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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 11, NOVEMBER 2006 4261 Discrete Generalized Fresnel Functions and
 

Summary: IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 11, NOVEMBER 2006 4261
Discrete Generalized Fresnel Functions and
Transforms in an Arbitrary Discrete Basis
Igor Aizenberg, Senior Member, IEEE, and Jaakko T. Astola, Fellow, IEEE
Abstract--The idea of generalized Fresnel functions, which
traces back to expressing a discrete transform as a linear convolu-
tion, is developed in this paper. The generalized discrete Fresnel
functions and the generalized discrete Fresnel transforms for an
arbitrary basis are considered. This problem is studied using a
general algebraic approach to signal processing in an arbitrary
basis. The generalized Fresnel functions for the discrete Fourier
transform (DFT) are found, and it is shown that DFT of even
order has two generalized Fresnel functions, while DFT of odd
order has a single generalized Fresnel function. The generalized
Fresnel functions for the conjunctive and Walsh transforms and
the generalized Fresnel transforms induced by these functions are
considered. It is shown that the generalized Fresnel transforms
induced by the Walsh basis and the corresponding generalized
Fresnel functions are unitary and that the generalized Fresnel
transforms induced by the conjunctive basis and the corre-

  

Source: Aizenberg, Igor - College of Science, Technology, Engineering, and Mathematics, Texas A&M University at Texarkana

 

Collections: Computer Technologies and Information Sciences