The novel method of the estimation of the fourier transform based on noisy measurements

Abstract

This article refers to the problem of the analysis of spectrum of signals observed in the presence of noise. We propose a new concept of estimation of the frequency content in the signal. The method is derived from the nonparametric methodology of function estimation. We refer to the model of the system yi = R (xi) + ϵi, i = 1, 2,… n, where xi is assumed to be the set of deterministic inputs, xi ∈ D, yi is the set of probabilistic outputs, and ϵi is a measurement noise with zero mean and bounded variance. R(.) is a completely unknown function. In this paper we are interested in a question about frequency spectrum of unknown function. Finding of unknown function in the model could be realized using algorithms based on the Parzen kernel. The alternative approach is based on the orthogonal series expansions. Nonparametric methodology could also be used in the task of implicit estimation of its spectrum. The main aim of this paper is to propose an original integral version of nonparametric estimation of spectrum based on trigonometric series - referring to the classic Fourier transform. The results of numerical experiments are presented.