The linear canonical transformation : definition and properties

Martin J. Bastiaans, Tatiana Alieva

Onderzoeksoutput: Hoofdstuk in Boek/Rapport/CongresprocedureHoofdstukAcademicpeer review

4 Citaties (Scopus)

Uittreksel

In this chapter we introduce the class of linear canonical transformations, which includes as particular cases the Fourier transformation (and its generalization: the fractional Fourier transformation), the Fresnel transformation, and magnifier, rotation and shearing operations. The basic properties of these transformations—such as cascadability, scaling, shift, phase modulation, coordinate multiplication and differentiation—are considered. We demonstrate that any linear canonical transformation is associated with affine transformations in phase space, defined by time-frequency or position-momentum coordinates. The affine transformation is described by a symplectic matrix, which defines the parameters of the transformation kernel. This alternative matrix description of linear canonical transformations is widely used along the chapter and allows simplifying the classification of such transformations, their eigenfunction identification, the interpretation of the related Wigner distribution and ambiguity function transformations, among many other tasks. Special attention is paid to the consideration of one- and two-dimensional linear canonical transformations, which are more often used in signal processing, optics and mechanics. Analytic expressions for the transforms of some selected functions are provided.
TaalEngels
TitelLinear Canonical Transforms: Theory and Applications
RedacteurenJ.J. Healy, M.A. Kutay, H.M. Ozaktas, J.T. Sheridan
Plaats van productieNew York
UitgeverijSpringer
Pagina's29-80
Aantal pagina's52
ISBN van geprinte versie978-1-4939-3027-2
DOI's
StatusGepubliceerd - 2016

Publicatie series

NaamSpringer Series in Optical Sciences
UitgeverijSpringer
Volume198
ISSN van geprinte versie0342-4111

Vingerafdruk

Fourier transformation
matrices
magnification
shearing
multiplication
phase modulation
ambiguity
signal processing
eigenvectors
distribution functions
optics
momentum
scaling
shift

Citeer dit

Bastiaans, M. J., & Alieva, T. (2016). The linear canonical transformation : definition and properties. In J. J. Healy, M. A. Kutay, H. M. Ozaktas, & J. T. Sheridan (editors), Linear Canonical Transforms: Theory and Applications (blz. 29-80). (Springer Series in Optical Sciences; Vol. 198). New York: Springer. DOI: 10.1007/978-1-4939-3028-9_2
Bastiaans, Martin J. ; Alieva, Tatiana. / The linear canonical transformation : definition and properties. Linear Canonical Transforms: Theory and Applications. redacteur / J.J. Healy ; M.A. Kutay ; H.M. Ozaktas ; J.T. Sheridan. New York : Springer, 2016. blz. 29-80 (Springer Series in Optical Sciences).
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Bastiaans, MJ & Alieva, T 2016, The linear canonical transformation : definition and properties. in JJ Healy, MA Kutay, HM Ozaktas & JT Sheridan (redactie), Linear Canonical Transforms: Theory and Applications. Springer Series in Optical Sciences, vol. 198, Springer, New York, blz. 29-80. DOI: 10.1007/978-1-4939-3028-9_2

The linear canonical transformation : definition and properties. / Bastiaans, Martin J.; Alieva, Tatiana.

Linear Canonical Transforms: Theory and Applications. redactie / J.J. Healy; M.A. Kutay; H.M. Ozaktas; J.T. Sheridan. New York : Springer, 2016. blz. 29-80 (Springer Series in Optical Sciences; Vol. 198).

Onderzoeksoutput: Hoofdstuk in Boek/Rapport/CongresprocedureHoofdstukAcademicpeer review

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Bastiaans MJ, Alieva T. The linear canonical transformation : definition and properties. In Healy JJ, Kutay MA, Ozaktas HM, Sheridan JT, redacteurs, Linear Canonical Transforms: Theory and Applications. New York: Springer. 2016. blz. 29-80. (Springer Series in Optical Sciences). Beschikbaar vanaf, DOI: 10.1007/978-1-4939-3028-9_2