Frequency estimation of multi-sinusoidal signal by multiple integrals A. Bonaventura, L. Coluccio, G. Fedele Dip. Elettronica Informatica e Sistemistica, Universit` a degli Studi della Calabria 87036 Rende (CS), Italy Abstract—In this paper, a simple and robust method for frequency estimation of multi-sinusoidal signal from a given discrete sequence of noisy data is proposed. The approach used herein is based on the algebraic derivative method in the frequency domain and it does not make any assumption on the way to collect the samples of the considered signal. The frequen- cies are estimated without iterations, and explicit relationship between the elementary symmetric functions on the frequencies of the signal and its multiple integrals is given. Due to the use of the process output integrals, the resulting parameters estimation is very robust in the face of large measurement noise in the output. Moreover, for an easy time realization of the estimation, a time-varying linear (unstable) filter is proposed. Finally, the effectiveness of the proposed method is demonstrated through practical and simulated experiments. Keywords— frequency estimation, elementary symmetric functions, least-squares. I. I NTRODUCTION The frequency estimation problem is an important task because of its wide applications in science and engineering, such as power systems [1], [2], [3], signal processing [4], [5], [6], biomedical engineering [7], global position systems [8], etc. Because of its importance, many frequency schemes and many solutions which offer different approaches to the problem have been suggested in literature. Some of them are more general since they are able to handle multiple signals [9], [10], while others focus only on the estimation frequency of a biased (or unbiased) single sine wave [11]-[17]. The key problem is to find a method that improves speed of convergence, accuracy, ability to handle multiple signals, noise re- jection, etc. In [18] (and the references therein), a list of several algorithms is reported: adaptive notch filter, time frequency representation (TFR) based method, phase locked loop (PLL) based method, eigensubspace tracking estimation, extended Kalman filter frequency estimation, internal model based method. Presented in this paper, is a method for estimating the frequencies of multi-sinusoidal signal. The frequency estimation problem can be stated as the approximation of functions on a finite real interval by linear combination of sine waves, that is, the fitting of a weighted sum of n p sine waves of the form: y(t)= np  k=1 A k sin(ω k t +Φ k ) (1) with unknown parameters {A i ,ω i , Φ i } np i=1 , which are to be found from a given discrete sequence of noisy data {y(k)} n k=1 obtained from some experiment. In this paper, we will focus only on the estimation of the parameters {ω i } np i=1 . The approach used herein, is based on the algebraic derivative method in the frequency domain [19], which allows to yield exact formula in terms of multiple integrals of the signal when placed in the time domain. We would like to note that since such method is used, an advantage of our approach is that we can use it for irregular sampling (yet we need to approximate the integrals, and for that approximation some sampling might be better than other). Our method is close to the proposed one in [11], where an algebraic approach for fast and reliable on line identification of the amplitude, frequency and phase parameters in unknown noisy sinu- soidal signal is presented. As regards what is proposed in [11], our method provides a relation between the elementary symmetric functions on the frequencies of multi-sine wave signal and multiple integrals directly on the signal. Due to the use of the process output integrals, the resulting parameters estimation is very robust in the face of large measurement noise and from the performed experiments it seems that our approach is also able to handle signals distorted by harmonic components. The remainder of the paper is organized as follows. In Section II , the main results with analytical expressions of the estimator are derived. Section III is devoted to laboratory and simulated experiments. Finally, Section IV contains some conclusions. II. MAIN RESULTS Let Y (s)= np  k=1 A k cos(φ k )ω k + sin(φ k )s s 2 + ω 2 k (2) be the Laplace transform of y(t). Define the polynomial B(s)= np  k=1 (s 2 + ω 2 k ) (3) of degree 2n p and the following function  σ(np,k)= ∑ 1≤π 1 <...<π k ≤np ω 2 π 1 ω 2 π 2 ...ω 2 π k , k =1, ..., np, σ(np, 0) = 1, (4) that is the kth order elementary symmetric function asso- ciated with  ω 2 i ,  np i=1 , then B(s)= np  i=0 σ(n p ,n p - i)s 2i . (5) By using an approach similar to the proposed one in [20], a relation between the elementary symmetric function defined in eq. (4) and the derivative of the Laplace transform of y(t) is obtained.