A property of the elementary symmetric functions on the frequencies of sinusoidal signals

In this paper, a relation between the elementary symmetric functions on the frequencies of multi-sine wave signal and its multiple integrals is proposed. In particular, such relation is useful to obtain a closed-form expression for the frequencies estimation. The approach used herein is based on the algebraic derivative method in the frequency domain, which allows to yield exact formula in terms of multiple integrals of the signal when placed in the time domain. Moreover, it allows to free oneself from the hypothesis of uniform sampling. Two different ways to approach the estimation are advised, the first is based on least-squares estimation, while the second one is based on the solution of a linear system of dimension equal to the number of sinusoidal components involved. For an easy time realization of such formula, a time-varying filter is proposed. Due to use of multiple integrals of the signal, the resulting parameters estimation is accurate in the face of large measurement noise. To corroborate the theoretical analysis and to investigate the performance of the developed algorithm, computer simulated and laboratory experiments data records are processed.