Open AccessNumerical Application of a Technique for Recovering the Spectrum of a Time Function *
Author(s) -
Abramovici Flavian
Publication year - 1973
Publication title -
geophysical journal of the royal astronomical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0016-8009
DOI - 10.1111/j.1365-246x.1973.tb06520.x
Subject(s) - eigenfunction , window function , spectrum (functional analysis) , convolution (computer science) , function (biology) , operator (biology) , mathematics , mathematical analysis , fourier transform , series (stratigraphy) , simple (philosophy) , fourier series , interval (graph theory) , kernel (algebra) , transformation (genetics) , pure mathematics , eigenvalues and eigenvectors , physics , computer science , spectral density , combinatorics , quantum mechanics , biology , evolutionary biology , philosophy , repressor , chemistry , paleontology , biochemistry , epistemology , machine learning , artificial neural network , transcription factor , statistics , gene
Summary Consider a time function f ( t ) representing a physical quantity, e.g. a seismic record. This function is known only in the finite time‐interval ( O, T ) so that its spectrum F (ω) cannot be calculated exactly by Fourier transformation, but only its convolution φ(ω) with the spectrum of the windowing function. It has been shown by Barnes and Zagalski that for the simple case of a boxcar window, one can recover F (ω) if one uses an expansion in series of eigenfunctions of the integral operator corresponding to the Fourier kernel. This paper deals with the numerical aspects of this method. After showing some difficulties and how they may be overcome, there are given some examples of time functions that are superpositions of either harmonic or damped oscillations.