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The Fractional Order Fourier Transform and Its Application to Quantum Mechanics
Abstract
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.
... The fractional Fourier transform (FrFT), a broader version of the traditional Fourier transform (FT), was introduced seven decades ago by Namias [13]. However, it has only recently gained traction in fields such as signal processing, optics, and quantum mechanics [3,11,26]. ...
... Let f P L 1 pRq. Recall from [13] that the FrFT of order α is defined by ...
... The function F α f exhibits 2π periodicity with respect to α, and, thus, we will consistently consider α within the interval r0, 2πq. Notice that when n P Z, F nπ{2 f " F n f , where F n is the n-th power of the FT (1.2), i.e., F α is the sth power of the FT for s " 2α{π, for α within the interval r0, 2πq, [13]. So, the FT is of order 1, while the identity operator is of order 0. Negative orders correspond to inverse transforms. ...
In this paper, we presented Tauberian type results that intricately link the quasi-asymptotic behavior of both even and odd distributions to the corresponding asymptotic properties of their fractional Fourier cosine and sine transforms. We also obtained a structural theorem of Abelian type for the quasi-asymptotic boundedness of even (resp. odd) distributions with respect to their fractional Fourier cosine transform (FrFCT) (resp. fractional Fourier sine transform (FrFST)). In both cases, we quantified the scaling asymptotic properties of distributions by asymptotic comparisons with Karamata regularly varying functions.
... Fractional Fourier transform can be considered as a generalization of the traditional Fourier transform. In 1980, Namias first introduced the mathematical definition of the FrFT [4]. Then, Almeida analyzed the relationship between the FrFT and the Wigner-Ville Distribution (WVD), and interpreted it as a rotation operator in the time-frequency plane. ...
... Fractional Fou Fractional Fou generalization of eneralization of Namias first in ias f FrFT [4]. T [4] the FrFT the FrFT interp interp pla traditional speech processing method, these parameters are considered as constant within a frame. ...
... Fractional Fou Fractional Fou generalization of eneralization of Namias first in ias f FrFT [4]. T [4] the FrFT the FrFT interp interp pla traditional speech processing method, these parameters are considered as constant within a frame. Since voiced speech has a harmonic structure, it can be modeled as: ...
... Therefore, an inverse chirp is not required in the decoding process. Data bits can be decoded using noncoherent techniques [8][9][10][11]. The non-coherent detection scheme for the two proposed coding will be presented. The remainder of this paper is organized as follows: Section I provides a brief introduction of this study. ...
... The single-slope cyclic-shift chirp signal cannot be decoded using a noncoherent technique [8][9][10][11] and requires a reference symbol for decoding [13]. Hence, two forms of frequency shift chirp symbols based on a dual-slope frequency shift chirp are proposed herein. ...
One of the most significant concerns in serial digital communication is the synchronization of data bits between the transmitter and receiver. Therefore, the preferred coding symbols should contain an inherited clock signal. The cyclic-shift chirp, which is primarily used in long-range systems, employs a single slope in a symbol, which contains few information for synchronization. Hence, two new forms of cyclic-shift chirp symbols are proposed herein. The associated coding schemes are based on the use of a dual slope within one symbol. Synchronization is inherited in the coding symbol using a dual slope. Additionally, both proposed coding schemes utilize the same decoding technique. The decoded chirp symbol is in the form of a pulse width modulation signal, which can be used for synchronization and retrieving data bits from their duty cycle. The simulation results show that the coding and decoding processes of the proposed coding schemes align with those of a theoretical framework. The error performance analysis for the proposed detection scheme is given and the results of error probability analysis are consistent with the simulation results. Moreover, under a Rayleigh fading channel, the proposed coding schemes provide superior performance compared with other comparable coding schemes.
... He obtained an integral operator which is the general form of Fourier transform (FT) [40]. It was studied again until 1980 s by Namias [21] for solving a Cauchy problem for the forced Schrödinger equation in quantum mechanics, ...
... The solution of problem (1) can be represented with the semigroup of operators e −it H ( f )(x) formally [41]. The author in [21] obtained the integral representation of e −it H ( f )(x) by Hermite polynomials and Mehler's formula, which is exactly the form of frac-FT with a multiplicative constant. The integral kernel of frac-FT is called harmonic oscillator propagator in quantum mechanics [3,4,38]. ...
The paper promotes a new sparse approximation for fractional Fourier transform, which is based on adaptive Fourier decomposition in Hardy-Hilbert space on the upper half-plane. Under this methodology, the local polynomial Fourier transform characterization of Hardy space is established, which is an analog of the Paley-Wiener theorem. Meanwhile, a sparse fractional Fourier series for chirp L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ L^2 $$\end{document} function is proposed, which is based on adaptive Fourier decomposition in Hardy-Hilbert space on the unit disk. Besides the establishment of the theoretical foundation, the proposed approximation provides a sparse solution for a forced Schro¨\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\textrm{o}}$$\end{document}dinger equations with a harmonic oscillator.
... In 1987, McBride and Kerr provided an explicit definition of the fractional Fourier transform (1) on the Schwartz space on R as an integral transform, due to a modification of Namias' fractional operators [17,19]. Both definitions coincide for h 2 RnpZ. ...
... This integral transform has since been explored extensively, with applications spanning a various of topics linked with engineering and science. We state, for instance, some developments include digital communications [16], signal processing [22], quantum mechanics [19], harmonic analysis [11,23,31,32,[35][36][37][38], renewable energy [29] and time frequency analysis [1,24]. The McBride-Kerr's definition of the fractional Fourier transform [17] is given by Recently, the harmonic analysis for the Opdam-Cherednik transform was well developed and investigated. ...
In the present paper, we study a new harmonic analysis in the setting of the Opdam–Cherednik. We establish the fractional Opdam–Cherednik transform in order to generalize the classical fractional Fourier transform. At first, we establish the associated harmonic analysis and we introduce some of its basic properties, such as Parseval identity and inversion formula. Next, we study the fractional Opdam–Cherednik transform on the corresponding Schwartz space and the tempered distribution. In the end, we develop the scope of this work by investigating the fractional Opdam–Cherednik transform for transport equation.
... In 1980, Namias [1] formulated the fractional Fourier transform as a path to find out the solutions of certain differential equations which occasionally appear in quantum mechanics. Later on, his results were polished by McBride and Kerr [2], who developed an operational calculus for the fractional Fourier transform. ...
... Due to numerous applications in the area of image processing, signal analysis and optics, fractional Fourier transform has received more attention in the last several years. This transform plays an important role for solving various problems in quantum physics [1,3], signal processing and optics [4][5][6][7][8][9]. The fractional Fourier transform, which is a generalization of usual Fourier transform, has been studied in several areas of mathematical analysis, for instances wavelets [10,11], pseudo-differential operators [12] and generalized functions [13][14][15]. ...
The main aim of this article is to derive certain continuity and boundedness properties of the coupled fractional Fourier transform on Schwartz-like spaces. We extend the domain of the coupled fractional Fourier transform to the space of tempered distributions and then study the mapping properties of pseudo-differential operators associated with the coupled fractional Fourier transform on a Schwartz-like space. We conclude the article by applying some of the results to obtain an analytical solution of a generalized heat equation.
... Notably, the FrFT offers advantages in digital communications [4] and radar systems [5], where chirp-like basis functions (see (2)) are more effective than traditional sinusoidal basis (Fourier). Other areas of application include quantum mechanics [6], harmonic analysis [7], [8], time-frequency representations [9], [10], and optics and imaging [11]. ...
... The solution to Dq = 0 → q allows for construction of Q (z) in (9); its roots then lead to the estimates of {t k } K−1 k=0 . It remains to estimate {c k } K−1 k=0 which can be obtained by solving linear system of equations in in (7), thus leading to exact recovery of unknown s K (t) in (6). ■ Note that when θ = π/2, Theorem 2 reduces to the conventional Fourier domain case and * θ , θ = π/2 reduces to the conventional notion of low-pass filtering or convolution. ...
Sampling theory in fractional Fourier Transform (FrFT) domain has been studied extensively in the last decades. This interest stems from the ability of the FrFT to generalize the traditional Fourier Transform, broadening the traditional concept of bandwidth and accommodating a wider range of functions that may not be bandlimited in the Fourier sense. Beyond bandlimited functions, sampling and recovery of sparse signals has also been studied in the FrFT domain. Existing methods for sparse recovery typically operate in the transform domain, capitalizing on the spectral features of spikes in the FrFT domain. Our paper contributes two new theoretical advancements in this area. First, we introduce a novel time-domain sparse recovery method that avoids the typical bottlenecks of transform domain methods, such as spectral leakage. This method is backed by a sparse sampling theorem applicable to arbitrary FrFT-bandlimited kernels and is validated through a hardware experiment. Second, we present Cramér-Rao Bounds for the sparse sampling problem, addressing a gap in existing literature.
... Condon [3] and Namias [11], independently, introduced the fractional Fourier transform which generalizes the Fourier transform. The fractional Fourier transform, which depends on a parameter α, has an interpretation of rotation of time-frequency plane by an angle α and this transform reduces to the ordinary Fourier transform for α = π 2 . ...
... The fractional Fourier transform, which depends on a parameter α, has an interpretation of rotation of time-frequency plane by an angle α and this transform reduces to the ordinary Fourier transform for α = π 2 . Based on the Namias's fractional Fourier transform [11], many fractional integral transforms were introduced. Some of them are fractional wavelet transform [10,17], short time fractional Fourier transform [18,20], fractional Hartley transform [7] and fractional Stockwell transform [5,21]. ...
In this paper, we introduce a new multidimensional fractional S transform Sϕ,α,λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\phi ,\varvec{\alpha },\lambda }$$\end{document} using a generalized fractional convolution ⋆α,λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\star _{\varvec{\alpha },\lambda }$$\end{document} and a general window function ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document} satisfying some admissibility condition. The value of Sϕ,α,λf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\phi ,\varvec{\alpha },\lambda }f$$\end{document} is also written in the form of inner product of the input function f with a suitable function ϕt,uαλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{\textbf{t},\textbf{u}}^{\varvec{\alpha }_{\lambda }}$$\end{document}. The representation of Sϕ,α,λf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\phi ,\varvec{\alpha },\lambda }f$$\end{document} in terms of the generalized fractional convolution helps us to obtain the Parseval’s formula for Sϕ,α,λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\phi ,\varvec{\alpha },\lambda }$$\end{document} using the generalized fractional convolution theorem. Then, the inversion theorem is proved as a consequence of the Parseval’s identity. Using a generalized window function in the kernel of Sϕ,α,λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\phi ,\varvec{\alpha },\lambda }$$\end{document} gives option to choose window function whose Fourier transform as a compactly supported smooth function or a rapidly decreasing function. We also discuss about the characterization of range of Sϕ,α,λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\phi ,\varvec{\alpha },\lambda }$$\end{document} on L2(RN,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\mathbb {R}^N, \mathbb {C})$$\end{document}. Finally, we extend the transform to a class of quaternion valued functions consistently.
... Then R θ ∈ SL(2, R) and the corresponding metaplectic operator R θ is the so-called fractional Fourier transform of order θ [9,22]. For S = R θ , the constants p and q in Theorem 4.1 satisfy p = 1 and q = 0. Thus, for every θ ∈ R, every 0 < α < 1 2π e , and every f , h ∈ L 2 (R) the following statements are equivalent: We notice, that the previous equivalence is valid if ϕ gets replaced by an arbitrary Hermite basis function. ...
... For S = R θ , the constants p and q in Theorem 4.1 satisfy p = 1 and q = 0. Thus, for every θ ∈ R, every 0 < α < 1 2π e , and every f , h ∈ L 2 (R) the following statements are equivalent: We notice, that the previous equivalence is valid if ϕ gets replaced by an arbitrary Hermite basis function. This follows from the fact, that Hermite functions are eigenfunctions for the Fractional Fourier transform corresponding to an eigenvalue ν ∈ T [20,22]. Example 4.3 (Sheared square-root lattices) Let σ ∈ R, and let A σ ∈ SL 2 (R) be given by ...
Due to its appearance in a remarkably wide field of applications, such as audio processing and coherent diffraction imaging, the short-time Fourier transform (STFT) phase retrieval problem has seen a great deal of attention in recent years. A central problem in STFT phase retrieval concerns the question for which window functions \(g \in {L^2({\mathbb R}^d)}\) and which sampling sets \(\Lambda \subseteq {\mathbb R}^{2d}\) is every \(f \in {L^2({\mathbb R}^d)}\) uniquely determined (up to a global phase factor) by phaseless samples of the form
where \(V_gf\) denotes the STFT of f with respect to g. The investigation of this question constitutes a key step towards making the problem computationally tractable. However, it deviates from ordinary sampling tasks in a fundamental and subtle manner: recent results demonstrate that uniqueness is unachievable if \(\Lambda \) is a lattice, i.e \(\Lambda = A{\mathbb Z}^{2d}, A \in \textrm{GL}(2d,{\mathbb R})\). Driven by this discretization barrier, the present article centers around the initiation of a novel sampling scheme which allows for unique recovery of any square-integrable function via phaseless STFT-sampling. Specifically, we show that square-root lattices, i.e., sets of the form
guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians
... where f ∈ L 2 (R, C), ∧ = (A, B, C, D, E), B = 0 which generalizes the classical Fourier transform (FT). Several other important integral transforms like fractional Fourier transform (FrFT) [1,41], linear canonical transform (LCT) [28], offset linear canonical transform (OLCT) [9,48], Fresnel transform [31] and Lorentz transform can be obtained by choosing ∧ appropriately and amplifying (1) with suitable constants. Along with several important properties like the Riemann-Lebesgue lemma and Plancherel theorem, authors in [12] have given several convolutions and obtained the convolution theorem associated with the QPFT. ...
In this paper, we extend the quadratic phase Fourier transform of a complex valued functions to that of the quaternion-valued functions of two variables. We call it the quaternion quadratic phase Fourier transform (QQPFT). Based on the relation between the QQPFT and the quaternion Fourier transform (QFT) we obtain the sharp Hausdorff–Young inequality for QQPFT, which in particular sharpens the constant in the inequality for the quaternion offset linear canonical transform (QOLCT). We define the short time quaternion quadratic phase Fourier transform (STQQPFT) and explore some of its properties including inner product relation and inversion formula. We find its relation with that of the 2D quaternion ambiguity function and the quaternion Wigner–Ville distribution associated with QQPFT and obtain the Lieb’s uncertainty and entropy uncertainty principles for these three transforms.
... The tomographic representation (1) is strictly connected with the so-called Fractional Fourier Transformation (FRFT) (see, e g. [11,12]), which is widely used for example in quantum optics. From the other side it is the same as the Radon transformation [13]. ...
... , and F i f = F f is the classical Fourier transform [30,35]. If σ ∈ {−1, 1}, then F σ is a quadratic Fourier transform with the explicit formula ...
Metaplectic Wigner distributions are joint time-frequency representations that are parametrized by a symplectic matrix and generalize the short-time Fourier transform and the Wigner distribution. We investigate the question which metaplectic Wigner distributions satisfy an uncertainty principle in the style of Benedicks and Amrein-Berthier. That is, if the metaplectic Wigner distribution is supported on a set of finite measure, must the functions then be zero? While this statement holds for the short-time Fourier transform, it is false for some other natural time-frequency representations. We provide a full characterization of the class of metaplectic Wigner distributions which exhibit an uncertainty principle of this type, both for sesquilinear and quadratic versions.
... Certain optical systems can rotate a signal by an arbitrary angle. Mathematically this is implemented by the fractional Fourier transform (FRFT) [5,26,41,42]. The FRFT is one of several transforms called linear canonical transforms (LCT) which are widely used in optics and signal processing [1]. ...
We derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function, by resorting to the fact that these transforms are just metaplectic operators associated with free symplectic matrices. The results obtained synthesize and generalize previous results found in the literature, because they apply to all signals, in arbitrary dimension and any metaplectic operator (which includes Linear Canonical Transforms as particular cases). Moreover, we also obtain a generalization of the Robertson-Schr\"odinger uncertainty principle for Linear Canonical Transforms. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane. The marginal distributions are always non-negative and permit a simple interpretation in terms of the Radon transform. We also give a geometric interpretation of this quadratic phase-space representation as a Wigner distribution obtained upon Weyl quantization on a non-standard symplectic vector space. Finally, we derive the multidimensional version of the Hardy uncertainty principle for metaplectic operators and the Paley-Wiener theorem for Linear Canonical Transforms.
... Fractional Fourier Transform is an operator that provides representations within a given domain, either temporal or spatial, and the frequency domain; the representations obtained by this operator are called the fractional Fourier domains. 72,73 The operator is denoted as F α , and its definition is shown in Eq. (1), ...
This paper proposes a methodology for the diagnosis of electrical system conditions using fractional-order integral transforms for feature extraction. This work proposes three feature extraction algorithms using the Fractional Fourier Transform (FRFT), the Fourier Transform combined with the Mittag-Leffler function, and the Wavelet Transform (WT). Each algorithm extracts data from an electrical system to obtain a set of features that are classified by an Artificial Neural Network to determine the system’s condition. The algorithms are utilized in diagnosing two types of electrical machine faults, one in a photovoltaic system, and another in classifying the power quality disturbances (PQDs). An optimization algorithm is suggested to find the optimal orders of integral transforms. The obtained results demonstrate the system’s effective diagnosis, displaying superior performance in classifying the faults and PQDs with complex signals.
... A generalization of the Fourier transform (FT), the fractional Fourier transform (FrFT), was first proposed by Namias [18]. It finds widespread use in a variety of fields, such as differential equations, quantum mechanics, neural networks, optics, pattern recognition, communication systems, radar, sonar, signal and image processing, and more [19][20][21][22][23][24][25][26]. ...
The traditional scaled Wigner distribution (SWD) is extended to a novel one inspired by merits of fractional instantaneous autocorrelation present in the definition of fractional bi-spectrum and the fractional Fourier transform (FrFT). We begin by examining the basic characteristics of the novel fractional scaled Wigner distribution (Fr-SWD), such as its nonlinearity, marginality, shifting, conjugate symmetry, and antiderivative nature. Following that, a thorough analysis of Moyal’s formula is also conducted. The proposed distribution is used to detect both single-component and multi-component linear frequency-modulated signals in order to demonstrate its effectiveness. The results of the simulation clearly show that the novel fractional scaled Wigner distribution performs exceptionally well in comparison with the conventional Wigner distribution and its scaled version.
... As a similar transform, the fractional Fourier transform was introduced in 1929, in [5]. The fundamental properties of the fractional Fourier transform can be seen in [6,7], and studied in several papers and books, for example [8,9]. ...
The gyrator transform is an integral transform that has attracted much attention in the field of optics and other engineering fields. We consider the image of the gyrator transform of the Gelfand-Shilov space and its dual space. While the gyrator transform is closely related to the fractional Fourier transform, we discuss the difference between these two transforms. Moreover, we show the relation between the above spaces and the eigenfunctions of the gyrator transform.
... Data were gathered from hospitals in Sao Paulo, Brazil. In this case, the data consist of 60 patients infected with coronavirus; 28 of them were female, and 32 were male [42]. The size of the images in this database was different; the size of all the images equalized before the pre-processing stage ( Figure 2). ...
COVID-19 is a lung disease caused by a coronavirus family virus. Due to its extraordinary prevalence and associated death rates, it has spread quickly to every country in the world. Thus, achieving peaks and outlines and curing different types of relapses is extremely important. Given the worldwide prevalence of coronavirus and the participation of physicians in all countries, information has been gathered regarding the properties of the virus, its diverse types, and the means of analyzing it. Numerous approaches have been used to identify this evolving virus. It is generally considered the most accurate and acceptable method of examining the patient’s lungs and chest through a CT scan. As part of the feature extraction process, a method known as fractional Fourier transform (FrFT) has been applied as one of the time-frequency domain transformations. The proposed method was applied to a database consisting of 2481 CT images. Following the transformation of all images into equal sizes and the removal of non-lung areas, multiple combination windows are used to reduce the number of features extracted from the images. In this paper, the results obtained for KNN and SVM classification have been obtained with accuracy values of 99.84% and 99.90%, respectively.
... The concept of the fractional-order Fourier transform (FrFT) was first introduced by Namias in mathematics [10]. The optical realization of fractional Fourier transform was proposed by Mendlovic and Ozaktas [11,12], and meanwhile, an optical system only using one lens and setting equal object and image distances was presented by Lohmann to implement the fractional Fourier transform [13]. ...
A fractional Fourier-transform digital holographic imaging method with resolution enhancement features is presented. In an optical configuration, an extended fractional Fourier-transform optical setup is set in the object arm of an off-axis digital holographic recording system to record a fractional Fourier-transform hologram via the optical interference of the fractional Fourier-transform wavefront of an object wave with a reference wave. For reconstruction imaging, the reconstruction approach for fractional Fourier-transform holograms is given. In the experiment, the fractional Fourier-transform digital holograms are recorded under the different recording parameters, and their amplitude images are effectively reconstructed. The imaging results demonstrate that the reconstruction-imaging resolution of fractional-order Fourier-transform holograms is obviously enhanced compared to that of conventional image-plane holograms. The presented fractional Fourier-transform digital holographic imaging with resolution enhancement and optical configuration flexibility provides, to our knowledge, a novel way for off-axis digital holographic imaging.
... It can detect the frequency contents within the signal, and it has been widely applied in speech processing, quantum physics, and remote sensing images. As a generalization of the FT, the fractional FT (FRFT) was designed in the mathematics literature [3,4]. Both the FT and FRFT are global transformations that cannot describe the time location of a frequency or fractional frequency. ...
Time–frequency analysis is an important tool used for the processing and interpretation of non-stationary signals, such as seismic data and remote sensing data. In this paper, based on the novel short-time fractional Fourier transform (STFRFT), a new modified STFRFT is first proposed which can also generalize the properties of the modified short-time Fourier transform (STFT). Then, in the modified STFRFT domain, we derive the instantaneous frequency estimator for the chirp signal and present a new type of synchrosqueezing STFRFT (FRSST). The proposed FRSST presents many results similar to those of the synchrosqueezing STFT (FSST), and it extends the harmonic signal to a chirp signal that offers attractive new features. Furthermore, we provide a detailed analysis of the signal reconstruction, theories, and some properties of the proposed FRSST. Several experiments are conducted, and all of the results illustrate that the proposed FRSST is more effective than the FSST. Finally, based on the linear amplitude modulation and frequency modulation signal, we present a derivation for analyzing the limitations of the FRSST.
... During the last few years, Namias (Namias 1980) introduced the fractional Fourier transform (FRFT) as a mathematical technique to solve problems in quantum mechanics. Since then, researchers have employed the technique in various applications including laser optics, signal processing, and image encryption Ozaktas and Mendlovic 1993;Lohmann 1993;Mendlovic et al. 1994;Dorsch et al. 1995;Zhang et al 1998;Kutay and Ozaktas 1998;Xue 2001;Torre 2002;Hannelly and Sheridan 2003;Wang and Zhao 2013). ...
We theoretically investigate the propagation properties of a Laguerre higher order cosh Gaussian beam (LHOchGB) in a fractional Fourier transform (FRFT) optical system. Based on the Collins formula and the expansion of the hard aperture function into a finite sum of Gaussian functions, we derive analytical expressions for a LHOchGB propagating through apertured and unapertured FRFT systems. The analysis of the evolution of the intensity distribution at the output plane has shown from the obtained expressions, using illustrative numerical examples. The results show that the intensity distribution of the considered beam propagating in FRFT is significantly influenced by the source beam parameters and the parameters of the FRFT system. It is possible to demonstrate the potential benefits of the results obtained for applications in laser beam shaping, optical trapping, and micro-particle manipulation.
... Due to it accumulates the global information of function in the form of weighting, it is often applied to describe the actual phenomenon with memory characteristics or historical dependencies. Therefore, fractional calculus is widely employed in image processing [5], quantum mechanics [6], anomalous diffusion [7], viscoelastic material [8], and so on. Complex dynamic networks are composed of a large number of interconnected dynamic nodes, each of which is a unit for storing specific content. ...
This paper investigates the cluster synchronization of fractional‐order complex networks. Considering that impulsive control can reduce the update of controller, and the appearance of impulse is always dependent on each node in the networks instead of appearing at fixed instant, thus we design a variable‐time impulsive controller to control the considered networks. Foremost, several assumptions are proposed to guarantee the every solution of coupled error networks intersect each discontinuous impulsive surface exactly once. In addition, by utilizing the B‐equivalence method and the theory of fractional calculus, the variable‐time impulsive fractional‐order system is reduced to a fixed‐time impulsive fractional‐order system, which can be regarded as the comparison system of the former. Next, under the framework of 1‐norm, some sufficient conditions are presented to ensure that fractional‐order system and target trajectory ultimately achieve cluster synchronization. In the end, a numerical example is designed to illustrate the validity and feasibility of theoretical results.
... As a consequence, we may naturally introduce the fractional squeezing transformation (FrST), which is the non-trivial generalization of the fractional Fourier transformation (FrFT) [5][6][7]. By fractionality we mean that two successive integration transformations, the th transformation ℑ and then the th transformation ℑ , are equal to that of the ( + )th transformation, ℑ ℑ = ℑ + . ...
Based on the usual Wigner-Weyl transformation theory we find that the Wigner hyperbolic rotation in phase space will map onto fractional squeezing operator in Hilbert space. The merit of Weyl ordering and the coherent state representation of Fresnel operator is used in our derivation.
... From last theree decades in signal processing and in any other scientific studies/streams, Fourier analysis is a most frequently used tools [1,2,3,4]. In the literature of the pure mathematics and apllied mathematics, a generalized concept of the Fourier transform well known as the fractional Fourier transform was considered in 1980-1987, by Mcbride, Kerr and Namias [5,6], The Kernels of the two operators differ in a ratation by an angle α in the time-frequency domain. From 1980s, a number of research workers and faculty members across the world have independently reinvented in the related field of the fractional Fourier transform . ...
... Additionally, the FrFT can also be applied to solve the FSP JTC cryptosystem variant, as the FDI can be addressed using a FrFT [26,27]. The concept of the FrFT was initially introduced by Namias [28] in the context of quantum physics and has since been applied to optics by several groups [26,27,[29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. Ozaktas and Mendlovic [30] found an interpretation of the FrFT for light propagating in a quadratic graded index media, which was later deeply studied by them in subsequent contributions [32,45]. ...
In this contribution, we introduce a new practical approach to apply the fractional Fourier transform in the modelling of two optical systems: free space propagation and a single lens processor. This formulation presents a simple way to stablish a direct relationship between physical parameters of the two optical systems and a real-valued fractional order. Furthermore, we employ and compare two numerical methods for evaluating the fractional Fourier transform: the convolution and the Fresnel transform. Consequently, we apply this innovative approach to the digital decryption process in an opto-digital joint transform correlator cryptosystem, considering both the free space propagation and the single lens processor variants. We analyse both numerically and experimentally encrypted data to support our proposed method and to investigate the sensitivity of the decryption process with the fractional order. Notably, we obtain similar decryption results for both numerically and experimentally encrypted objects, demonstrating excellent agreement between the theoretical model, the numerical test, and the experiment.
... Like the traditional Fourier transform (FT) case, the fractional Fourier transform (FrFT) is an effective mathematical tool that has been broadly utilized in quantum mechanics, neural networks, differential equations, optics, pattern recognition, radar, sonar, and other communication systems; see, for example, [1][2][3][4][5]. It can be thought of as an expansion of the traditional FT which is initiated in 1980 by Namias [6]. Recently, there have been a number of attempts to extend the FrFT to a new generalized transformation called the coupled FrFT, and immediately, it received much attention in theoretical study of mathematics [7,8]. ...
The windowed coupled fractional Fourier transform was recently proposed in the literature. It may be considered as a generalized version of the windowed fractional Fourier transform. In this study, we first present various basic properties of the windowed coupled fractional Fourier transform including linearity, shifting, modulation, parity, orthogonality relation, and inversion formula. Further, the close relation of the windowed coupled fractional Fourier transform with the two‐dimensional Fourier transform and the windowed fractional Fourier transform is studied. By combining the properties and relation, we derive several versions of the uncertainty inequalities related to the windowed coupled fractional Fourier transform.
... Fractional Fourier Transform (FRT) is a natural extension of a Fourier transform -a transformation fundamental for (optical) signal processing [38][39][40][41]. The definition of FRT as an integral transform (see, e.g., Ref. [39]) is not directly interpretable in terms of physical operations and as such we shall not repeat it here. ...
The Fractional Fourier Transform (FRT) corresponds to an arbitrary-angle rotation in the phase space, e.g., the time-frequency (TF) space, and generalizes the fundamentally important Fourier Transform. FRT applications range from classical signal processing (e.g., time-correlated noise optimal filtering) to emerging quantum technologies (e.g., super-resolution TF sensing) which rely on or benefit from coherent low-noise TF operations. Here a versatile low-noise single-photon-compatible implementation of the FRT is presented. Optical TF FRT can be synthesized as a series of a spectral disperser, a time-lens, and another spectral disperser. Relying on the state-of-the-art electro-optic modulators (EOM) for the time-lens, our method avoids added noise inherent to the alternatives based on non-linear optical interactions (such as wave-mixing, cross-phase modulation, or parametric processes). Precise control of the EOM-driving radio-frequency signal enables fast all-electronic control of the FRT angle. In the experiment, we demonstrate FRT angles of up to 1.63 rad for pairs of coherent temporally separated 11.5 ps-wide pulses in the near-infrared (800 nm). We observe a good agreement between the simulated and measured output spectra in the bright-light and single-photon-level regimes, and for a range of pulse separations (20 ps to 26.7 ps). Furthermore, a tradeoff is established between the maximal FRT angle and optical bandwidth, with the current setup accommodating up to 248 GHz of bandwidth. With the ongoing progress in EOM on-chip integration, we envisage excellent scalability and vast applications in all-optical TF processing both in the classical and quantum regimes.
... The FrFT result of a signal x (t) with an order p is denoted as X α (u). The mathematical definition of the FrFT is as follows [29]: The relationship between the order p of the FrFT and rotation angle α is p = 2α/π. We can also understand the FrFT from the perspective of time-frequency physical transformation [30]. ...
To comprehensively assess the conditions of an optical fiber communication system, it is essential to implement joint estimation of the following four critical impairments: nonlinear signal-to-noise ratio (SNR NL ), optical signal-to-noise ratio (OSNR), chromatic dispersion (CD), and differential group delay (DGD). However, current studies only achieve identifying a limited number of impairments within a narrow range, due to a lack of high-performance computing algorithms and a unified representation of impairments. To address these challenges, we adopt time-frequency signal processing based on the fractional Fourier transform (FrFT) to achieve the unified representation of impairments, while employing a Transformer-based neural network (NN) to break through network performance limitations. To verify the effectiveness of the proposed estimation method, numerical simulations were conducted on a five-channel polarization-division-multiplexed quadrature phase shift keying (PDM-QPSK) long haul optical transmission system with the symbol rate of 50 GBaud per channel. The mean absolute error (MAE) for SNR NL , OSNR, CD, and DGD estimation is 0.091 dB, 0.058 dB, 117 ps/nm, and 0.38 ps, and the monitoring window ranges from 0−20dB, 10−30dB, 1700−51,000ps/nm, and 0−100ps, respectively. Our proposed method achieves accurate estimation of linear and nonlinear impairments over a broad range, representing a significant advancement in the field of optical performance monitoring (OPM).
... In parallel with the developments in wavelet-based MRA, Fourier transform (FT)-based approaches are also being developed. This framework includes the fractional FT [9][10][11], and the relatively recently developed quadratic-phase FT [12,13], which offers a unified approach to processing transient and non-transient signals and is very useful for analyzing signals with time-varying properties. Although many FT-based MRA methods have been developed, in modern digital receiver development the conventional FT is still widely used. ...
The discrete Fourier transform (DFT) is the most commonly used signal processing method in modern digital sensor design for signal study and analysis. It is often implemented in hardware, such as a field programmable gate array (FPGA), using the fast Fourier transform (FFT) algorithm. The frequency resolution (i.e., frequency bin size) is determined by the number of time samples used in the DFT, when the digital sensor’s bandwidth is fixed. One can vary the sensitivity of a radio frequency receiver by changing the number of time samples used in the DFT. As the number of samples increases, the frequency bin width decreases, and the digital receiver sensitivity increases. In some applications, it is useful to compute an ensemble of FFT lengths; e.g., 2P−j for j=0, 1, 2, …, J, where j is defined as the spectrum level with frequency resolution 2j·Δf. Here Δf is the frequency resolution at j=0. However, calculating all of these spectra one by one using the conventional FFT method would be prohibitively time-consuming, even on a modern FPGA. This is especially true for large values of P; e.g., P≥20. The goal of this communication is to introduce a new method that can produce multi-resolution spectrum lines corresponding to sample lengths 2P−j for all J+1 levels, concurrently, while one long 2P-length FFT is being calculated. That is, the lower resolution spectra are generated naturally as by-products during the computation of the 2P-length FFT, so there is no need to perform additional calculations in order to obtain them.
... This leads to what is called sixty years later fractional Fourier transform (FrFT) (see e.g. [19]). ...
We provide a concrete characterization of the Bergman space of bicomplex-valued bc-meromorphic functions with a strong pole at the origin of the bicomplex discus. The explicit expression of its reproducing kernel is given, and its integral representation as the range of the bicomplex version of the generalized second Bargmann transform is also considered. In addition, we construct the bicomplex analog of the fractional Hankel transform as well as its dual transform. Its range is described and its reproducing kernel is given. Such description involves the zeros of the generalized Laguerre polynomials.
... During the last few years, Namias (Namias, 1980) introduced the fractional Fourier transform (FRFT) as a mathematical technique to solve problems in quantum mechanics. Since then, researchers have employed the technique in various applications including laser optics, signal processing, and image encryption Ozaktas and Mendlovic, 1993;Lohmann, 1993;Mendlovic et al., 1994;Dorsch et al., 1995;Zhang et al 1998;Kutay and Ozaktas, 1998;Xue, 2001;Torre, 2002;Hannelly and Sheridan, 2003;Wang and Zhao, 2013). ...
We theoretically investigate the propagation properties of a Laguerre higher order cosh Gaussian beam (LHOchGB) in a fractional Fourier transform (FRFT) optical system. Based on the Collins formula and the expansion of the hard aperture function into a finite sum of Gaussian functions, we derive analytical expressions for a LHOchGB propagating through apertured and unapertured FRFT systems. The analysis of the evolution of the intensity distribution at the output plane has shown from the obtained expressions, using illustrative numerical examples. The results show that the intensity distribution of the considered beam propagating in FRFT is significantly influenced by the source beam parameters and the parameters of the FRFT system. It is possible to demonstrate the potential benefits of the results obtained for applications in laser beam shaping, optical trapping, and micro-particle manipulation.
... In 1980, Namias described an incomplete form of the fractional Fourier transform [30], which is a generalization of the Fourier transform (FT). In 1987, McBride and Kerr published an extended analysis of the FrFT [31], on which recent work is based. ...
The advancement of spatial interaction technology has greatly enriched the domain of consumer electronics. Traditional solutions based on optical technologies suffers high power consumption and significant costs, making them less ideal in lightweight implementations. In contrast, ultrasonic solutions stand out due to their lower power consumption and cost-effectiveness, capturing widespread attention and interest. This paper addresses the challenges associated with the application of ultrasound sensors in spatial localization. Traditional ultrasound systems are hindered by blind spots, large physical dimensions, and constrained measurement ranges, limiting their practical applicability. To overcome these limitations, this paper proposes a miniature ultrasonic spatial localization module employing piezoelectric micromechanical ultrasonic transducers (PMUTs). The module is comprised of three devices each with dimension of 1.2 mm × 1.2 mm × 0.5 mm, operating at a frequency of around 180 kHz. This configuration facilitates a comprehensive distance detection range of 0–800 mm within 80° directivity, devoid of blind spot. The error rate and failure range of measurement as well as their relationship with the SNR (signal-to-noise ratio) are also thoroughly investigated. This work heralds a significant enhancement in hand spatial localization capabilities, propelling advancements in acoustic sensor applications of the meta-universe.
We introduce the continuous and discrete versions of multi-dimensional linear canonical wave packet transform with the non-separable kernel. At first, we establish a relationship between the non-separable linear canonical wave packet transform (NSLCWPT) and n-dimensional windowed Fourier transform (WFT) and then establish the fundamental results like reconstruction formula, orthogonality relation and some bounds for the NSLCWPT. Some examples were also provided for the enhancement of our proposed transform. We apply the discrete version of the NSLCWPT in the context of the almost periodic functions. At last, a simple conclusion is given, which provides the idea for future work.
The coupled fractional Wigner–Ville distribution is a more general version of the fractional Wigner–Ville distribution. Main properties including boundedness, Moyal’s formula and inversion formula are studied in detail for the transformation. Additionally, the relation of the coupled fractional Wigner–Ville distribution with the two-dimensional Fourier transform is studied. We also present the relationship between the coupled fractional Wigner–Ville distribution with the two-dimensional Wigner–Ville distribution. We show how the properties and relations allow us to derive several versions of the uncertainty inequalities related to the coupled fractional Wigner–Ville distribution.
In this paper, an efficient method for parameter estimation of polynomial phase signal (PPS) is proposed. Instead of the existing methods based on parametric ergodic search or phase differentiation, the proposed method adaptively tracks the PPS phase through extended Kalman filtering (EKF), termed as APT-EKF. Firstly, based on the smoothness assumption of the local phase, a state-space model describing the PPS phase is constructed. Then, by solving the state-space model through EKF, the PPS phase can be tracked. Finally, the least square estimation (LSE) is performed for the inversion of PPS coefficients, and the O’Shea refinement strategy is implemented to enhance the estimation accuracy, thereby achieving the Cramér-Rao lower bound (CRLB). Compared with most existing studies, the proposed method occupies an obvious advantage in signal-to-noise ratio (SNR) threshold and computational efficiency. It is suitable for the arbitrary order PPS. Moreover, this article provides a comprehensive analysis of the parameter initialization, performance bounds, and computational complexity of the proposed method. Both simulation and experiment results are provided to demonstrate the effectiveness of the proposed method.
Una aproximación para entender los fenómenos con dinámica compleja es el análisis de datos. El escalado multidimensional permite visualizar el comportamiento de los sistemas y capturar su evolución espaciotemporal. Mientras que el cálculo fraccional, aplicado mediante Fractional State Space Portrait permite identificar clústeres en grupos de datos, incluyendo variables meteorológicas como la temperatura. Para ello se ha utilizado la información mutua multivariante para encontrar el orden de derivada óptima que ha dado como resultado una visualización mejorada del sistema dinámico de temperaturas en 11 estaciones meteorológicas de la provincia de Chimborazo durante el año 2015. En el mapa del Fractional State Space Potrait se ha logrado identificar dos grandes clústeres que representan las dos estaciones típicas de un clima tropical ecuatorial. Tales clústeres se encuentran fuertemente influenciados por los diversos microclimas presentes en un territorio heterogéneo.
In this work, an n-dimensional pseudo-differential operator involving the n-dimensional linear canonical transform associated with the symbol ?(x1,..., xn; y1,..., yn) ? C?(Rn ? Rn) is defined. We have introduced various properties of the n-dimensional pseudo-differential operator on the Schwartz space using linear canonical transform. It has been shown that the product of two n-dimensional pseudodifferential operators is an n-dimensional pseudo-differential operator. Further, we have investigated formal adjoint operators with a symbol ? ? Sm using the n-dimensional linear canonical transform, and the Lp(Rn) boundedness property of the n-dimensional pseudo-differential operator is provided. Furthermore, some applications of the n-dimensional linear canonical transform are given to solve generalized partial differential equations and their particular cases that reduce to well-known n-dimensional time-dependent Schr?dinger-type-I/Schr?dinger-type-II/Schr?dinger equations in quantum mechanics for one particle with a constant potential.
The Fractional Fourier transform has a good energy aggregation effect for linear frequency modulation (LFM) signals commonly used in radar systems. Therefore, this paper proposes a mainlobe interference suppression method based on Fractional Fourier transform (FRFT). Firstly, the mixed radar echo signal containing main lobe interference is processed by FRFT transform with specific LFM signal characteristics, then the interference and most noise energy are removed by filtering in the FRFT domain. Finally, FRFT inverse transformation recovers the target signal. Simulation verifies the effectiveness of the algorithm.
In this study, we describe a linear space A_(α,p(.))^(w,ν) (R^d ) of functions f∈L_w^1 (R^d ) whose fractional Fourier transforms F_α f belong to L_ν^p(.) (R^d ) for p^+<∞. We show that A_(α,p(.))^(w,ν) (R^d ) becomes a Banach algebra with the sum norm ‖f‖_(A_(α,p(.))^(w,ν) )=‖f‖_(1,w)+‖F_α f‖_(p(.),ν) and under Θ (fractional convolution) convolution operation. Besides, we indicate that the space A_(α,p(.))^(w,ν) (R^d ) is an abstract Segal algebra, where w is weight function of regular growth. Moreover, we find an approximate identity for A_(α,p(.))^(w,ν) (R^d ). We also discuss some other properties of A_(α,p(.))^(w,ν) (R^d ). Finally, we investigate some inclusions of this space.
Sparkers are widely-used sources of high-resolution images for shallow sub-bottom seismic monitoring. Sparker seismic data contain high random noise levels caused by the high voltage and electrical current required to power the sparker source. Random noise is distributed over the entire frequency and wavenumber, and is thereby difficult to suppress. Numerous studies have been conducted to develop random noise suppression by using convolutional neural networks (CNNs). However, because CNNs learn common characteristics of random noise without considering the sequence while training, learning the sequence relationship between useful signal and random noise in time-series data is difficult. As an alternative, we proposed a recurrent neural network (RNN)-based denoising method that considers seismic data sequences. Effective denoising of random noise using RNNs requires features that distinguish between useful signals and random noise during network training. We proposed a denoising method for sparker seismic data that uses bidirectional long short-term memory RNN in fractional Fourier transform (FrFT). The time-frequency distribution in the FrFT domain was utilized for network training, and useful signals were effectively separated from random noise. The proposed method was applied to multi-channel sparker seismic data acquired in Yeongil, Pohang (offshore eastern Korea) by the Korea Maritime & Ocean University. Numerical results confirmed that the proposed method effectively suppressed random noise while maintaining a useful signal in sparker seismic data.
