The fractional fourier transform

Abstract

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.

Fundamental structure of Fresnel diffraction: longitudinal uniformity with respect to fractional Fourier order

“…(The constant phase terms e i2πd∕λ e −iaπ∕4 are not of significance.) Equation (2) holds true regardless of the choice of s.…”

“…The fractional Fourier transform (FRT) of a function f u is denoted as f a u , where a is the FRT order [2]. The Fresnel integral describes the propagation of light from one transverse plane along the optical axis to another.…”

“…If we choose to observe the diffracted light on a spherical reference surface of radius R, the chirp multiplication can be dispensed with, and we simply observe the FRT of the input, magnified by M [2]. (The constant phase terms e i2πd∕λ e −iaπ∕4 are not of significance.)…”

“…We introduce dimensionless coordinates [2] and work with a single transverse dimension. Letf x andF σ x denote the space and frequency representation of a signal.…”

“…1). It is known that the Fresnel integral can be decomposed into an FRT followed by magnification followed by chirp multiplication [2][3][4][5][6]:…”