Signal Processing 77 (1999) 111}114 Fast communication New sampling formulae for the fractional Fourier transform Ahmed I. Zayed!, Antonio G. Garcm H a",* !Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA "Departamento de Matema & ticas, Universidad Carlos III de Madrid, 28911 Legane & s, Spain Received 8 April 1999 Abstract In this note we obtain two new sampling formulae for reconstructing signals that are band limited or time limited in the fractional Fourier transform sense. In both cases, we use samples from both the signal and its Hilbert transform, but each taken at half the Nyquist rate. ( 1999 Elsevier Science B.V. All rights reserved. 1. Introduction The fractional Fourier transform (FRFT) has been investigated in a number of papers [1}3,12,17] and has proved to be useful in solving some prob- lems in quantum physics, optics, and signal pro- cessing [4,7}13]. The operational properties of the FRFT have also been the subject of some recent papers [3,17]. The Hilbert transform is also known to play an important role in signal analysis and optics. An optical implementation of the Hilbert transform was introduced in 1950 by Kastler [6], who used it for image processing, especially for edge enhance- ment. In 1996, Lohmann et al. [7] generalized the Hilbert transform by introducing two di!erent de"nitions of what they called the fractional Hilbert transform. The two de"nitions are not equivalent. The "rst is a modi"cation of the spatial "lter with a fractional parameter, while the second * Corresponding author. E-mail: agarcia@math.uc3m.es is based on the author's work on the fractional Fourier transform. They also showed how these fractional Hilbert transforms can be easily imple- mented optically. In [18], Zayed introduced an- other generalization of the Hilbert transform in order to obtain the analytic part of a signal that is associated with the signal's FRFT, i.e., that part of the signal that is obtained by suppressing the nega- tive frequencies of the signal's FRFT. Sampling expansions for the fractional Fourier transform of band-limited and time-limited signals have been derived in [11,16] and they can be used to reconstruct the signal or its fractional Fourier transform from their samples at a discrete set of points satisfying the Nyquist rate. The purpose of this letter is to derive two new sampling expansions to reconstruct the fractional Fourier transform of a time-limited or band- limited signal using samples of the signal and its Hilbert transform, each at half the Nyquist rate. This is an analogue of Goldman's classical result on reconstructing a band-pass signal using samples of the signal and its Hilbert transform, each taken at half the Nyquist rate [5]; see also [15, pp. 66}67]. 0165-1684/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 6 8 4 ( 9 9 ) 0 0 0 6 4 - X