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A concise introduction to the concept of fractional Fourier transforms is followed by a discussion of their relation to chirp and wavelet transforms. The notion of fractional Fourier domains is developed in conjunction with the Wigner distribution of a signal. Convolution, filtering, and multiplexing of signals in fractional domains are discussed, revealing… (More)

- David Mendlovic
- NSIP
- 1999

- A W Lohmann, D Mendlovic, Z Zalevsky
- Optics letters
- 1996

We have generalized the Hilbert transform by defining the fractional Hilbert transform (FHT) operation. In the first stage, two different approaches for defining the FHT are suggested. One is based on modifying only the spatial filter, and the other proposes using the fractional Fourier plane for filtering. In the second stage, the two definitions are… (More)

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… (More)

The original definition of the derivatives of a function makes sense only for integral orders, i.e. we can speak of the first or second derivative and so on. However, it is possible to extend the definition of the derivative to noninteger orders by using an elementary property of Fourier transforms. Bracewell shows how fractional derivatives can be used to… (More)

It is often the case that an operation originally defined for integer orders can be generalized to fractional or even complex orders in a meaningful and useful way. A basic example is the power operation. The ath power of x, denoted by Xa, might be defined as the number obtained by multiplying unity a times with x. Thus X = X X X X X. This definition makes… (More)

There exists a fractional Fourier-transform relation between the amplitude distributions of light on two spherical surfaces of given radii and separation. The propagation of light can be viewed as a process of continual fractional Fourier transformation. As light propagates, its amplitude distribution evolves through fractional transforms of increasing… (More)

- D Mendlovic, H M Ozaktas, A W Lohmann
- Applied optics
- 1995

Recently, optical interpretations of the fractional-Fourier-transform operator have been introduced. On the basis of this operator the fractional correlation operator is defined in two different ways that are both consistent with the definition of conventional correlation. Fractional correlation is not always a shift-invariant operation. This property leads… (More)

- H M Ozaktas, D Mendlovic
- Optics letters
- 1994

The complex amplitude distributions on two spherical reference surfaces of given curvature and spacing are simply related by a fractional Fourier transform. The order of the fractional Fourier transform is proportional to the Gouy phase shift between the two surfaces. This result provides new insight into wave propagation and spherical mirror resonators as… (More)

- J Garcia, D Mendlovic, Z Zalevsky, A Lohmann
- Applied optics
- 1996

A fractional correlator that is based on the anamorphic fractional Fourier transform is defined. This new, to our knowledge, correlator has been extended to work with multiple filters. The novelty introduced by the suggested system is the possibility of the simultaneous detection of several objects in different parts of the input scene (when anamorphic… (More)