Abstract
The principal aim of a spectral observer is twofold: the reconstruction of a signal of time via state estimation and the decomposition of such a signal into the frequencies that make it up. A spectral observer can be catalogued as an online algorithm for time-frequency analysis because is a method that can compute on the fly the Fourier Transform (FT) of a signal, without having the entire signal available from the start. In this regard, this paper presents a novel spectral observer with an adjustable constant gain for reconstructing a given signal by means of the recursive identification of the coefficients of a Fourier series. The reconstruction or estimation of a signal in the context of this work means to find the coefficients of a linear combination of sines a cosines that fits a signal such that it can be reproduced. The design procedure of the spectral observer is presented along with the following applications: (1) the reconstruction of a simple periodical signal, (2) the approximation of both a square and a triangular signal, (3) the edge detection in signals by using the Fourier coefficients, (4) the fitting of the historical Bitcoin market data from 1 December 2014 to 8 January 2018 and (5) the estimation of a input force acting upon a Duffing oscillator. To round out this paper, we present a detailed discussion about the results of the applications as well as a comparative analysis of the proposed spectral observer vis-à-vis the Short Time Fourier Transform (STFT), which is a well-known method for time-frequency analysis.
Highlights
The term spectral observer was proposed by Hostetter in his pioneering work [1] to name the algorithm that permits the recursive calculation of the Fourier Transform (FT) of a band-limited signal via state estimation
A spectral observer can be catalogued as an online algorithm to compute the Fourier Transform (FT) during a time window which slides along the signal, i.e., an algorithm to compute the Short Time Fourier Transform (STFT)
It is necessary to point out here that both the frequency resolution and the time resolution do depend on the parameters τ and λ, the parameters hynfft used in the STFT algorithm and ω and n used in the spectral observer, play an important factor; the adjustment of these parameters directly affects in the computational burden and the amount of data to be processed
Summary
The term spectral observer was proposed by Hostetter in his pioneering work [1] to name the algorithm that permits the recursive calculation of the Fourier Transform (FT) of a band-limited signal via state estimation. The main goals of a spectral observer are both the estimation of a given signal and the transformation of such a signal to the frequency domain by means of the recursive identification of the coefficients of a Fourier series [5]. The estimation provided by the observer are both the reconstruction of the original signal and the Fourier coefficients to compute the signal frequency components. This paper is organized as follows: Section 2 presents the core of the proposed method which is the formulation of the spectral observer from the Fourier series.
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