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CRC Press eBooks

Unified view of tomographic and other transforms in signal analysis

Margarita Man'ko

Published Nov 30, 2003• 0 citations
Publisher

Abstract

Unified construction of tomograms, wavelets and quasidistributions is discussed. Relation of the construction to Klauder’s coherent states (continuous representation) is elucidated. Tomographic representation of nonlinear equations is presented.
DOI
10.1201/9781482269079-180

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A general framework is presented which unifies the treatment of wavelet-like, quasidistribution and tomographic transforms. Explicit formulae relating the three types of transforms are obtained. The case of transforms associated with the symplectic and affine groups is treated in some detail. Special emphasis is given to the properties of the scale-time and scale-frequency tomograms. Tomograms are interpreted as a tool to sample the signal space by a family of curves or as the matrix element of a projector.
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We develop the tomographic representation of wavefunctions which are solutions of the generalized nonlinear Schrödinger equation (NLSE) and show its connection with the Weyl–Wigner map. The generalized NLSE is presented in the form of a nonlinear Fokker–Planck-type equation for the standard probability distribution function (tomogram). In particular, this theory is applied to solitons, where tomograms for envelope bright solitons of a family of modified NLSEs are presented and numerically evaluated. Examples of symplectic tomography and Fresnel tomography of linear and nonlinear signals are discussed.