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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 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.

  • digital computation of the fractional Fourier Transform
    IEEE Transactions on Signal Processing, 1996
    Co-Authors: Haldun M. Ozaktas, Orhan Arikan, M A Kutay, G Bozdagt
    Abstract:

    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.

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

  • sequency ordered generalized walsh Fourier Transform
    Signal Processing, 2013
    Co-Authors: Soochang Pei, Chiachang Wen, Jian-jiun Ding
    Abstract:

    A new Transform family, called the sequency-ordered generalized Walsh-Fourier Transform (SGWFT), is proposed in this paper. Using the kernel matrix generation process and the controllable phase quantization parameter, the Walsh-Hadamard Transform (WHT), the sequency-ordered Hadamard Transform (SCHT), and the discrete Fourier Transform (DFT) become special cases of the SGWFT. The SGWFT can be adjusted by a single parameter to become the WHT, the SCHT, and the DFT. In addition, the SGWFT also has the radix-2 and the split-radix fast algorithms. Compared with the WHT and the SCHT, the properties and the performance of the SGWFT are more similar to those of the DFT. On the other hand, compared with the DFT, the number of multiplications in the SGWFT is less. We also show that the proposed SGWFT has better performance in the applications of DS-CDMA sequence design and Transform coding.

  • 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.

  • 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.

  • 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.

Soontorn Oraintara - One of the best experts on this subject based on the ideXlab platform.

  • integer fast Fourier Transform
    IEEE Transactions on Signal Processing, 2002
    Co-Authors: Soontorn Oraintara, Ying-jui Chen, T Q Nguyen
    Abstract:

    A concept of integer fast Fourier Transform (IntFFT) for approximating the discrete Fourier Transform is introduced. Unlike the fixed-point fast Fourier Transform (FxpFFT), the new Transform has the properties that it is an integer-to-integer mapping, is power adaptable and is reversible. The lifting scheme is used to approximate complex multiplications appearing in the FFT lattice structures where the dynamic range of the lifting coefficients can be controlled by proper choices of lifting factorizations. Split-radix FFT is used to illustrate the approach for the case of 2/sup N/-point FFT, in which case, an upper bound of the minimal dynamic range of the internal nodes, which is required by the reversibility of the Transform, is presented and confirmed by a simulation. The Transform can be implemented by using only bit shifts and additions but no multiplication. A method for minimizing the number of additions required is presented. While preserving the reversibility, the IntFFT is shown experimentally to yield the same accuracy as the FxpFFT when their coefficients are quantized to a certain number of bits. Complexity of the IntFFT is shown to be much lower than that of the FxpFFT in terms of the numbers of additions and shifts. Finally, they are applied to noise reduction applications, where the IntFFT provides significantly improvement over the FxpFFT at low power and maintains similar results at high power.

  • Integer fast Fourier Transform (INTFFT)
    2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221), 2001
    Co-Authors: Soontorn Oraintara, Ying-jui Chen, T. Nguyen
    Abstract:

    The concept of integer fast Fourier Transform (IntFFT) for approximating the discrete Fourier Transform is introduced. Unlike the fixed-point fast Fourier Transform (FxpFFT), the new Transform has properties that it is an integer-to-integer mapping, power-adaptable and also reversible. A lifting scheme is used to approximate complex multiplications appearing in the FFT lattice structures. Split-radix FFT is used to illustrate the approach for the case of 2N-point FFT. The Transform can be implemented by using only bit shifts and additions but no multiplication. While preserving the reversibility, the IntFFT is shown experimentally to yield the same accuracy as the FxpFFT when their coefficients are quantized to a certain number of bits. Complexity of the IntFFT is shown to be much lower than that of the FxpFFT in terms of the numbers of additions and shifts

T Q Nguyen - One of the best experts on this subject based on the ideXlab platform.

  • integer fast Fourier Transform
    IEEE Transactions on Signal Processing, 2002
    Co-Authors: Soontorn Oraintara, Ying-jui Chen, T Q Nguyen
    Abstract:

    A concept of integer fast Fourier Transform (IntFFT) for approximating the discrete Fourier Transform is introduced. Unlike the fixed-point fast Fourier Transform (FxpFFT), the new Transform has the properties that it is an integer-to-integer mapping, is power adaptable and is reversible. The lifting scheme is used to approximate complex multiplications appearing in the FFT lattice structures where the dynamic range of the lifting coefficients can be controlled by proper choices of lifting factorizations. Split-radix FFT is used to illustrate the approach for the case of 2/sup N/-point FFT, in which case, an upper bound of the minimal dynamic range of the internal nodes, which is required by the reversibility of the Transform, is presented and confirmed by a simulation. The Transform can be implemented by using only bit shifts and additions but no multiplication. A method for minimizing the number of additions required is presented. While preserving the reversibility, the IntFFT is shown experimentally to yield the same accuracy as the FxpFFT when their coefficients are quantized to a certain number of bits. Complexity of the IntFFT is shown to be much lower than that of the FxpFFT in terms of the numbers of additions and shifts. Finally, they are applied to noise reduction applications, where the IntFFT provides significantly improvement over the FxpFFT at low power and maintains similar results at high power.