Fourier transform moments

paper_content:

Based on the relation between the ambiguity function represented in a quasipolar coordinate system and the fractional power spectra, the fractional Fourier transform (FT) moments are introduced. Important equalities for the global second order fractional FT moments are derived, and their applications for signal analysis are discussed. The connection between the local moments and the angle derivative of the fractional power spectra is established. This permits us to solve the phase retrieval problem if only two close fractional power spectra are known.
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paper_title: New signal representation based on the fractional Fourier transform: definitions

paper_content:

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.
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paper_title: Relationships between the Radon–Wigner and fractional Fourier transforms

paper_content:

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.
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paper_title: The fractional fourier transform and the wigner distribution

paper_content:

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.
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paper_title: Radon transformation of time-frequency distributions for analysis of multicomponent signals

paper_content:

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. >
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paper_title: Time-frequency signal analysis based on the windowed fractional Fourier transform

paper_content:

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.
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paper_title: Kernel design for time-frequency signal analysis using the Radon transform

paper_content:

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. >
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paper_title: Relations between fractional operations and time-frequency distributions, and their applications

paper_content:

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.
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paper_title: On rotated time-frequency kernels

paper_content:

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.
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paper_title: Image rotation, Wigner rotation, and the fractional Fourier transform

paper_content:

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.
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paper_title: Effect of fractional Fourier transformation on time-frequency distributions belonging to the Cohen class

paper_content:

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.
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paper_title: A new form of the Fourier transform for time-varying frequency estimation

paper_content:

The local polynomial Fourier transform (LPFT) and the local polynomial periodogram (LPP) are proposed in order to estimate the rapidly time-varying instantaneous frequency (IF) /spl Omega/(t) of a harmonic signal. The LPFT gives the time-frequency power distribution over the t-(/spl Omega/(t), d/spl Omega/(t)/dt,...,d/sup m-1//spl Omega/(t)/dt/sup m-1/) space, where m is a degree of the LPFT. The LPFT enables one to estimate both the time-varying frequency and its derivatives. The technique is based on fitting the local polynomial approximation of the frequency which implements a high-order nonparametric regression. The a priori information about bounds for the frequency and its derivatives can be incorporated to improve the accuracy of the estimation. The asymptotic mean square errors, bias and covariance, of the estimators of d/sup s//spl Omega/(t)/dt/sup s/, s=0, 1,2,...,m/sup -1/, are obtained. The considered estimators are high-order generalization of the short-time Fourier transform. The comparative study of the asymptotic variance and bias of the estimates is presented.
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paper_title: Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing

paper_content:

Many useful relations between the Gabor transform (GT) and the fractional Fourier transform (FRFT) have been derived. First, we find that, like the Wigner distribution function (WDF), the FRFT is also equivalent to the rotation operation of the GT. Then, we show that performing the scaled inverse Fourier transform (IFT) along an oblique line of the GT of f(t) can yield its FRFT. Since the GT is closely related to the FRFT, we can use it for analyzing the characteristics of the FRFT. Compared with the WDF, the GT does not have the cross-term problem. This advantage is important for the applications of filter design, sampling, and multiplexing in the FRFT domain. Moreover, we find that if the GT is combined with the WDF, the resultant operation [called the Gabor-Wigner transform (GWT)] also has rotation relation with the FRFT. We also derive the general form of the linear distribution that has rotation relation with the FRFT.
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paper_title: Discrete-time local polynomial approximation of the instantaneous frequency

paper_content:

The local polynomial approximation (LPA) of the time-varying phase is used to develop a new form of the Fourier transform and the local polynomial periodogram (LPP) as an estimator of the instantaneous frequency (IF) /spl Omega/(t) of a harmonic complex-valued signal. The LPP is interpreted as a time-frequency energy distribution over the t-(/spl Omega/(t), /spl Omega//sup 1/(t)),...,/spl Omega//sup m-1/(t) space, where m is a degree of the LPA. The variance and bias of the estimate are studied for the short- and long-time asymptotic behavior of the IF estimates. In particular, it is shown that the optimal asymptotic mean squared errors of the estimates of /spl Omega//sup k-1/(t) have orders O(N/sup -(2k+1)/) and O(N/sup -/2(m-k+1)/2m+3), k=1.2,...,m, respectively, for a polynomial /spl Omega/(t) of the degree m-1 and arbitrary smooth /spl Omega/(t) with a bounded mth derivative.
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paper_title: On rotated time-frequency kernels

paper_content:

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.
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paper_title: Fractional Gabor transform

paper_content:

A fractional Gabor transform (FRGT) is proposed. This new transform is a generalization of the conventional Gabor transform (GT) based on the Fourier transform to the windowed fractional Fourier transform (FRFT). The FRGT provides analyses of signals in both the real space and the FRFT frequency domain simultaneously. The space–FRFT frequency pattern can be rotated as the fractional order changes. The FRGT has an additional freedom, compared with the conventional GT, i.e., the transform order. The FRGT may offer a useful tool for guiding optimal filter design in the FRFT domain in signal processing.
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paper_title: Time-frequency feature representation using energy concentration: An overview of recent advances

paper_content:

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.
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paper_title: An uncertainty principle for real signals in the fractional Fourier transform domain

paper_content:

The fractional Fourier transform (FrFT) can be thought of as a generalization of the Fourier transform to rotate a signal representation by an arbitrary angle /spl alpha/ in the time-frequency plane. A lower bound on the uncertainty product of signal representations in two FrFT domains for real signals is obtained, and it is shown that a Gaussian signal achieves the lower bound. The effect of shifting and scaling the signal on the uncertainty relation is discussed. An example is given in which the uncertainty relation for a real signal is obtained, and it is shown that this relation matches with that given by the uncertainty relation derived.
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paper_title: Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms

paper_content:

A concise introduction to the concept of fractional Fourier transforms is followed by a discussion of their relation to chirp and wavelet transforms. The notion of fractional Fourier domains is developed in conjunction with the Wigner distribution of a signal. Convolution, filtering, and multiplexing of signals in fractional domains are discussed, revealing that under certain conditions one can improve on the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing.
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paper_title: Three uncertainty relations for real signals associated with linear canonical transform

paper_content:

Uncertainty principle plays an important role in signal processing, physics and mathematics, and it represents the relations between time spread and frequency spread (or position and velocity). Linear canonical transform (LCT) is one generalisation of Fresnel transform, fractional Fourier transform and others. The LCT has been used in physical optics and signal processing. Three novel results of uncertainty principle in the LCT domains are obtained here, in which one is connected with parameters a and b and the other one is connected with c and d; the last one is connected with the four transformation parameters a, b, c and d. Their physical meanings are given as well. These results disclose the inequalities' relations between two spreads, between two group delays and between one spread and one group delay in the LCT domains. It also shows that any one of the three cases can reduce to classical uncertainty principle in time/frequency domain. The effects of time scaling on these results' bounds are also involved.
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paper_title: Time-frequency signal analysis based on the windowed fractional Fourier transform

paper_content:

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.
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paper_title: A fractional Gabor expansion

paper_content:

We present a fractional Gabor expansion on a non-rectangular time–frequency lattice. Sinusoidal analysis used in the traditional Gabor expansion is not appropriate for a compact representation for chirp-type signals. Basis functions of the proposed expansion are obtained via fractional Fourier basis. Completeness and biorthogonality conditions of the new expansion are derived.
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paper_title: Tomography time-frequency transform

paper_content:

The paper shows that the fractional Fourier transform (FRFT) of a signal is the Radon transform of the time-frequency distribution of the same signal. Therefore, a time-frequency distribution known as the tomography time-frequency transform (TTFT) is defined as the inverse Radon transform of the FRFT of the signal. Because the computation of the TTFT does not explicitly require any window or kernel function, high resolutions in both the frequency and time domains can be achieved. When the signal contains multiple components, the cross terms can be effectively removed by an adaptive filtering process that is applied on the FRFT rather than the final result. Therefore, distortions made by the filtering process on the desired signal components can be minimized.
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paper_title: Fractional-Fourier-domain weighted Wigner distribution

paper_content:

A fractional-Fourier-domain realization of the weighted Wigner distribution (or S-method), producing auto-terms close to the ones in the Wigner distribution itself, but with reduced cross-terms, is presented. 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 short-time 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 analysis of the second-order fractional Fourier transform moments. Numerical simulations show a qualitative advantage in the time-frequency representation, when the calculation is done in the optimal fractional domain.
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paper_title: FRACTIONAL FOURIER-TRANSFORMS, WAVELET TRANSFORMS, AND ADAPTIVE NEURAL NETWORKS

paper_content:

A new optical architecture is developed, based on fractional Fourier transforms, that compromises between shift-invariant (frequency) and position-dependent filtering. The analogy of this architecture to wavelet transforms and adaptive neural networks is also presented. The ambiguity and Wigner distribution functions are obtainable from special cases of the filter. The filter design corresponds to the training of the neural networks, and an adaptive learning algorithm is developed based on gradient-descent error minimization and error back propagation. The extension to multilayer architecture is straightforward.
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paper_title: Discrete rotational Gabor transform

paper_content:

In this paper, we present a discrete rotational Gabor transform and apply it to signal representation on a non-rectangular time-frequency plane tiling. We use a modified version of the recently proposed discrete rotational Fourier transform. The rotational Gabor transform decomposes a signal using as basis functions scaled, translated and modulated by linear chirps windows. As a result, the time-varying frequency content of a signal is represented better than with sinusoidal modulated expansions. For a multi-component signal, the Gabor coefficients are obtained by combining different tilings to maximize a local energy concentration measure. This permits us to achieve a highly localized time-frequency signal representation. Examples are given to illustrate the transformation.
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paper_title: Fractional Fourier transform of the Gaussian and fractional domain signal support

paper_content:

The fractional Fourier transform (FrFT) provides an important extension to conventional Fourier theory for the analysis and synthesis of linear chirp signals. It is a parameterised transform which can be used to provide extremely compact representations. The representation is maximally compressed when the transform parameter, /spl alpha/, is matched to the chirp rate of the input signal. Existing proofs are extended to demonstrate that the fractional Fourier transform of the Gaussian function also has Gaussian support. Furthermore, expressions are developed which allow calculation of the spread of the signal representation for a Gaussian windowed linear chirp signal in any fractional domain. Both continuous and discrete cases are considered. The fractional domains exhibiting minimum and maximum support for a given signal define the limit on joint time-frequency resolution available under the FrFT. This is equated with a restatement of the uncertainty principle for linear chirp signals and the fractional Fourier domains. The calculated values for the fractional domain support are tested empirically through comparison with the discrete transform output for a synthetic signal with known parameters. It is shown that the same expressions are appropriate for predicting the support of the ordinary Fourier transform of a Gaussian windowed linear chirp signal.
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paper_title: The Fractional Fourier Transform: With Applications in Optics and Signal Processing

paper_content:

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.
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paper_title: Eigenfunctions of linear canonical transform

paper_content:

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.
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paper_title: Simplified fractional Fourier transforms.

paper_content:

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.
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paper_title: Design of higher order polynomial Wigner-Ville distributions

paper_content:

The polynomial Wigner-Ville distribution (PWVD) is a time-frequency signal analysis tool for representing polynomial frequency modulated (FM) signals. We present an optimization procedure to design PWVDs and discuss the need for higher order PWVDs.
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paper_title: A multitime definition of the Wigner higher order distribution: L-Wigner distribution

paper_content:

A dual form of the Wigner higher order spectra is introduced. Its analysis in the case of multicomponent signals is performed. An efficient distribution for time-frequency signal analysis (L-Wigner distribution) is derived from that analysis. The theory is illustrated on a numerical example. >
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paper_title: Effect of fractional Fourier transformation on time-frequency distributions belonging to the Cohen class

paper_content:

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.
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paper_title: Numerical calculation of fractional Fourier transforms with a single fast-Fourier-transform algorithm

paper_content:

A numerical algorithm based on a single fast Fourier transform is proposed. Its precision and calculation efficiency show better performance than those of previously published algorithms. It is also shown that if specific conditions are met, the numerical calculations of two successive fractional Fourier transforms produce results that are similar to the analytical solution.
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paper_title: On the relationship between the Fourier and fractional Fourier transforms

paper_content:

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.
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paper_title: Digital computation of the fractional Fourier transform

paper_content:

An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N, the presented algorithm computes the fractional transform in O(NlogN) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed.
---
paper_title: On bandlimited signals with fractional Fourier transform

paper_content:

We study bandlimited signals with fractional Fourier transform (FRFT). We show that if a nonzero signal f is bandlimited with FRFT F/sub /spl alpha// for a certain real /spl alpha/, then it is not bandlimited with FRFT F/sub /spl beta// for any /spl beta/ with /spl beta//spl ne//spl plusmn//spl alpha/+n/spl pi/ for any integer n. This is a generalization of the fact that a nonzero signal can not be both timelimited and bandlimited. We also provide sampling theorems for bandlimited signals with FRFT that are similar to the Shannon sampling theorem.
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paper_title: Sampling and series expansion theorems for fractional Fourier and other transforms

paper_content:

We present much briefer and more direct and transparent derivations of some sampling and series expansion relations for fractional Fourier and other transforms. In addition to the fractional Fourier transform, the method can also be applied to the Fresnel, Hartley, and scale transform and other relatives of the Fourier transform.
---
paper_title: Improved discrete fractional Fourier transform

paper_content:

The fractional Fourier transform is a useful mathematical operation that generalizes the well-known continuous Fourier transform. Several discrete fractional Fourier transforms (DFRFT’s) have been developed, but their results do not match those of the continuous case. We propose a new DFRFT. This improved DFRFT provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties.
---
paper_title: New sampling formulae for the fractional Fourier transform

paper_content:

In this note we obtain two new sampling formulae for reconstructing signals that are band limited or time limited in the fractional Fourier transform sense. In both cases, we use samples from both the signal and its Hilbert transform, but each taken at half the Nyquist rate.
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paper_title: The discrete fractional Fourier transform

paper_content:

We propose and consolidate a definition of the discrete fractional Fourier transform which generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform (FRT) generalizes the continuous ordinary Fourier Transform. This definition is based on a particular set of eigenvectors of the DFT which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The fact that this definition satisfies all the desirable properties expected of the discrete FRT, supports our confidence that it will be accepted as the definitive definition of this transform.
---
paper_title: Closed-form discrete fractional and affine Fourier transforms

paper_content:

The discrete fractional Fourier transform (DFRFT) is the generalization of discrete Fourier transform. Many types of DFRFT have been derived and are useful for signal processing applications. We introduce a new type of DFRFT, which are unitary, reversible, and flexible; in addition, the closed-form analytic expression can be obtained. It works in performance similar to the continuous fractional Fourier transform (FRFT) and can be efficiently calculated by the FFT. Since the continuous FRFT can be generalized into the continuous affine Fourier transform (AFT) (the so-called canonical transform), we also extend the DFRFT into the discrete affine Fourier transform (DAFT). We derive two types of the DFRFT and DAFT. Type 1 is similar to the continuous FRFT and AFT and can be used for computing the continuous FRFT and AFT. Type 2 is the improved form of type 1 and can be used for other applications of digital signal processing. Meanwhile, many important properties continuous FRFT and AFT are kept in the closed-form DFRFT and DAFT, and some applications, such as filter design and pattern recognition, are also discussed. The closed-form DFRFT we introduce has the lowest complexity among all current DFRFTs that is still similar to the continuous FRFT.
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paper_title: Unified fractional Fourier transform and sampling theorem

paper_content:

The fractional Fourier transform (FRT) is an extension of the ordinary Fourier transform (FT). Applying the language of the unified FT, we develop FRT expressions for discrete and continuous signals, introducing a particular form of periodicity: chirp-periodicity. The FRT sampling theorem is derived as an extension of its ordinary counterpart.
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paper_title: Fractional Fourier transform of bandlimited periodic signals and its sampling theorems

paper_content:

Abstract In this correspondence, some features of the fractional Fourier transform (FRFT) of the bandlimited periodic signals are discussed. Two sampling relations in the context of FRFT domain bandlimited, periodic signals are also presented. More specifically, it is shown that only 2 K + 1 coefficients are sufficient to reconstruct any FRFT domain bandlimited periodic signal, where K is the order of the highest nonzero harmonic component in the particular FRFT domain in which the signal is bandlimited.
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paper_title: Discrete fractional Fourier transform based on orthogonal projections

paper_content:

The continuous fractional Fourier transform (FRFT) performs a spectrum rotation of signal in the time-frequency plane, and it becomes an important tool for time-varying signal analysis. A discrete fractional Fourier transform has been developed by Santhanam and McClellan (see ibid., vol.42, p.994-98, 1996) but its results do not match those of the corresponding continuous fractional Fourier transforms. We propose a new discrete fractional Fourier transform (DFRFT). The new DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT. To obtain DFT Hermite eigenvectors, two orthogonal projection methods are introduced. Thus, the new DFRFT will provide similar transform and rotational properties as those of continuous fractional Fourier transforms. Moreover, the relationship between FRFT and the proposed DFRFT has been established in the same way as the conventional DFT-to-continuous-Fourier transform.
---
paper_title: Sampling and Sampling Rate Conversion of Band Limited Signals in the Fractional Fourier Transform Domain

paper_content:

The fractional Fourier transform (FRFT) has become a very active area in signal processing community in recent years, with many applications in radar, communication, information security, etc., This study carefully investigates the sampling of a continuous-time band limited signal to obtain its discrete-time version, as well as sampling rate conversion, for the FRFT. Firstly, based on product theorem for the FRFT, the sampling theorems and reconstruction formulas are derived, which explain how to sample a continuous-time signal to obtain its discrete-time version for band limited signals in the fractional Fourier domain. Secondly, the formulas and significance of decimation and interpolation are studied in the fractional Fourier domain. Using the results, the sampling rate conversion theory for the FRFT with a rational fraction as conversion factor is deduced, which illustrates how to sample the discrete-time version without aliasing. The theorems proposed in this study are the generalizations of the conventional versions for the Fourier transform. Finally, the theory introduced in this paper is validated by simulations.
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paper_title: The discrete rotational Fourier transform

paper_content:

We define a discrete version of the angular Fourier transform and present the properties of the transform that show it to be a rotation in time-frequency space, this new transform is a generalization of the DFT. Efficient algorithms for its computation can then be based on the FFT and the eigenstructure of the DFT.
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paper_title: Eigenvectors and functions of the discrete Fourier transform

paper_content:

A method is presented for computing an orthonormal set of eigenvectors for the discrete Fourier transform (DFT). The technique is based on a detailed analysis of the eigenstructure of a special matrix which commutes with the DFT. It is also shown how fractional powers of the DFT can be efficiently computed, and possible applications to multiplexing and transform coding are suggested.
---
paper_title: Discrete fractional Fourier transform based on orthogonal projections

paper_content:

The continuous fractional Fourier transform (FRFT) performs a spectrum rotation of signal in the time-frequency plane, and it becomes an important tool for time-varying signal analysis. A discrete fractional Fourier transform has been developed by Santhanam and McClellan (see ibid., vol.42, p.994-98, 1996) but its results do not match those of the corresponding continuous fractional Fourier transforms. We propose a new discrete fractional Fourier transform (DFRFT). The new DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT. To obtain DFT Hermite eigenvectors, two orthogonal projection methods are introduced. Thus, the new DFRFT will provide similar transform and rotational properties as those of continuous fractional Fourier transforms. Moreover, the relationship between FRFT and the proposed DFRFT has been established in the same way as the conventional DFT-to-continuous-Fourier transform.
---
paper_title: Discrete fractional Hartley and Fourier transforms

paper_content:

This paper is concerned with the definitions of the discrete fractional Hartley transform (DFRHT) and the discrete fractional Fourier transform (DFRFT). First, the eigenvalues and eigenvectors of the discrete Fourier and Hartley transform matrices are investigated. Then, the results of the eigendecompositions of the transform matrices are used to define DFRHT and DFRFT. Also, an important relationship between DFRHT and DFRFT is described, and numerical examples are illustrated to demonstrate that the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT. Finally, a filtering technique in the fractional Fourier transform domain is applied to remove chirp interference.
---
paper_title: Digital computation of the fractional Fourier transform

paper_content:

An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N, the presented algorithm computes the fractional transform in O(NlogN) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed.
---
paper_title: The multiple-parameter discrete fractional Fourier transform

paper_content:

The discrete fractional Fourier transform (DFRFT) is a generalization of the discrete Fourier transform (DFT) with one additional order parameter. In this letter, we extend the DFRFT to have N order parameters, where N is the number of the input data points. The proposed multiple-parameter discrete fractional Fourier transform (MPDFRFT) is shown to have all of the desired properties for fractional transforms. In fact, the MPDFRFT reduces to the DFRFT when all of its order parameters are the same. To show an application example of the MPDFRFT, we exploit its multiple-parameter feature and propose the double random phase encoding in the MPDFRFT domain for encrypting digital data. The proposed encoding scheme in the MPDFRFT domain significantly enhances data security.
---
paper_title: Improved discrete fractional Fourier transform

paper_content:

The fractional Fourier transform is a useful mathematical operation that generalizes the well-known continuous Fourier transform. Several discrete fractional Fourier transforms (DFRFT’s) have been developed, but their results do not match those of the continuous case. We propose a new DFRFT. This improved DFRFT provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties.
---
paper_title: Fractional discrete Fourier transforms

paper_content:

Direct calculation of fractional Fourier transforms from the expressions derived for their optical implementation is laborious. An extension of the discrete Fourier transform would have only O(N2) computational complexity. We define such a system, offer a general way to compute the fractional discrete Fourier transform matrix, and numerically validate the algorithm.
---
paper_title: The discrete fractional Fourier transform

paper_content:

We propose and consolidate a definition of the discrete fractional Fourier transform which generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform (FRT) generalizes the continuous ordinary Fourier Transform. This definition is based on a particular set of eigenvectors of the DFT which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The fact that this definition satisfies all the desirable properties expected of the discrete FRT, supports our confidence that it will be accepted as the definitive definition of this transform.
---
paper_title: On Higher Order Approximations for Hermite–Gaussian Functions and Discrete Fractional Fourier Transforms

paper_content:

Discrete equivalents of Hermite-Gaussian functions play a critical role in the definition of a discrete fractional Fourier transform. The discrete equivalents are typically calculated through the eigendecomposition of a commutator matrix. In this letter, we first characterize the space of DFT-commuting matrices and then construct matrices approximating the Hermite-Gaussian generating differential equation and use the matrices to accurately generate the discrete equivalents of Hermite-Gaussians.
---
paper_title: Discrete fractional Fourier transform based on orthogonal projections

paper_content:

The continuous fractional Fourier transform (FRFT) performs a spectrum rotation of signal in the time-frequency plane, and it becomes an important tool for time-varying signal analysis. A discrete fractional Fourier transform has been developed by Santhanam and McClellan (see ibid., vol.42, p.994-98, 1996) but its results do not match those of the corresponding continuous fractional Fourier transforms. We propose a new discrete fractional Fourier transform (DFRFT). The new DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT. To obtain DFT Hermite eigenvectors, two orthogonal projection methods are introduced. Thus, the new DFRFT will provide similar transform and rotational properties as those of continuous fractional Fourier transforms. Moreover, the relationship between FRFT and the proposed DFRFT has been established in the same way as the conventional DFT-to-continuous-Fourier transform.
---
paper_title: Discrete fractional Hartley and Fourier transforms

paper_content:

This paper is concerned with the definitions of the discrete fractional Hartley transform (DFRHT) and the discrete fractional Fourier transform (DFRFT). First, the eigenvalues and eigenvectors of the discrete Fourier and Hartley transform matrices are investigated. Then, the results of the eigendecompositions of the transform matrices are used to define DFRHT and DFRFT. Also, an important relationship between DFRHT and DFRFT is described, and numerical examples are illustrated to demonstrate that the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT. Finally, a filtering technique in the fractional Fourier transform domain is applied to remove chirp interference.
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paper_title: A method for the discrete fractional Fourier transform computation

paper_content:

A new method for the discrete fractional Fourier transform (DFRFT) computation is given in this paper. With the help of this method, the DFRFT of any angle can be computed by a weighted summation of the DFRFTs with the special angles.
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paper_title: Discrete Fractional Fourier Transform Based on New Nearly Tridiagonal Commuting Matrices

paper_content:

Based on discrete Hermite-Gaussian like functions, a discrete fractional Fourier transform (DFRFT) which provides sample approximations of the continuous fractional Fourier transform was defined and investigated recently. In this paper, we propose a new nearly tridiagonal matrix which commutes with the discrete Fourier transform (DFT) matrix. The eigenvectors of the new nearly tridiagonal matrix are shown to be better discrete Hermite-Gaussian like functions than those developed before. Furthermore, by appropriately combining two linearly independent matrices which both commute with the DFT matrix, we develop a method to obtain even better discrete Hermite-Gaussian like functions. Then, new versions of DFRFT produce their transform outputs more close to the samples of the continuous fractional Fourier transform, and their application is illustrated.
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paper_title: On the multiangle centered discrete fractional Fourier transform

paper_content:

Existing versions of the discrete fractional Fourier transform (DFRFT) are based on the discrete Fourier transform (DFT). These approaches need a full basis of DFT eigenvectors that serve as discrete versions of Hermite-Gauss functions. In this letter, we define a DFRFT based on a centered version of the DFT (CDFRFT) using eigenvectors derived from the Gru/spl uml/nbaum tridiagonal commutor that serve as excellent discrete approximations to the Hermite-Gauss functions. We develop a fast and efficient way to compute the multiangle version of the CDFRFT for a discrete set of angles using the FFT algorithm. We then show that the associated chirp-frequency representation is a useful analysis tool for multicomponent chirp signals.
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paper_title: Understanding discrete rotations

paper_content:

The concept of rotations in continuous-time, continuous-frequency is extended to discrete-time, discrete-frequency as it applies to the Wigner distribution. As in the continuous domain, discrete rotations are defined to be elements of the special orthogonal group over the appropriate (discrete) field. Use of this definition ensures that discrete rotations will share many of the same mathematical properties as continuous ones. A formula is given for the number of possible rotations of a prime-length signal, and an example is provided to illustrate what such rotations look like. In addition, by studying a 90 degree rotation, we formulate an algorithm to compute a prime-length discrete Fourier transform (DFT) based on convolutions and multiplications of discrete, periodic chirps. This algorithm provides a further connection between the DFT and the discrete Wigner distribution based on group theory.
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paper_title: Fast Algorithms for Polynomial Time-Frequency Transforms of Real-Valued Sequences

paper_content:

This paper presents fast algorithms for computing the polynomial time-frequency transform that deals with a real-valued sequence of length-apb, where a, b and p are positive integers. In particular, it shows that the polynomial time-frequency transform has a conjugate symmetric property, similar to that of the discrete Fourier transform, if the input sequence is real-valued. The computational complexities needed by these proposed algorithms are analyzed in terms of the numbers of real additions and real multiplications. When a=3,4, and 8, comparisons show that the computational complexities required by the proposed algorithms are less than 60% of those needed by the fast algorithms for complex-valued sequences.
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paper_title: Fast Quadratic Phase Transform for Estimating the Parameters of Multicomponent Chirp Signals,

paper_content:

Abstract We present a technique based on the quadratic phase transform (QPT) for joint phase parameter estimation of multicomponent chirp signals. The QPT has the ability to analyze chirp signals in the same way as Fourier transform analyzes sinusoidal signals. In view of its practical implementation, a discrete version of the QPT, called the discrete quadratic phase transform (DQPT), is introduced. A novel fast computational procedure of the DQPT is developed. Explicit expressions for the arithmetic operation count are derived. Simulation results are presented to illustrate the usefulness of the DQPT.
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paper_title: Discrete Chirp-Fourier Transform and Its Application to Chirp Rate Estimation

paper_content:

The discrete Fourier transform (DFT) has found tremendous applications in almost all fields, mainly because it can be used to match the multiple frequencies of a stationary signal with multiple harmonics. In many applications, wideband and nonstationary signals, however, often occur. One of the typical examples of such signals is chirp-type signals that are usually encountered in radar signal processing, such as synthetic aperture radar (SAR) and inverse SAR imaging. Due to the motion of a target, the radar return signals are usually chirps, and their chirp rates include the information about the target, such as the location and the velocity. In this paper, we study discrete chirp-Fourier transform (DCFT), which is analogous to the DFT. Besides the multiple frequency matching similar to the DFT, the DCFT can be used to match the multiple chirp rates in a chirp-type signal with multiple chirp components. We show that when the signal length N is prime, the magnitudes of all the sidelobes of the DCFT of a quadratic chirp signal are 1, whereas the magnitude of the mainlobe of the DCFT is /spl radic/N. With this result, an upper bound for the number of the detectable chirp components using the DCFT is provided in terms of signal length and signal and noise powers. We also show that the N-point DCFT performs optimally when N is a prime.
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paper_title: A fast algorithm for fractional Fourier transforms

paper_content:

Abstract A fast algorithm for numerical calculation of arbitrary real order of fractional Fourier transforms is presented. Its computation complexity is comparable to that of fast convolution Fourier transform. Furthermore, it allows one to freely choose the sampling resolutions in both x - and u -space under the restriction of the Nyquist sampling theorem.
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paper_title: Eigenvectors and functions of the discrete Fourier transform

paper_content:

A method is presented for computing an orthonormal set of eigenvectors for the discrete Fourier transform (DFT). The technique is based on a detailed analysis of the eigenstructure of a special matrix which commutes with the DFT. It is also shown how fractional powers of the DFT can be efficiently computed, and possible applications to multiplexing and transform coding are suggested.
---
paper_title: Generalized Fast Algorithms for the Polynomial Time-Frequency Transform

paper_content:

This paper presents a general class of fast algorithms for computing the polynomial time-frequency transform (PTFT) of length-apb, where a, b, and p are positive integers. The process of derivation shows some interesting properties that are effectively used for minimization of the computational complexity. By assigning values of a, b, and p, various algorithms, for example, radix-o and split-radix-2/(2a), can be easily obtained to provide the flexibility supporting polynomial time-frequency transforms of various sequence lengths. The detailed analysis on the computational complexities needed by these algorithms is also presented in terms of the numbers of additions and multiplications. It is shown that the proposed algorithms significantly reduce the computational complexity for applications that deal with polynomial phase signals.
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paper_title: Time-frequency feature representation using energy concentration: An overview of recent advances

paper_content:

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.
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paper_title: The Fractional Fourier Transform: With Applications in Optics and Signal Processing

paper_content:

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.
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paper_title: Adaptive fractional Fourier domain filtering

paper_content:

A novel adaptive filtering scheme based on fractional Fourier transform (FrFT) is introduced and characterized. As a generalization of the ordinary Fourier transform, FrFT is a powerful analysis tool to describe signals with chirp-type components. We show that in case of linear frequency modulated (LFM) signals fractional Fourier domain adaptive filtering schemes provide less error and faster convergence. The adaptation algorithm is employed in an active noise control system application with various LFM signals and a real multi-component signal. Simulation results present that the adaptation performance in FrFT-domain adaptive filtering is superior compared to time-domain adaptation.
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paper_title: Optimal filtering in fractional Fourier domains

paper_content:

The ordinary Fourier transform is suited best for analysis and processing of time-invariant signals and systems. When we are dealing with time-varying signals and systems, filtering in fractional Fourier domains might allow us to estimate signals with smaller minimum mean square error (MSE). We derive the optimal fractional Fourier domain filter that minimizes the MSE for given non-stationary signal and noise statistics, and time-varying distortion kernel. We present an example for which the MSE is reduced by a factor of 50 as a result of filtering in the fractional Fourier domain, as compared to filtering in the conventional Fourier or time domains. We also discuss how the fractional Fourier transformation can be computed in O(N log N) time, so that the improvement in performance is achieved with little or no increase in computational complexity.
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paper_title: Convolution and Filtering in Fractional Fourier Domains

paper_content:

Fractional Fourier transforms, which are related to chirp and wavelet transforms, lead to the notion of fractional Fourier domains. The concept of filtering of signals in fractional domains is developed, revealing that under certain conditions one can improve upon the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing.
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paper_title: Watermarking in the space/spatial-frequency domain using two-dimensional Radon-Wigner distribution

paper_content:

A two-dimensional (2-D) signal with a variable spatial frequency is proposed as a watermark in the spatial domain. This watermark is characterized by a linear frequency change. It can be efficiently detected by using 2-D space/spatial-frequency distributions. The projections of the 2-D Wigner distribution-the 2-D Radon-Wigner distribution, are used in order to emphasize the watermark detection process. The watermark robustness with respect to some very important image processing attacks, such as for example, the translation, rotation, cropping, JPEG compression, and filtering, is demonstrated and tested by using Stirmark 3.1.
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paper_title: Noise-resistant watermarking in the fractional Fourier domain utilizing moment-based image representation

paper_content:

This paper presents a novel noise-robust scheme for watermark embedding and extraction, applicable on the broad scientific field of information security, including digital encryption for secure transmission. The proposed scheme employs Fourier coefficients for moment-based image analysis and the fractional Fourier transformation for watermark embedding. This approach maintains the advantages of spatial and frequency domain representations, offers two additional degrees of freedom associated with the fractional Fourier transformation angles, whereas it provides increased detectability of the embedded watermarks at the received end. The experimental evaluation of the proposed scheme leads to the conclusion that it is more robust in the presence of noise than previous schemes for watermark embedding and extraction.
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paper_title: Quickly tracing detection for spread spectrum watermark based on effect estimation of the affine transform

paper_content:

Recently, watermarking synchronization and spread spectrum watermarking schemes have become two interesting, new focuses of study. In this paper, we propose a method based on the synchronization pre-estimation in order to resist geometric attacks. First, we utilize the global features by testing them against translation, rotation and scaling, and present a simple method to estimate the geometric transform kernel of the two-dimensional gray image. Maps, applying some correlative barycenters to the original are produced that can accurately reflect the affine transform feature of the image. Furthermore, unknown variables existing in the kernel of the transform are calculated. Secondly, the watermark is directly embedded in the spatial domain. Watermarking tracing is well performed on the watermarked image instead of performing an inverse geometrical transform on it. Experimental results show that the proposed scheme can work well under several kinds of geometrical attacks.
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paper_title: A content-based image authentication scheme based on singular value decomposition

paper_content:

In this paper, we propose an image authentication scheme in which image features are embedded for copyright protection and content-tampering detection. The features are based on the invariance of the relationship among SVD coefficients. The proposed method is sensitive to malicious manipulations and robust to lossy compressions or regular image operations, such as brightening, shifting, averaging, rotation, and so on. Several experiments demonstrated that the proposed scheme could efficiently detect locations which have been tampered with and effectively resist several types of attacks. Moreover, stego images based on our proposed method have high visual quality.
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paper_title: Handbook Of Image And Video Processing

paper_content:

Thank you for reading handbook of image and video processing. As you may know, people have look numerous times for their favorite novels like this handbook of image and video processing, but end up in harmful downloads. Rather than enjoying a good book with a cup of coffee in the afternoon, instead they juggled with some harmful virus inside their computer. handbook of image and video processing is available in our book collection an online access to it is set as public so you can get it instantly. Our books collection hosts in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the handbook of image and video processing is universally compatible with any devices to read.
---
paper_title: Digital watermarking in the fractional Fourier transformation domain

paper_content:

An application of the fractional Fourier transform for the multimedia copyright protection is proposed in the paper. The watermark robustness as well as statistical performance are considered.
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paper_title: A Digital Watermarking Algorithm for Image Based on Fractional Fourier Transform

paper_content:

A novel digital watermaking embedding and detecting algorithm for image is presented, which uses the chrip signal as a watermark and embeds in the fractional Fourier transform (FRFT) domain of the image, and the watermark position and the transform order are used as the encryption keys. With the help of the property of the impulse characteristic in the FRFT domain for chirp signal, the watermark can be detected conveniently. The experiments results have show that the proposed algorithm not only is of good imperceptibility and security and is very robust to JPEG compression and to noise attacks, but also can provide protection under the cropping and the filtering.
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paper_title: Media hash-dependent image watermarking resilient against both geometric attacks and estimation attacks based on false positive-oriented detection

paper_content:

The major disadvantage of existing watermarking methods is their limited resistance to extensive geometric attacks. In addition, we have found that the weakness of multiple watermark embedding methods that were initially designed to resist geometric attacks is their inability to withstand the watermark-estimation attacks (WEAs), leading to reduce resistance to geometric attacks. In view of these facts, this paper proposes a robust image watermarking scheme that can withstand geometric distortions and WEAs simultaneously. Our scheme is mainly composed of three components: 1) robust mesh generation and mesh-based watermarking to resist geometric distortions; 2) construction of media hash-based content-dependent watermark to resist WEAs; and 3) a mechanism of false positive-oriented watermark detection, which can be used to determine the existence of a watermark so as to achieve a tradeoff between correct detection and false detection. Furthermore, extensive experimental results obtained using the standard benchmark (i.e., Stirmark) and WEAs, and comparisons with relevant watermarking methods confirm the excellent performance of our method in improving robustness. To our knowledge, such a thorough evaluation has not been reported in the literature before
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paper_title: Robust Watermarking in the Wigner Domain

paper_content:

In this paper, a new watermarking scheme in the joint time-frequency domain is introduced. Wigner distribution is used to transform an image into the spatial-spectral domain. The proposed method selects the time-frequency cells to be watermarked based on the particular image's energy distribution in the joint domain. This approach ensures the imperceptibility of the embedded watermark. It is shown that embedding in the time-frequency domain is equivalent to a nonlinear embedding function in the spatial domain. A corresponding watermark detection algorithm is also introduced. The performance of the proposed watermarking algorithm under possible attacks, such as noise, re-sampling, rotation, filtering, and JPEG compression is illustrated.
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paper_title: Image quality assessment using the joint spatial/spatial-frequency representation

paper_content:

This paper demonstrates the usefulness of spatial/spatial-frequency representations in image quality assessment by introducing a new image dissimilarity measure based on 2D Wigner-Ville distribution (WVD). The properties of 2D WVD are shortly reviewed, and the important issue of choosing the analytic image is emphasized. The WVD-based measure is shown to be correlated with subjective human evaluation, which is the premise towards an image quality assessor developed on this principle.
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paper_title: A Novel Blind Multiple Watermarking Technique for Images

paper_content:

Three novel blind watermarking techniques are proposed to embed watermarks into digital images for different purposes. The watermarks are designed to be decoded or detected without the original images. The first one, called single watermark embedding (SWE), is used to embed a watermark bit sequence into digital images using two secret keys. The second technique, called multiple watermark embedding (MWE), extends SWE to embed multiple watermarks simultaneously in the same watermark space while minimizing the watermark (distortion) energy. The third technique, called iterative watermark embedding (IWE), embeds watermarks into JPEG-compressed images. The iterative approach of IWE can prevent the potential removal of a watermark in the JPEG recompression process. Experimental results show that embedded watermarks using the proposed techniques can give good image quality and are robust in varying degree to JPEG compression, low-pass filtering, noise contamination, and print-and-scan.
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paper_title: Dual Digital Watermarking Algorithm for Image Based on Fractional Fourier Transform

paper_content:

In this paper, a dual digital watermarking embedding and detecting algorithm for image is presented, which uses the robust watermark and fragile watermark embed in the fractional Fourier transform (FRFT) domain of the image, and the watermark position and the transform order are used as the encryption keys. The experiment results show that the embedded watermark is perceptually invisible and robust to various image processing operations. Most importantly, the tampered place can be identified accurately.
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paper_title: Watermarking based on discrete fractional random transform

paper_content:

We propose a novel watermarking algorithm based on discrete fractional random transform. Random block selection and high amplitude selection techniques have been employed to improve the robustness of the watermarking algorithm. The imperceptibility of the watermarking is improved by adjusting the intensity of phase shift keying. Simulations under several conditions are given to verify the effectiveness of the proposed scheme. The results have shown the proposed algorithm is robust against the attacks of cropping, noising and low-pass filtering.
---
paper_title: Robust Image Watermarking Based on Generalized Radon Transformations

paper_content:

The paper presents a novel watermarking scheme able to resist geometric attacks. The proposed method performs imperceptible watermarking of images in the spatial domain. To generate resistance to scaling and rotation attacks, two generalized Radon transformations of the image are introduced, while resistance to translation is accomplished through a localization of the watermarking method based on feature points of the image. The original image is not required for the detection process. Experimental evaluation demonstrates that the proposed scheme is able to withstand a variety of attacks including common geometric attacks.
---
paper_title: Digital watermarking of images in the fractional Fourier domain

paper_content:

We describe the implementation of a watermark embedding technique for images using the discrete fractional Fourier transform. The idea is that a 2D discrete fractional Fourier transform of the image is computed. In this transform domain, a normal distributed random sequence with certain characteristics is used to modify the transform coefficients in the middle range of magnitude. Then the inverse transform is applied to give the watermarked image. The detection is based on performing the same transform operation. The watermark is recognized if there is a strong correlation with the embedded watermark. That is when the correlation is above some threshold.
---
paper_title: Application of the fractional Fourier transform to moving target detection in airborne SAR

paper_content:

As a useful signal processing technique, the fractional Fourier transform (FrFT) is largely unknown to the radar signal processing community. In this correspondence, the FrFT is applied to airborne synthetic aperture radar (SAR) slow-moving target detection. For airborne SAR, the echo from a ground moving target can be regarded approximately as a chirp signal, and the FrFT is a way to concentrate the energy of a chirp signal. Therefore, the FrFT presents a potentially effective technique for ground moving target detection in airborne SAR. Compared with the common Wigner-Ville distribution (WVD) algorithm, the FrFT is a linear operator, and will not be influenced by cross-terms even if multiple moving targets exist. Moreover, to solve the problem whereby weak targets are shadowed by the sidelobes of strong ones, a new implementation of the CLEAN technique is proposed based on filtering in the fractional Fourier domain. In this way strong moving targets and weak ones can be detected iteratively. This combined method is demonstrated by using raw clutter data combined with simulated moving targets.
---
paper_title: Time-Frequency Analysis for Sar and Isar Imaging

paper_content:

In the past 10 years the Time-Frequency (TF) techniques have found significant importance for improvement and refinement of radar images (both SAR and ISAR) and in extraction of objects' features from radar images. The main contributions in this direction are given by V. Chen (Chen and Ling, 2002). Here, we will concentrate our attention on the radar imaging and two novel strategies: S-Method (SM) imaging and Local Polynomial Fourier Transform (LPFT) imaging. The application of these ideas to radar imaging, as well as a procedure for setting design parameter of the algorithms, will be presented. Transformations will be compared in terms of calculation complexity and quality of the resulted radar images. Moreover, it will be demonstrated that the LPFT-based algorithm has better robustness to additive noise than the SM or the adaptive SM.
---
paper_title: SAR imaging of moving targets using polynomial Fourier transform

paper_content:

The polynomial Fourier transform (FT) is employed as a tool for the SAR imaging of moving targets. An efficient algorithm is proposed that can be used for radar images containing both moving and stationary targets. The proposed algorithm can be used in the form of the second-order polynomial FT, but it can also be extended to higher-order and order-adaptive polynomial FT forms.
---
paper_title: Autofocusing of SAR images based on the product high-order ambiguity function

paper_content:

An autofocusing technique is proposed, based on the modelling of the instantaneous phase, induced by the relative radar/target motion, as a generic order polynomial and on the estimation of the polynomial coefficients by the so-called product high-order ambiguity function. The proposed estimation method is able to deal with multicomponent signals and is robust against model order mismatching, provided that the selected order is greater than the true order. The proposed method is compared with the phase gradient autofocusing method, and applications are presented of focusing of extended scenes and detection and imaging of moving targets.
---
paper_title: Novel approach for ISAR image cross-range scaling

paper_content:

Inverse synthetic aperture radar (ISAR) imaging systems produce electromagnetic images of targets in the range-Doppler domain. In order to rescale the image in a homogeneous range-cross range domain (meters by meters), the modulus of the target effective rotation vector must be known. Although in some cases it can be retrieved by means of ancillary data, in most cases the modulus of the target effective rotation vector must be estimated. A blind technique is proposed for estimating the modulus of the target effective rotation vector that exploits information carried by the chirp rate of scattering centres. A technique based on image segmentation, local polynomial Fourier transform (LPFT), and image contrast (IC) maximisation is used in order to extract the scattering centres and estimate their chirp rate. Simulated and real data analyses are provided to confirm the effectiveness of the proposed algorithm.
---
paper_title: Quantitative SNR Analysis for ISAR Imaging using LPFT

paper_content:

This paper presents the quantitative analysis of signal-to-noise ratio (SNR) for the local polynomial Fourier transform (LPFT) used for application of inverse synthetic aperture radar (ISAR) imaging. The relationship between the LPFT and Wigner-Ville distribution (WVD) is derived and theoretical analysis on the SNR achieved by using the LPFT is given. Comparisons on the SNR performances achieved by using the LPFT, the short-time Fourier transform (STFT) and the Fourier transform (FT) are presented to illustrate the merits of using the LPFT. Measures are also taken to minimize the required computational complexity by reducing the overlap length between adjacent segments of input data. Simulation results of processing radar data show the performance improvements achieved by using the LPFT.
---
paper_title: Adaptive local polynomial fourier transform in ISAR

paper_content:

The adaptive local polynomial Fourier transform is employed for improvement of the ISAR images in complex reflector geometry cases, as well as in cases of fast maneuvering targets. It has been shown that this simple technique can produce significantly improved results with a relatively modest calculation burden. Two forms of the adaptive LPFT are proposed. Adaptive parameter in the first form is calculated for each radar chirp. Additional refinement is performed by using information from the adjacent chirps. The second technique is based on determination of the adaptive parameter for different parts of the radar image. Numerical analysis demonstrates accuracy of the proposed techniques.
---
paper_title: Autofocusing of SAR images based on parameters estimated from the PHAF

paper_content:

The local polynomial Fourier transform (LPFT) based algorithm for auto-focusing SAR images has recently been proposed by the authors. It produces a well focused image of moving targets, without defocusing stationary targets or inducing undesired cross-terms. The drawback of this algorithm is its high computational burden caused by the large number of elements in the set of used chirp-rates. We propose an algorithm with decreased number of elements used for the LPFT-based SAR imaging. The product high-order ambiguity function (PHAF) is applied to estimate parameters of a radar signal. The estimated chirp-rate is used as an initial value for forming the set of chirp-rates. The proposed algorithm has significantly smaller set of chirp-rate values (tens comparing to several hundreds or thousands used in the previous algorithm version). In this manner, the calculation complexity is significantly reduced. The proposed procedure is fully automated, meaning that it follows the change of motion parameters. In addition, our procedure considers the third-order phase compensation.
---
paper_title: A new chirp scaling algorithm based on the fractional Fourier transform

paper_content:

The fractional Fourier transform (FrFT), which is a generalized form of the well-known Fourier transform, has only recently started to appear in the field of signal processing. This has opened up the possibility of a new range of potentially promising and useful applications. In this letter, we develop a new FrFT-based chirp scaling algorithm (CSA) and compare its performance with the classical CSA based on the fast Fourier transform (FFT). Simulation results show that the FrFT-based CSA can offer significantly enhanced features compared to the classical FFT-based approach.
---
paper_title: Interference analysis of multicarrier systems based on affine fourier transform

paper_content:

Multicarrier techniques based on affine Fourier transform (AFT) have been recently proposed for transmission in the wireless channels. The AFT represents a generalization of the Fourier and fractional Fourier transform. We derive the exact and approximated interference power, upper bound and measure of applicability for the AFT based multicarrier (AFT-MC) system. It is demonstrated that the AFT-MC system effectively minimizes interference in time-varying multipath channels with line-of-sight component and narrow beamwidth of scattered components that often occurs in aeronautical and satellite communications.
---
paper_title: A multicarrier system based on the fractional Fourier transform for time-frequency-selective channels

paper_content:

Traditional multicarrier techniques perform a frequency-domain decomposition of a channel characterized by frequency-selective distortion in a plurality of subchannels that are affected by frequency flat distortion. The distortion in each independent subchannel can then be easily compensated by simple gain and phase adjustments. Typically, digital Fourier transform schemes make the implementation of the multicarrier system feasible and attractive with respect to single-carrier systems. However, when the channel is time-frequency-selective, as it usually happens in the rapidly fading wireless channel, this traditional methodology fails. Since the channel frequency response is rapidly time-varying, the optimal transmission/reception methodology should be able to process nonstationary signals. In other words, the subchannel carrier frequencies should be time-varying and ideally decompose the frequency distortion of the channel perfectly at any instant in time. However, this ideally optimal approach presents significant challenges both in terms of conceptual and computational complexity. The idea disclosed in this work is that a nonstationary approach can be approximated using signal bases that are especially suited for the analysis/synthesis of nonstationary signals. We propose in fact the use of a multicarrier system that employs orthogonal signal bases of the chirp type that in practice correspond to the fractional Fourier transform signal basis. The significance of the methodology relies on the important practical consideration that analysis/synthesis methods of the fractional Fourier type can be implemented with a complexity that is equivalent to the traditional fast Fourier transform.
---
paper_title: Improved Fractional Fourier Transform Based Receiver for Spatial Multiplexed MIMO Antenna Systems

paper_content:

In this work the performance of a Fractional Fourier transform (FrFT) based Minimum Mean Squared Error receiver for MIMO systems with space time processing over Rayleigh faded channels is presented. The proposed receiver called Optimum FrFT based MIMO receiver (OFMR) shows improved performance outperforming the simple MMSE receiver in Rayleigh faded channel.
---
paper_title: Improved Chirp Modulation Spread Spectrum Receiver Based on Fractional Fourier Transform for Multiple Access

paper_content:

The fractional Fourier transform (FRFT) is one among the most efficient techniques for digital signal processing of linear frequency modulated (LFM) signal applications. Chirp (or LFM) modulation spread spectrum (CMSS) with FRFT technique is employed for demodulation of the multiple access System. The concept of multiple access is based on the use of multiple chirp signals, with different chirp rate, as carrier signal for transmission. Corresponding receiver is designed with optimal ordered FRFT block with multiplicative filter to reduce the interference. Additive white Gaussian noise (AWGN) is considered as a channel noise for evaluation of performance. Bit error rate (BER) is taken as a measure of performance with different signal to noise ratio and the results are compared with those reported in literatures. The details of the simulation techniques employed and the results of the novel receiver are presented and discussed.
---
paper_title: A multicarrier architecture based upon the affine Fourier transform

paper_content:

Recently, innovative multicarrier schemes have been proposed that exploit the transmission of chirp-shaped waves e/sup -j2/spl pi/ct2/ by optimally choosing the chirp parameter c on the basis of the channel characteristics and are more robust to time-varying channels than ordinary OFDM schemes. This concept was applied to continuous-time and to discrete-time systems. In the present paper, we aim at developing those ideas using the affine Fourier transform (AFT), which is a very general formulation of chirp transforms. We present a multicarrier modulation based upon the discrete form of the AFT that is therefore inherently discrete and strictly invertible. Moreover, it allows to define a circular prefix concept that is coherent with the chirp nature of the transmission. The system can be efficiently implemented by adding a simple phase-correction block to standard OFDM modulators/demodulators and can effectively combat interchannel interference when the propagation channel is made of few multipath components affected by independent frequency offsets. Our discrete-time multicarrier scheme is an improved version of Martone's approach (as we also show by simulation results), and exhibits analogous characteristics to Barbarossa's continuous-time system.
---
paper_title: Signal compression using discrete fractional Fourier transform and set partitioning in hierarchical tree

paper_content:

Signal compression has been characterized as the removal of redundancy and irrelevancy. Fractional Fourier Transform (FRFT), an orthogonal, linear transform, is known to decompose the signal in terms of chirps. In this paper we propose a scheme for signal compression based on the combination of discrete FRFT (DFRFT) and set partitioning in hierarchical tree (SPIHT). The application of the scheme to different types of signals demonstrates significant reduction in bits leading to high signal compression ratio. The results are compared with those obtained with discrete cosine transform (DCT).
---
paper_title: Fractional Fourier transform pre-processing for neural networks and its application to object recognition.

paper_content:

This study investigates fractional Fourier transform pre-processing of input signals to neural networks. The fractional Fourier transform is a generalization of the ordinary Fourier transform with an order parameter a. Judicious choice of this parameter can lead to overall improvement of the neural network performance. As an illustrative example, we consider recognition and position estimation of different types of objects based on their sonar returns. Raw amplitude and time-of-flight patterns acquired from a real sonar system are processed, demonstrating reduced error in both recognition and position estimation of objects.
---
paper_title: Synthesis of pattern recognition filters for fractional Fourier processing

paper_content:

Abstract Pattern recognition, based on matched filtering, can be generalized by replacing the standard Fourier operations by fractional Fourier operations. One important consequence of this is that the system is no longer space invariant. We study the impact of space variance, and point out that space variance may be desirable under certain circumstances. We illustrate our theoretical conclusions by simulated experiments.
---
paper_title: The multiple-parameter discrete fractional Fourier transform

paper_content:

The discrete fractional Fourier transform (DFRFT) is a generalization of the discrete Fourier transform (DFT) with one additional order parameter. In this letter, we extend the DFRFT to have N order parameters, where N is the number of the input data points. The proposed multiple-parameter discrete fractional Fourier transform (MPDFRFT) is shown to have all of the desired properties for fractional transforms. In fact, the MPDFRFT reduces to the DFRFT when all of its order parameters are the same. To show an application example of the MPDFRFT, we exploit its multiple-parameter feature and propose the double random phase encoding in the MPDFRFT domain for encrypting digital data. The proposed encoding scheme in the MPDFRFT domain significantly enhances data security.
---
paper_title: Deficiencies of the cryptography based on multiple-parameter fractional Fourier transform

paper_content:

Methods of image encryption based on fractional Fourier transform have an incipient flaw in security. We show that the schemes have the deficiency that one group of encryption keys has many groups of keys to decrypt the encrypted image correctly for several reasons. In some schemes, many factors result in the deficiencies, such as the encryption scheme based on multiple-parameter fractional Fourier transform [Opt. Lett.33, 581 (2008)]. A modified method is proposed to avoid all the deficiencies. Security and reliability are greatly improved without increasing the complexity of the encryption process. © 2009 Optical Society of America
---
paper_title: Optical image encryption based on the multiple-parameter fractional Fourier transform.

paper_content:

A novel image encryption algorithm is proposed based on the multiple-parameter fractional Fourier transform, which is a generalized fractional Fourier transform, without the use of phase keys. The image is encrypted simply by performing a multiple-parameter fractional Fourier transform with four keys. Optical implementation is suggested. The method has been compared with existing methods and shows superior robustness to blind decryption.
---
paper_title: On the Security of a Cryptosystem Based on Multiple-Parameters Discrete Fractional Fourier Transform

paper_content:

Pei and Hsue (IEEE Signal Processing Letters, Vol. 13, No. 6, pp. 329-332, June 2006) proposed an encryption scheme based on multiple-parameter discrete fractional Fourier transform. We show that all the building blocks of this scheme are linear, and hence, breaking this scheme, using a known plaintext attack, is equivalent to solving a set of linear equations.
---
paper_title: Random Discrete Fractional Fourier Transform

paper_content:

In this letter, a new commuting matrix with random discrete Fourier transform (DFT) eigenvectors is first constructed. A random discrete fractional Fourier transform (RDFRFT) kernel matrix with random DFT eigenvectors and eigenvalues is then proposed. The RDFRFT has an important feature that the magnitude and phase of its transform output are both random. As an application example, a security-enhanced image encryption scheme based on the RDFRFT is illustrated.
---
paper_title: An Improved Hurst Parameter Estimator Based on Fractional Fourier Transform

paper_content:

A fractional Fourier transform (FrFT) based estimation method is introduced in this paper to analyze the long range dependence (LRD) in time series. The degree of LRD can be characterized by the Hurst parameter. The FrFT-based estimation of Hurst parameter proposed in this paper can be implemented efficiently allowing very large data set. We used fractional Gaussian noises (FGN) which typically possesses long-range dependence with known Hurst parameters to test the accuracy of the proposed Hurst parameter estimator. For justifying the advantage of the proposed estimator, some other existing Hurst parameter estimation methods, such as wavelet-based method and a global estimator based on dispersional analysis, are compared. The proposed estimator can process the very long experimental time series locally to achieve a reliable estimation of the Hurst parameter.Copyright © 2007 by ASME
---
paper_title: Fractional Fourier transform as a tool for investigation of fractal objects

paper_content:

The effect of the fractional Fourier transform on homogeneous functions is discussed. The fractional Fourier transform is used for the determination of the main parameters of fractals. An application is made to the Mandelbrot–Weierstrass cosine fractal. The results are applied to describe the propagation of a fractal field through quadratic refractive-index media and to determine its fractal dimension.
---
paper_title: Diagnosis of Induction Motor Faults in the Fractional Fourier Domain

paper_content:

Motor current signature analysis (MCSA) is a well-established method for the diagnosis of induction motor faults. It is based on the analysis of the spectral content of a motor current, which is sampled while a motor runs in steady state, to detect the harmonic components that characterize each type of fault. The Fourier transform (FT) plays a prominent role as a tool for identifying these spectral components. Recently, MCSA has also been applied during the transient regime (TMCSA) using the whole transient speed range to create a unique stamp of each harmonic as it evolves in the time-frequency plane. This method greatly enhances the reliability of the diagnostic process compared with the traditional method, which relies on spectral analysis at a single speed. However, the FT cannot be used in this case because the fault harmonics are not stationary signals. This paper proposes the use of the fractional FT (FrFT) instead of the FT to perform TMCSA. This paper also proposes the optimization of the FrFT to generate a spectrum where the frequency-varying fault harmonics appear as single spectral lines and, therefore, facilitate the diagnostic process. A discrete wavelet transform (DWT) is used as a conditioning tool to filter the motor current prior to its processing by the FrFT. Experimental results that are obtained with a 1.1-kW three-phase squirrel-cage induction motor with broken bars are presented to validate the proposed method.
---
paper_title: Relation between computed tomography and diffraction tomography

paper_content:

The relationship between computed tomography (CAT) and diffraction tomography (DT) is investigated. A simple condition with a clear physical meaning is derived for the applicability of CAT. Corrections due to scattering are incorporated into CAT, and it is shown that the effect of scattering may be characterized by a two-dimensional fractional Fourier transform. The implications of these results for the three-dimensional imaging of weakly scattering objects are also discussed.
---
paper_title: The Fractional Fourier Transform: With Applications in Optics and Signal Processing

paper_content:

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.
---
paper_title: The Fractional Order Fourier Transform and its Application to Quantum Mechanics

paper_content:

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.
---
paper_title: An alternative point of view to the theory of fractional Fourier transform

paper_content:

The concept of the fractional Fourier transform is framed within the context of quantum evolution operators. This point of view yields an extension of the above concept and greatly simplifies the underlying operational algebra. It is also proved that a multidimensional extension can be performed by using a biorthogonal multiindex harmonic oscillator basis. It t is finally shown that most of the proposed physical interpretations of the fractional Fourier transform are just trivial consequences of the analysis developed in this paper.
---
paper_title: Diagnosis of Induction Motor Faults in the Fractional Fourier Domain

paper_content:

Motor current signature analysis (MCSA) is a well-established method for the diagnosis of induction motor faults. It is based on the analysis of the spectral content of a motor current, which is sampled while a motor runs in steady state, to detect the harmonic components that characterize each type of fault. The Fourier transform (FT) plays a prominent role as a tool for identifying these spectral components. Recently, MCSA has also been applied during the transient regime (TMCSA) using the whole transient speed range to create a unique stamp of each harmonic as it evolves in the time-frequency plane. This method greatly enhances the reliability of the diagnostic process compared with the traditional method, which relies on spectral analysis at a single speed. However, the FT cannot be used in this case because the fault harmonics are not stationary signals. This paper proposes the use of the fractional FT (FrFT) instead of the FT to perform TMCSA. This paper also proposes the optimization of the FrFT to generate a spectrum where the frequency-varying fault harmonics appear as single spectral lines and, therefore, facilitate the diagnostic process. A discrete wavelet transform (DWT) is used as a conditioning tool to filter the motor current prior to its processing by the FrFT. Experimental results that are obtained with a 1.1-kW three-phase squirrel-cage induction motor with broken bars are presented to validate the proposed method.
---
paper_title: Relation between computed tomography and diffraction tomography

paper_content:

The relationship between computed tomography (CAT) and diffraction tomography (DT) is investigated. A simple condition with a clear physical meaning is derived for the applicability of CAT. Corrections due to scattering are incorporated into CAT, and it is shown that the effect of scattering may be characterized by a two-dimensional fractional Fourier transform. The implications of these results for the three-dimensional imaging of weakly scattering objects are also discussed.
---
paper_title: The Fractional Order Fourier Transform and its Application to Quantum Mechanics

paper_content:

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.
---
paper_title: An alternative point of view to the theory of fractional Fourier transform

paper_content:

The concept of the fractional Fourier transform is framed within the context of quantum evolution operators. This point of view yields an extension of the above concept and greatly simplifies the underlying operational algebra. It is also proved that a multidimensional extension can be performed by using a biorthogonal multiindex harmonic oscillator basis. It t is finally shown that most of the proposed physical interpretations of the fractional Fourier transform are just trivial consequences of the analysis developed in this paper.
---

```
Title: Fractional Fourier Transform as a Signal Processing Tool: An Overview of Recent Developments

Section 1: Introduction
Description 1: Introduce the concept of the Fractional Fourier Transform (FRFT), its historical background, and the main goal of the paper.

Section 2: The fractional Fourier transform
Description 2: Define the FRFT, its mathematical properties, and its relation to the ordinary Fourier transform and other transforms.

Section 3: FRFT as fractional powers of Fourier transform
Description 3: Discuss the derivation of FRFT from the eigenfunctions of the Fourier transform and its spectral decomposition.

Section 4: Relations to time-frequency representations
Description 4: Explore the relationship between FRFT and various time-frequency analyses, including ambiguity functions and Wigner distributions.

Section 5: Local polynomial Fourier transform
Description 5: Establish the relationship between the FRFT and the local polynomial Fourier transform (LPFT) and local polynomial periodogram (LPP).

Section 6: Other time-frequency representations and properties
Description 6: Discuss the relationship of FRFT with wavelet transforms and other time-frequency representations.

Section 7: FRFT and linear canonical transform
Description 7: Explain the connection between FRFT and linear canonical transform (LCT) and various special cases of LCT.

Section 8: Coordinate transformation of other higher-order time-frequency representations
Description 8: Generalize the FRFT and LCT framework to higher-order time-frequency representations.

Section 9: Multiparameter coordinate transformations
Description 9: Introduce further generalizations of FRFT and LPFT by adding additional parameters.

Section 10: Practical realizations of FRFT
Description 10: Provide an overview of various practical implementations of the Discrete Fractional Fourier Transform (DFRFT).

Section 11: DFRFT through sampling of FRFT
Description 11: Discuss different approaches for obtaining DFRFT through direct sampling of FRFT.

Section 12: Linear combination-type DFRFT
Description 12: Examine the method of DFRFT derived by linear combination of identity operation and ordinary Fourier operators.

Section 13: DFRFT based on eigenvectors
Description 13: Describe the approach of DFRFT based on eigenvectors and eigenvalues of the DFT matrix.

Section 14: Other approaches
Description 14: Highlight other methods for computing DFRFT, including those based on the quadratic phase transform.

Section 15: Applications
Description 15: Review practical applications of FRFT and DFRFT in various fields such as filtering, watermarking, radar applications, communications, compression, pattern recognition, cryptography, fractal signal processing, and other areas.

Section 16: Conclusion
Description 16: Summarize the major findings and implications of the paper, outline the connections between FRFT and other transforms, and predict future developments and applications.
```