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Optimal Filtering Applied to the Inversion of the Laplace Transform
Author(s) -
Ross Douglas A.
Publication year - 1988
Publication title -
particle and particle systems characterization
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.877
H-Index - 56
eISSN - 1521-4117
pISSN - 0934-0866
DOI - 10.1002/ppsc.19880050302
Subject(s) - laplace transform , two sided laplace transform , laplace transform applied to differential equations , inverse laplace transform , eigenfunction , mathematics , mellin transform , inversion (geology) , fourier transform , integral transform , mathematical analysis , inverse transform sampling , fractional fourier transform , physics , fourier analysis , optics , eigenvalues and eigenvectors , quantum mechanics , paleontology , structural basin , surface wave , biology
Inversion of the Laplace transform, used in the laser scattering measurement of colloidal particle size distributions, presents severe numerical difficulties. In the presence of noise the variance of the inversion integral is infinite, indicating maximum uncertainty in the inversion. This paper applies the method of minimum variance, or “optimal”, filtering to the eigenfunction spectrum of the Laplace transform, giving an inversion which has finite variance. Spectral decomposition using the eigenfunctions of the Laplace transform gives a representation of the noise and desired signals analogous to the Fourier spectrum used in linear system theory. It is possible to obtain a filtered estimate of the unknown linewidth distribution. The requirement that the variance of this filtered estimate is minimum leads to a Wiener‐Hopf integral equation defining the optimal filter. The results of this paper provide a basis of comparison of all methods of inversion of the Laplace transform, including the extensive literature of colloidal particle sizing by laser scattering or photon correlation.