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Fractionalization of a Discrete Hankel Transform Based on an Involutory Symmetric Kernel Matrix.

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Název: Fractionalization of a Discrete Hankel Transform Based on an Involutory Symmetric Kernel Matrix.
Autoři: Hanna, Magdy Tawfik
Zdroj: Circuits, Systems & Signal Processing; May2022, Vol. 41 Issue 5, p2750-2778, 29p
Témata: SYMMETRIC matrices, SINGULAR value decomposition, ORTHOGONAL decompositions, LAGUERRE polynomials, DIGITAL watermarking, ORTHOGRAPHIC projection
Abstrakt: The paper marks the emergence of a discrete fractional Hankel transform (DFRHT) based on the eigen decomposition of a symmetric involutory kernel matrix T of a discrete Hankel transform (DHT). Matrix T has only two distinct eigenvalues because it is an involutory matrix. Simple explicit expressions are derived for the orthogonal projection matrices on the eigen spaces of T, and expressions are derived for the dimensions of these two spaces in terms of the trace of matrix T. Since the Hankel transform (HT) is self-reciprocating, taking the HT once can be viewed as a rotation by an angle π radians in the space—Hankel frequency plane. The fractional Hankel transform (FRHT) of fractional order a corresponds to a rotation by an arbitrary angle α where α = π a . The FRHT was introduced by Namias to have the same eigen functions as the HT, namely the product of generalized Laguerre polynomials, a Gaussian function and a power function. Initial orthonormal bases are generated for the two eigen spaces of T by the singular value decomposition of the orthogonal projection matrices. In order for the DFRHT to approximate its continuous counterpart, namely the FRHT, it is preferable to define it in terms of Laguerre–Gaussian-power-like eigenvectors, i.e., eigenvectors that are as close as possible to samples of the eigen functions of the FRHT. A sampling scheme is proposed such that a finite number of samples of the eigen functions will be representative of the behavior of those functions. Final superior orthonormal eigenvectors of matrix T are individually generated for the two eigen spaces by means of either the orthogonal procrustes algorithm (OPA) or the Gram–Schmidt algorithm (GSA). [ABSTRACT FROM AUTHOR]
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Databáze: Complementary Index
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Abstrakt:The paper marks the emergence of a discrete fractional Hankel transform (DFRHT) based on the eigen decomposition of a symmetric involutory kernel matrix T of a discrete Hankel transform (DHT). Matrix T has only two distinct eigenvalues because it is an involutory matrix. Simple explicit expressions are derived for the orthogonal projection matrices on the eigen spaces of T, and expressions are derived for the dimensions of these two spaces in terms of the trace of matrix T. Since the Hankel transform (HT) is self-reciprocating, taking the HT once can be viewed as a rotation by an angle π radians in the space—Hankel frequency plane. The fractional Hankel transform (FRHT) of fractional order a corresponds to a rotation by an arbitrary angle α where α = π a . The FRHT was introduced by Namias to have the same eigen functions as the HT, namely the product of generalized Laguerre polynomials, a Gaussian function and a power function. Initial orthonormal bases are generated for the two eigen spaces of T by the singular value decomposition of the orthogonal projection matrices. In order for the DFRHT to approximate its continuous counterpart, namely the FRHT, it is preferable to define it in terms of Laguerre–Gaussian-power-like eigenvectors, i.e., eigenvectors that are as close as possible to samples of the eigen functions of the FRHT. A sampling scheme is proposed such that a finite number of samples of the eigen functions will be representative of the behavior of those functions. Final superior orthonormal eigenvectors of matrix T are individually generated for the two eigen spaces by means of either the orthogonal procrustes algorithm (OPA) or the Gram–Schmidt algorithm (GSA). [ABSTRACT FROM AUTHOR]
ISSN:0278081X
DOI:10.1007/s00034-021-01911-6