CopYright © IL\C Stochastic Control \·ilnius. Lithuanian SSR. l·SSR. I <.J H6 ON THE FOUNDATION OF PRONY'S METHOD v. Slivinskas and V. Simonyte /1/. \ 1111111' of ,\/alhmtalnS and Cybnneli cs, Lilhllanian AwdflllY of Scitll(fS, Fill/illS, L' SSR Abstract. The paper concerns the foundation of Prony's method whose basis is the groundless assumption that the investigated function can be expanded into the sum of complex exponents. To begin with the class of continuous finite order functions is defined. The notion of the canonical expansion of a function into the formants is introduced. It is established that the formant's structure depends upon the type of the roots of the minimal annulating polynomials of the shift operators acting in the formant's shifts space. The formants over the field of complex numbers are obtained to be complex exponents in the case of simple roots and complex exponents multiplied by the polynomials in the case of multiple roots, what agrees with the assumptlon of Prony's method. The estimation of the spectral density of a stationary random process by Prony's method is considered. The presented foundation allows one not only to understand the method better, but to formulate its extensions. Keywords. Prony's method; formant analysis; minimal realization; identification; spectral analysers; random processes. INTRODUCTION Prony's method (I795) is widely used alongside with other modern methods of identification and spectral estimation (Crittenden and others, I983; Hildebrand, I956; Kay and Marple, I98I; Markel and Gray, I976; Pelikan, I984). The presump- tion that the investigated signal f(t), or the right side of the autocorre- lation function R(t), of a statio- nary random process can be expanded into the linear combination of complex expo- nents is assumed as a basis of the method. The presumption enables one to develop effi- cient algorithms for unknown parameters' estimation (Holt and Antill, I977; Kumaresan, Tufts and Scharf, I983; Van Blaricum and Mittra, I978). The fi- nite sequence of equally spaced samples f(O), f(lIt), ... , f«N-I)lIt) of the in- vestigated function is considered to be the initial data for these algorithms. However, up to now Prony's method is not founded theoretically, i.e., the class of functions for which the method gives the correct results is not determined; it is not quite clear why namely expo- nential functions are chosen as basic functions, whether the expansion of the form (I) is unique, how the sampling in- terval must be chosen, how many samples of the investigated function are re- quired. One can pick out the following Prony's method's distinguishing features: a comp- 121 lex function is expanded into the sum of elementary basic functions; the number of components is finite in the sum; the elementary functions' parameters are not fixed. By means of Prony's method the following problems are solved: the find- ing of the parametrical representation of the continuous function that interpo- lates and extrapolates a finite sequence of data; the estimation of the function's spectral density by means of its para- metrical representation. From the distinguishing features of Prony's method and the problems that the method solves it follows that for the method's foundation it is necessary: (i) to determine the class of the func- tions that can be expanded into the sum of the finite number of the elementary functions; (ii) to find the structure of the elemen- tary functions; (iii) to define the requirements the ini- tial data have to satisfy in order that a function of the determined class could be restored from these data. In this paper the named problems are solved on the basis of the minimal inter- polation theory (Slivinskas and Simonyte, I982, I983, I984a, I984b). THE DETERMINATION OF THE CLASS OF FUNCTIONS The class of the functions of Prony's