Discrete Generalized Fresnel Functions and Transforms in an Arbitrary Discrete Basis

The idea of generalized Fresnel functions, which traces back to expressing a discrete transform as a linear convolution, 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 studi...

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Bibliographic Details
Published in:IEEE transactions on signal processing 2006-11, Vol.54 (11), p.4261-4270
Main Authors: Aizenberg, I., Astola, J.T.
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Applied sciences
Chirp
Convolution
Digital convolution Fresnel function
Discrete Fourier transforms
Discrete transforms
Exact sciences and technology
Fourier transform
Fourier transforms
Holographic optical components
Holography
Information, signal and communications theory
Miscellaneous
Optical signal processing
Signal processing
Signal processing algorithms
Studies
Telecommunications and information theory
Transforms
Walsh transform
Walsh transforms
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Summary:The idea of generalized Fresnel functions, which traces back to expressing a discrete transform as a linear convolution, 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 corresponding generalized Fresnel functions consist of powers of the golden ratio. It is also shown that the Fresnel transforms induced by the generalized Fresnel functions for the Walsh and conjunctive transforms have fast algorithms
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2006.881189