Fractional Fourier Transform

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Haldun M. Ozaktas - One of the best experts on this subject based on the ideXlab platform.

  • the Fractional Fourier Transform and harmonic oscillation
    Nonlinear Dynamics, 2002
    Co-Authors: Alper M Kutay, Haldun M. Ozaktas
    Abstract:

    The ath-order Fractional Fourier Transform is a generalization ofthe ordinary Fourier Transform such that the zeroth-order FractionalFourier Transform operation is equal to the identity operation and thefirst-order Fractional Fourier Transform is equal to the ordinaryFourier Transform. This paper discusses the relationship of theFractional Fourier Transform to harmonic oscillation; both correspondto rotation in phase space. Various important properties of theTransform are discussed along with examples of commonTransforms. Some of the applications of the Transform are brieflyreviewed.

  • the Fractional Fourier Transform with applications in optics and signal processing
    2001
    Co-Authors: Haldun M. Ozaktas, M Kutayalper, Zeev Zalevsky
    Abstract:

    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.

  • the discrete Fractional Fourier Transform
    IEEE Transactions on Signal Processing, 2000
    Co-Authors: Cagatay Candan, M A Kutay, Haldun M. Ozaktas
    Abstract:

    We propose and consolidate a definition of the discrete Fractional Fourier Transform that generalizes the discrete Fourier Transform (DFT) in the same sense that the continuous Fractional Fourier Transform generalizes the continuous ordinary Fourier Transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete Fractional Fourier Transform supports our confidence that it will be accepted as the definitive definition of this Transform.

  • the discrete harmonic oscillator harper s equation and the discrete Fractional Fourier Transform
    Journal of Physics A, 2000
    Co-Authors: Laurence Barker, Cagatay Candan, T Hakioglu, Alper M Kutay, Haldun M. Ozaktas
    Abstract:

    Certain solutions to Harper's equation are discrete analogues of (and approximations to) the Hermite-Gaussian functions. They are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator, and they lead to a definition of a discrete Fractional Fourier Transform (FT). The discrete Fractional FT is essentially the time-evolution operator of the discrete harmonic oscillator.

  • the discrete Fractional Fourier Transform
    International Conference on Acoustics Speech and Signal Processing, 1999
    Co-Authors: Cagatay Candan, M A Kutay, Haldun M. Ozaktas
    Abstract:

    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.

Ran Tao - One of the best experts on this subject based on the ideXlab platform.

  • sliding 2d discrete Fractional Fourier Transform
    IEEE Signal Processing Letters, 2019
    Co-Authors: Yu Liu, Feng Zhang, Hongxia Miao, Ran Tao
    Abstract:

    The two-dimensional discrete Fractional Fourier Transform (2D DFrFT) has been shown to be a powerful tool for 2D signal processing. However, the existing discrete algorithms aren't the optimal for real-time applications, where the input signals are stream data arriving in a sequential manner. In this letter, a new sliding algorithm is proposed to solve this problem, termed as the 2D sliding DFrFT (2D SDFrFT). The proposed 2D SDFrFT algorithm directly computes the 2D DFrFT in current window using the results of previous window, which greatly reduces the computations. During the derivation, we find that the (m + δ, n)th DFrFT bin in previous window is needed for computing the (m, n)th DFrFT bin in current window, where the increment δ isn't always an integer. Further, a method is proposed to convert the increment δ to a certain integer by determining appropriate sampling interval. The theoretical analysis demonstrates that when compute the new 2D DFrFT in a shifted window in sliding process, our proposed algorithm has the lowest computational cost among existing 2D DFrFT algorithms.

  • Parameter estimation of optical fringes with quadratic phase using the Fractional Fourier Transform
    Optics and Lasers in Engineering, 2015
    Co-Authors: Feng Zhang, Ran Tao, Tingzhu Bai, Wen-ming Yang
    Abstract:

    Abstract Optical fringes with a quadratic phase are often encountered in optical metrology. Parameter estimation of such fringes plays an important role in interferometric measurements. A novel method is proposed for accurate and direct parameter estimation using the Fractional Fourier Transform (FRFT), even in the presence of noise and obstacles. We take Newton׳s rings fringe patterns and electronic speckle pattern interferometry (ESPI) interferograms as classic examples of optical fringes that have a quadratic phase and present simulation and experimental results demonstrating the performance of the proposed method.

  • sparse discrete Fractional Fourier Transform and its applications
    IEEE Transactions on Signal Processing, 2014
    Co-Authors: Shengheng Liu, Feng Zhang, Ran Tao, Tao Shan, Yimin Zhang, Guo Zhang, Yue Wang
    Abstract:

    The discrete Fractional Fourier Transform is a powerful signal processing tool with broad applications for nonstationary signals. In this paper, we propose a sparse discrete Fractional Fourier Transform (SDFrFT) algorithm to reduce the computational complexity when dealing with large data sets that are sparsely represented in the Fractional Fourier domain. The proposed technique achieves multicomponent resolution in addition to its low computational complexity and robustness against noise. In addition, we apply the SDFrFT to the synchronization of high dynamic direct-sequence spread-spectrum signals. Furthermore, a sparse Fractional cross ambiguity function (SFrCAF) is developed, and the application of SFrCAF to a passive coherent location system is presented. The experiment results confirm that the proposed approach can substantially reduce the computation complexity without degrading the precision.

  • short time Fractional Fourier Transform and its applications
    IEEE Transactions on Signal Processing, 2010
    Co-Authors: Ran Tao, Yue Wang
    Abstract:

    The Fractional Fourier Transform (FRFT) is a potent tool to analyze the chirp signal. However, it fails in locating the Fractional Fourier domain (FRFD)-frequency contents which is required in some applications. The short-time Fractional Fourier Transform (STFRFT) is proposed to solve this problem. It displays the time and FRFD-frequency information jointly in the short-time Fractional Fourier domain (STFRFD). Two aspects of its performance are considered: the 2-D resolution and the STFRFD support. The time-FRFD-bandwidth product (TFBP) is defined to measure the resolvable area and the STFRFD support. The optimal STFRFT is obtained with the criteria that maximize the 2-D resolution and minimize the STFRFD support. Its inverse Transform, properties and computational complexity are presented. Two applications are discussed: the estimations of the time-of-arrival (TOA) and pulsewidth (PW) of chirp signals, and the STFRFD filtering. Simulations verify the validity of the proposed algorithms.

  • optical image encryption based on the multiple parameter Fractional Fourier Transform
    Optics Letters, 2008
    Co-Authors: Ran Tao, Jun Lang, Yue Wang
    Abstract:

    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.

Soochang Pei - One of the best experts on this subject based on the ideXlab platform.

  • random discrete Fractional Fourier Transform
    IEEE Signal Processing Letters, 2009
    Co-Authors: Soochang Pei, Wenliang Hsue
    Abstract:

    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.

  • the multiple parameter discrete Fractional Fourier Transform
    IEEE Signal Processing Letters, 2006
    Co-Authors: Soochang Pei, Wenliang Hsue
    Abstract:

    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.

  • a new discrete Fractional Fourier Transform based on constrained eigendecomposition of dft matrix by lagrange multiplier method
    IEEE Transactions on Circuits and Systems Ii: Analog and Digital Signal Processing, 1999
    Co-Authors: Soochang Pei, Chiencheng Tseng, Minhung Yeh
    Abstract:

    This paper is concerned with the definition of the discrete Fractional Fourier Transform (DFRFT). First, an eigendecomposition of the discrete Fourier Transform (DFT) matrix is derived by sampling the Hermite Gauss functions, which are eigenfunctions of the continuous Fourier Transform and by performing a novel error-removal procedure. Then, the result of the eigendecomposition of the DFT matrix is used to define a new DFRFT. Finally, several 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.

  • discrete Fractional Fourier Transform based on orthogonal projections
    IEEE Transactions on Signal Processing, 1999
    Co-Authors: Soochang Pei, Minhung Yeh, Chiencheng Tseng
    Abstract:

    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.

  • two dimensional discrete Fractional Fourier Transform
    Signal Processing, 1998
    Co-Authors: Soochang Pei, Minhung Yeh
    Abstract:

    Abstract Fractional Fourier Transform (FRFT) performs a rotation of signals in the time–frequency plane, and it has many theories and applications in time-varying signal analysis. Because of the importance of Fractional Fourier Transform, the implementation of discrete Fractional Fourier Transform will be an important issue. Recently, a discrete Fractional Fourier Transform (DFRFT) with discrete Hermite eigenvectors has been proposed, and it can provide similar results to match the continuous outputs. On the other hand, the two dimensional continuous Fractional Fourier Transform is also proposed for 2D signal analysis. This paper develops a 2D DFRFT which can preserve the rotation properties and provide similar results to continuous FRFT.

Minhung Yeh - One of the best experts on this subject based on the ideXlab platform.

  • a new discrete Fractional Fourier Transform based on constrained eigendecomposition of dft matrix by lagrange multiplier method
    IEEE Transactions on Circuits and Systems Ii: Analog and Digital Signal Processing, 1999
    Co-Authors: Soochang Pei, Chiencheng Tseng, Minhung Yeh
    Abstract:

    This paper is concerned with the definition of the discrete Fractional Fourier Transform (DFRFT). First, an eigendecomposition of the discrete Fourier Transform (DFT) matrix is derived by sampling the Hermite Gauss functions, which are eigenfunctions of the continuous Fourier Transform and by performing a novel error-removal procedure. Then, the result of the eigendecomposition of the DFT matrix is used to define a new DFRFT. Finally, several 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.

  • discrete Fractional Fourier Transform based on orthogonal projections
    IEEE Transactions on Signal Processing, 1999
    Co-Authors: Soochang Pei, Minhung Yeh, Chiencheng Tseng
    Abstract:

    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.

  • two dimensional discrete Fractional Fourier Transform
    Signal Processing, 1998
    Co-Authors: Soochang Pei, Minhung Yeh
    Abstract:

    Abstract Fractional Fourier Transform (FRFT) performs a rotation of signals in the time–frequency plane, and it has many theories and applications in time-varying signal analysis. Because of the importance of Fractional Fourier Transform, the implementation of discrete Fractional Fourier Transform will be an important issue. Recently, a discrete Fractional Fourier Transform (DFRFT) with discrete Hermite eigenvectors has been proposed, and it can provide similar results to match the continuous outputs. On the other hand, the two dimensional continuous Fractional Fourier Transform is also proposed for 2D signal analysis. This paper develops a 2D DFRFT which can preserve the rotation properties and provide similar results to continuous FRFT.

  • improved discrete Fractional Fourier Transform
    Optics Letters, 1997
    Co-Authors: Soochang Pei, Minhung Yeh
    Abstract:

    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.

Jianjiun Ding - One of the best experts on this subject based on the ideXlab platform.

  • discrete to discrete prolate spheroidal wave functions and finite duration discrete Fractional Fourier Transform
    European Signal Processing Conference, 2007
    Co-Authors: Jianjiun Ding
    Abstract:

    In practical applications, the signal we deal with is usually a finite duration one. Continuous prolate spheroidal wave functions (PSWFs) were proposed by Slepian and are useful for analyzing the characters of the finite duration continuous Fourier Transform. Based on the PSWF, the finite Fractional Fourier Transform was developed. In this paper, for digital signal processing application, we derive discrete-to-discrete prolate spheroidal wave functions. Then, we define the finite duration discrete Fractional Fourier Transform (fi-DFRFT) based on it. We can use the fi-DFRFT for filter design, multiplexing, modulation, encryption, and optical system simulation. The fi-DFRFT has the advantage of less complexity and is useful for deal with the noise that is chirp-like and finite duration.

  • Discrete Fractional Fourier Transform Based on New Nearly Tridiagonal Commuting Matrices
    IEEE Transactions on Signal Processing, 2006
    Co-Authors: Wenliang Hsue, Jianjiun Ding
    Abstract:

    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 DFT eigenvectors, which are more similar to the continuous Hermite-Gaussian functions than those developed before. Rigorous discussions on the relations between the eigendecomposition of the newly proposed nearly tridiagonal matrix and the DFT matrix are described. Furthermore, by appropriately combining two linearly independent matrices that both commute with the DFT matrix, we develop a method to obtain DFT eigenvectors even more similar to the continuous Hermite-Gaussian functions (HGFs). Then, new versions of DFRFT produce their Transform outputs closer to the samples of the continuous Fractional Fourier Transform, and their applications are described. Related computer experiments are performed to illustrate the validity of the works in this paper

  • Discrete Fractional Fourier Transform based on new nearly tridiagonal commuting matrices
    Proceedings. (ICASSP '05). IEEE International Conference on Acoustics Speech and Signal Processing 2005., 2005
    Co-Authors: Wenliang Hsue, Jianjiun Ding
    Abstract:

    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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