# A Simple Spectral Observer

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## Abstract

**:**

## 1. Introduction

## 2. The Proposed Method

**Observability rank condition.**A system

## 3. Application Examples

#### 3.1. Example 1: A Simple Example

#### 3.2. Example 2: Reconstruction of Basic Signals

#### 3.3. Example 3: Edge Detection by Using the Fourier Coefficients

#### 3.4. Example 4: Fitting Complex Signal: The Bitcoin Price

#### 3.5. Example 5: Estimation of the Input Force on a Duffing Oscillator

## 4. Comparative Analysis Vis-à-Vis the STFT

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A. MATLAB CODES

#### Appendix A.1. Symbolic Computation of Matrix A_{ω}

syms w tn=10; %Order of the Fourier Seriesfor k=1:2*n;Aw(k,k)=((-1)^((mod(k-1,4)-mod(k-1,2))/2))*w^(k-1);end

#### Appendix A.2. Symbolic Computation of Matrix A_{ω}

syms w tn=10; %Order of the Fourier Seriesfor k=1:nfor m=1:nAk(2*m-1,2*k-1)=k^(2*m-2);Ak(2*m,2*k)=k^(2*m-1);endend

#### Appendix A.3. Symbolic Computation of Matrix Ω

syms w tn=10; %Order of the Fourier Seriesfor k=1:nfor m=2:2*nO(1,(2*k)-1)=cos(k*w*t);O(1,(2*k))=sin(k*w*t);O(m,(2*k)-1)=diff(O(m-1,2*k-1),’t’);O(m,(2*k))=diff(O(m-1,2*k),’t’);endend

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**MDPI and ACS Style**

Torres, L.; Jiménez-Cabas, J.; Gómez-Aguilar, J.F.; Pérez-Alcazar, P.
A Simple Spectral Observer. *Math. Comput. Appl.* **2018**, *23*, 23.
https://doi.org/10.3390/mca23020023

**AMA Style**

Torres L, Jiménez-Cabas J, Gómez-Aguilar JF, Pérez-Alcazar P.
A Simple Spectral Observer. *Mathematical and Computational Applications*. 2018; 23(2):23.
https://doi.org/10.3390/mca23020023

**Chicago/Turabian Style**

Torres, Lizeth, Javier Jiménez-Cabas, José Francisco Gómez-Aguilar, and Pablo Pérez-Alcazar.
2018. "A Simple Spectral Observer" *Mathematical and Computational Applications* 23, no. 2: 23.
https://doi.org/10.3390/mca23020023