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.

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( N log N ) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed.

Cataloged from PDF version of article.Fourier transforms of fractional order a are defined in a manner such that the common Fourier transform is a special case with order a = 1. An optical interpretation is provided in terms of quadratic graded index media and discussed from both wave and ray viewpoints. Several mathematical properties are derived

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 that under certain conditions one can improve on the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing.

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.

Fourier transforms of fractional order a are defined in a manner such that the common Fourier transform is a special case with order a = 1. An optical interpretation is provided in terms of quadratic graded index media and discussed from both wave and ray viewpoints. Several mathematical properties are derived.

It is customary to define the time-frequency plane such that time and frequency are mutually orthogonal coordinates. Representations of a signal in these domains are related by the Fourier transform. We consider a continuum of "fractional" domains making arbitrary angles with the time and frequency domains. Representations in these domains are related by the fractional Fourier transform. We derive transformation, commutation, and uncertainty relations among coordinate multiplication, differentiation, translation, and phase shift operators between domains making arbitrary angles with each other. These results have a simple geometric interpretation in time-frequency space. I1 est habitue1 de d&finir le plan temps-frkquence de telle sorte que le temps et la frkquence soient des coordonnkes orthogonales. Les reprtsentations d'un signal dans ces domaines sont relites par la transform& de Fourier. Nous considkrons un continuum de domaines "fractionnaires" faisant des angles arbitraires avec les domaines temporels et frtquentiels. Les reprksentations dans ces domaines sont reliCes par la transform& de Fourier fractionnaire. Nous dtrivons les rClations de transformation, de commutation et d'incertitude parmi les opbateurs de dkplacement de phase, de translation, de diffkrentiation, de multiplication de coordon&es entre des domaines faisant des angles arbitraires entre eux. Ces r&ultats ont une interprbtation gkomCtrique simple dans I'espace temps-frkquence.

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 order. This result allows us to pose the fractional Fourier transform as a tool for analyzing and describing optical systems composed of an arbitrary sequence of thin lenses and sections of free space and to arrive at a general class of fractional Fourier-transforming systems with variable input and output scale factors.