Applications of the fractional Fourier transform in optics and signal processing: a review

The fractional Fourier transform The fractional Fourier transform is a generalization of the common Fourier transform with an order parameter a. Mathematically, the ath order fractional Fourier transform is the ath power of the fractional Fourier transform operator. The a = 1st order fractional transform is the common Fourier transform. The a = 0th transform is the function itself. With the development of the fractional Fourier transform and related concepts, we see that the common frequency domain is merely a special case of a continuum of fractional domains, and arrive at a richer and more general theory of alternate signal representations, all of which are elegantly related to the notion of space-frequency distributions. Every property and application of the common Fourier transform becomes a special case of that for the fractional transform. In every area in which Fourier transforms and frequency domain concepts are used, there exists the potential for generalization and improvement by using the fractional transform.

[1]  Carlos Ferreira,et al.  Anamorphic fractional Fourier transforming , 1996, Other Conferences.

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

[3]  D Mendlovic,et al.  Fractional Hilbert transform. , 1996, Optics letters.

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

[5]  Sumiyoshi Abe,et al.  Almost-Fourier and almost-Fresnel transformations , 1995 .

[6]  Osvaldo J. Trabocchi,et al.  Fractional Fourier transform applied to spatial filtering in the Fresnel domain , 1995 .

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

[8]  A. Lohmann,et al.  Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform. , 1994, Applied optics.

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

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

[11]  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..

[12]  A. Lohmann,et al.  Chirp filtering in the fractional Fourier domain. , 1994, Applied optics.

[13]  Zeev Zalevsky,et al.  Some important fractional transformations for signal processing , 1996 .

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

[15]  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.

[16]  Haldun M. Ozaktas,et al.  Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms , 1995 .

[17]  M. Marhic.  Roots of the identity operator and optics , 1995 .

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

[19]  Haldun M. OZAKTAS,et al.  Convolution and Filtering in Fractional Fourier Domains , 1994 .

[20]  C. Shih,et al.  Optical interpretation of a complex-order Fourier transform. , 1995, Optics letters.

[21]  Levent Onural,et al.  Optimal filtering in fractional Fourier domains , 1997, IEEE Trans. Signal Process..

[22]  A. Lohmann.  A fake zoom lens for fractional Fourier experiments , 1995 .

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

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

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

[26]  David Mendlovic,et al.  Propagation of mutual intensity expressed in terms of the fractional Fourier transform , 1996 .