AN IDENTIFICATION OF THE FREQUENCIES AND AMPLITUDES OF THE TRIGONOMETRIC SERIES

In this paper, we propose an algorithm for identifying ! j 2 (0 ;1 ), a j ;b j 2 C and N of the following trigonometric series f ( t ) = a 0 +∑ Nj =1 [ a j cos ! j t + b j sin ! j t ]by means of the nite number of sample values. We prove that the fre-quency components are shown to be the solutions of some characteristicequation related to the inverse of a Hankel matrix derived from the samplevalues. 1. IntroductionIn this paper we consider the problem of identifying ! j 2 (0 ;1 ), a j ;b j 2 Cand N of the following trigonometric series(1) f ( t ) = a 0 +∑ Nj =1 [ a j cos ! j t + b j sin ! j t ]by means of the nite number of values f ( t 1 ) ;:::;f ( t L ) where the number L of the values depends on N .In engineering, it is well known that a (sound) signal can be represented asa trigonometric series as in (1). The algorithm developed in this paper thuscan be applied to analyze the signals from the engineering point of view.The main idea of this paper is originated from the paper [1] and [2] by ElBadia and Ha-Duong. In those papers, the authors established an algebraicalgorithm to solve inverse source problems for elliptic equations in 2D and 3Dwhose source terms are assumed to be the combination of either monopolesor dipoles. Applying the concept of the