Digital computation of the fractional Fourier transform

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.

[1]  G. S. Agarwal,et al.  A simple realization of fractional Fourier transform and relation to harmonic oscillator Green's function , 1994 .

[2]  L.R. Rabiner,et al.  Interpolation and decimation of digital signals—A tutorial review , 1981, Proceedings of the IEEE.

[3]  Gabor C. Temes,et al.  Interpolation and Decimation of Digital SignalsA Tutorial Review , 1992 .

[4]  H. Ozaktas,et al.  Fractional Fourier optics , 1995 .

[5]  Soo-Young Lee,et al.  Fractional Fourier transforms, wavelet transforms, and adaptive neural networks , 1994 .

[6]  Vogel,et al.  Phase distribution of a quantum state without using phase states. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[7]  H. Ozaktas,et al.  Fourier transforms of fractional order and their optical interpretation , 1993 .

[8]  P. Pellat-Finet.  Fresnel diffraction and the fractional-order Fourier transform. , 1994, Optics letters.

[9]  O. Soares,et al.  Fractional Fourier transforms and imaging , 1994 .

[10]  D. F. McAlister,et al.  Spatial and Temporal Optical Field Reconstruction Using Phase-Space Tomography , 1994 .

[11]  Beck,et al.  Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum. , 1993, Physical review letters.

[12]  Walls,et al.  Quantum superpositions generated by quantum nondemolition measurements. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[13]  John C. Wood,et al.  Radon transformation of time-frequency distributions for analysis of multicomponent signals , 1994, IEEE Trans. Signal Process..

[14]  John C. Wood,et al.  Tomographic time-frequency analysis and its application toward time-varying filtering and adaptive kernel design for multicomponent linear-FM signals , 1994, IEEE Trans. Signal Process..

[15]  Luís B. Almeida,et al.  The fractional Fourier transform and time-frequency representations , 1994, IEEE Trans. Signal Process..

[16]  H. Ozaktas,et al.  Fractional Fourier transform as a tool for analyzing beam propagation and spherical mirror resonators. , 1994, Optics letters.

[17]  F. Hlawatsch,et al.  Linear and quadratic time-frequency signal representations , 1992, IEEE Signal Processing Magazine.

[18]  P. Pellat-Finet,et al.  Fractional order Fourier transform and Fourier optics , 1994 .

[19]  Levent Onural,et al.  Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms , 1994 .

[20]  John C. Wood,et al.  Linear signal synthesis using the Radon-Wigner transform , 1994, IEEE Trans. Signal Process..

[21]  A. Lohmann.  Image rotation, Wigner rotation, and the fractional Fourier transform , 1993 .

[22]  R. Bracewell,et al.  Adaptive chirplet representation of signals on time-frequency plane , 1991 .

[23]  Chrysostomos L. Nikias,et al.  A new positive time-frequency distribution , 1994, Proceedings of ICASSP '94. IEEE International Conference on Acoustics, Speech and Signal Processing.

[24]  Ronald N. Bracewell,et al.  Whistler analysis in the time‐frequency plane using chirplets , 1992 .

[25]  James H. McClellan,et al.  The DRFT-a rotation in time-frequency space , 1995, 1995 International Conference on Acoustics, Speech, and Signal Processing.

[26]  Levent Onural,et al.  Optimal filtering in fractional Fourier domains , 1995, 1995 International Conference on Acoustics, Speech, and Signal Processing.

[27]  F. H. Kerr,et al.  On Namias's fractional Fourier transforms , 1987 .

[28]  O. Soares,et al.  Fractional Fourier transforms and optical systems , 1994 .

[29]  A. Lohmann,et al.  Fractional Correlation , 1995 .

[30]  M G Raymer,et al.  Chronocyclic tomography for measuring the amplitude and phase structure of optical pulses. , 1993, Optics letters.

[31]  B. Dickinson,et al.  Eigenvectors and functions of the discrete Fourier transform , 1982 .

[32]  Beck,et al.  Complex wave-field reconstruction using phase-space tomography. , 1994, Physical review letters.

[33]  Vogel,et al.  Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. , 1989, Physical review. A, General physics.

[34]  V. Namias.  The Fractional Order Fourier Transform and its Application to Quantum Mechanics , 1980 .

[35]  F. Smithies.  Linear Operators , 2019, Nature.

[36]  S. Haykin,et al.  'Chirplets' and 'warblets': novel time─frequency methods , 1992 .

[37]  Joseph Shamir,et al.  First-order optics—a canonical operator representation: lossless systems , 1982 .

[38]  H. Ozaktas,et al.  Fractional Fourier transforms and their optical implementation. II , 1993 .

[39]  L. Cohen,et al.  Time-frequency distributions-a review , 1989, Proc. IEEE.

[40]  Gene H. Golub,et al.  Matrix computations , 1983 .

[41]  A. Lohmann,et al.  RELATIONSHIPS BETWEEN THE RADON-WIGNER AND FRACTIONAL FOURIER TRANSFORMS , 1994 .