Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space-bandwidth product

Date
2010-07-30
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Source Title
Journal of the Optical Society of America A: Optics and Image Science, and Vision
Print ISSN
1084-7529
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Publisher
Optical Society of America
Volume
27
Issue
8
Pages
1885 - 1895
Language
English
Type
Article
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Abstract

Linear canonical transforms (LCTs) form a three-parameter family of integral transforms with wide application in optics. We show that LCT domains correspond to scaled fractional Fourier domains and thus to scaled oblique axes in the space-frequency plane. This allows LCT domains to be labeled and ordered by the corresponding fractional order parameter and provides insight into the evolution of light through an optical system modeled by LCTs. If a set of signals is highly confined to finite intervals in two arbitrary LCT domains, the space-frequency (phase space) support is a parallelogram. The number of degrees of freedom of this set of signals is given by the area of this parallelogram, which is equal to the bicanonical width product but usually smaller than the conventional space-bandwidth product. The bicanonical width product, which is a generalization of the space-bandwidth product, can provide a tighter measure of the actual number of degrees of freedom, and allows us to represent and process signals with fewer samples.

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Keywords
Eigenvalues and eigenfunctions, Fourier transforms, Integral equations, Mathematical transformations, Mechanics, Optical systems, Phase space methods, Finite intervals, Fractional Fourier domains, Fractional order, Integral transform, Linear canonical transform, Number of degrees of freedom, Phase spaces, Process signals, Space-bandwidth product, Space-frequency, Bandwidth
Citation
Published Version (Please cite this version)