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A Mathematical Investigation of a Continuous Covariance Function Fitting with Discrete Covariances of an AR Process^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Continuous Covariance Function

#### 2.1. Construction of a Continuous Covariance Function

#### 2.2. Properties of the Continuous Covariance Function

#### 2.2.1. Power Spectral Density

- The discrete covariances of an AR(p) process (${\mathsf{\Sigma}}_{j}$) are equivalent to the product of the Dirac comb with the continuous covariance function $\gamma \left(h\right)$.
- The convolution theorem shows that multiplication in time domain results in convolution in frequency domain.

#### 2.2.2. Positive Semi-Definite Function

## 3. Simulation

## 4. Conclusions and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. General Fourier Transform of an AR(p) Process

## Appendix B. Explicit Fourier Transform of the AR(1) Process and AR(2) Process with Two Complex Conjugated Zeros

#### Appendix B.1. Fourier Transform of the Continuous Covariance Function of AR(1) Processes

#### Appendix B.2. Fourier Transform of the Continuous Covariance Function of AR(2) Processes with Two Complex Conjugated Zeros

## Appendix C. Convolution of the Fourier Transform of a Continuous Covariance Function of an AR Process with a Dirac Comb

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**Figure 1.**Real part, imaginary part and sum of both parts of a complex covariance function of an AR process with pole −0.8.

**Figure 4.**The function $g\left(\varphi \right)=\varphi cot\left(\varphi \right)$ for $0\le \varphi \le \pi $, and an enlarged section of the beginning.

**Figure 5.**Magic square for a convolution of a continuous covariance function of an AR(3) process with a Dirac comb.

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**MDPI and ACS Style**

Korte, J.; Schubert, T.; Brockmann, J.M.; Schuh, W.-D.
A Mathematical Investigation of a Continuous Covariance Function Fitting with Discrete Covariances of an AR Process. *Eng. Proc.* **2021**, *5*, 18.
https://doi.org/10.3390/engproc2021005018

**AMA Style**

Korte J, Schubert T, Brockmann JM, Schuh W-D.
A Mathematical Investigation of a Continuous Covariance Function Fitting with Discrete Covariances of an AR Process. *Engineering Proceedings*. 2021; 5(1):18.
https://doi.org/10.3390/engproc2021005018

**Chicago/Turabian Style**

Korte, Johannes, Till Schubert, Jan Martin Brockmann, and Wolf-Dieter Schuh.
2021. "A Mathematical Investigation of a Continuous Covariance Function Fitting with Discrete Covariances of an AR Process" *Engineering Proceedings* 5, no. 1: 18.
https://doi.org/10.3390/engproc2021005018