The linear transform kernel for fractional Fourier transforms is derived. The spatial resolution and the space-bandwidth product for propagation in graded-index media are discussed in direct relation to fractional Fourier transforms, and numerical examples are presented. It is shown how fractional Fourier transforms can be made the basis of generalized spatial filtering systems: Several filters...

The propagation of mutual intensity through quadratic graded-index media or free space can be expressed in terms of two-dimensional fractional Fourier transforms for one-dimensional systems and in terms of fourdimensional fractional Fourier transforms for two-dimensional systems. As light propagates, its mutual intensity distribution is continually fractional Fourier transformed. These results ...

Discrete-time fractional derivative filters (1-D and 2-D) are shown to be well approximated from a small set of integer derivatives. A fractional derivative of arbitrary order (and, in 2-D, of arbitrary orientation) can therefore be efficiently computed from a linear combination of integer derivatives of the underlying signal or image.

In this article we provide integral representations for the Dirichlet beta and Riemann zeta functions, which are obtained by combining Mellin transform with the fractional Fourier transform. As an application of these integral formulas we derive tractable expansions of these L-functions in the series of Meixner-Pollaczek polynomials and rising factorials.

The fractional Fourier transform (FRFT) is a one-parametric generalization of the classical Fourier transform. The FRFT was introduced in the 80th and has found a lot of applications and is now used widely in signal processing. Both the space and the spatial frequency domains, respectively, are special cases of the fractional Fourier domains. They correspond to the 0th and 1st fractional Fourie...

The fractional Fourier transform (FrFT) can be thought of as a generalization of the Fourier transform to rotate a signal representation by an arbitrary angle in the time–frequency plane. A lower bound on the uncertainty product of signal representations in two FrFT domains for real signals is obtained, and it is shown that a Gaussian signal achieves the lower bound. The effect of shifting and ...

Filtering in a single time domain or in a single frequency domain has recently been generalized to filtering in a single fractional Fourier domain. In this correspondence, we further generalize this to repeated filtering in consecutive fractional Fourier domains and discuss its applications to signal restoration through an illustrative example.

Any system consisting of a sequence of multiplicative filters inserted between several fractional Fourier transform stages, is equivalent to a system composed of an appropriately chosen sequence of multiplicative filters inserted between appropriately scaled ordinary Fourier transform stages. Thus every operation that can be accomplished by repeated filtering in fractional Fourier domains can a...

In this communication we propose performing two-dimensional correlation operation between phase-space representations based on the fractional Fourier transform, instead of correlating the signals themselves. A numerical examples clearly indicates superior discrimination performance. Ó 2001 Published by Elsevier Science B.V.

In this paper, we consider a q-analogue of the Dunkl operator on R, we define and study its associated Fourier transform which is a q-analogue of the Dunkl transform. In addition to several properties, we establish an inversion formula and prove a Plancherel theorem for this q-Dunkl transform. Next, we study the q-Dunkl intertwining operator and its dual via the q-analogues of the Riemann-Liouv...