Journal of Applied Mathematics

Volume 2017 (2017), Article ID 5108946, 27 pages

https://doi.org/10.1155/2017/5108946

## A Guide on Spectral Methods Applied to Discrete Data in One Dimension

Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany

Correspondence should be addressed to Martin Seilmayer

Received 2 January 2017; Accepted 10 April 2017; Published 24 July 2017

Academic Editor: Zhichun Yang

Copyright © 2017 Martin Seilmayer and Matthias Ratajczak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper provides an overview about the usage of the Fourier transform and its related methods and focuses on the subtleties to which the users must pay attention. Typical questions, which are often addressed to the data, will be discussed. Such a problem can be the origin of frequency or band limitation of the signal or the source of artifacts, when a Fourier transform is carried out. Another topic is the processing of fragmented data. Here, the Lomb-Scargle method will be explained with an illustrative example to deal with this special type of signal. Furthermore, the time-dependent spectral analysis, with which one can evaluate the point in time when a certain frequency appears in the signal, is of interest. The goal of this paper is to collect the important information about the common methods to give the reader a guide on how to use these for application on one-dimensional data. The introduced methods are supported by the spectral package, which has been published for the statistical environment prior to this article.

#### 1. Introduction

Because the field of interest is as wide as the amount of information about spectral methods, it becomes necessary to compose this article out of the most important methods. It is split into three main parts, which point at different levels of abstraction and complexity. Section 2 will give a brief and short overview of the mathematical principles and ideas behind the common methods of spectral analysis. For further reading, the Appendix contains a selection of derivations. Remember, the intention of this work is not the completeness of mathematical proofs. Its focus lies on the application of methods on discrete data and the mistakes to be avoided. Section 3 will focus on simple one-dimensional methods, their properties, advantages, and drawbacks.

The individual sections are supported by examples which are programmed in the statistical language [1] with the help of Seilmayer’s spectral [2] package which has been published prior to this article. The main goal of the provided code examples and the introduced package is an easy-to-use user interface which computes scaled output data according to the input. This means if the input time vector would be in units of seconds the output of the spec.fft(x,y)-function would be the natural frequency unit . Besides the analysis of data sets it will be shown how spectral methods can be utilized to carry out (or even accelerate) several calculations on discrete data, like derivation, integration, or folding. Here, source code snippets of the spectral package highlight the implementation and try to underline the methodology.

#### 2. Mathematical Concepts

This chapter introduces the basic ideas and principles which lead to the methods of spectral analysis for discrete data. Thereby, the focus lays on measurements and time series of physical processes. For now, utilizing the time as variable makes it easier to understand the explained methods. Of course, the time variable might be replaced by the space variable , if a distribution of spatial events is of interest. So in this sense space and time are meant to be the* location* where the signal is defined.

The following definitions and explanations are short and sweet. For further reading, it is suggested that you refer to “The Scientist and Engineer’s Guide to Digital Signal Processing” by Smith [3] or to “Time-Frequency Analysis” by Cohen [4]. These two comprehensive works elaborate the underlying mathematical framework in detail. Both books are highly recommended if theoretical approaches and mathematical subtleties are of interest. But from now on the practical application to measured data and understanding of the complex connections is of interest.

##### 2.1. Signal Definition

In the context of this article, a signal corresponds to a physical measure and is therefore real-valued and causal. This means that with the measurement of the process the signal starts to exist at a certain point in time and ends later when the measurement is finished. With that in mind, the signal function represents a slice of the length for times . This can be properly defined as follows:The simple noisy signal in Figure 1 illustrates that. For this example, the underlying physical process could be the temperature, which is measured in the time range of . Evidently the temperature of an object exists before and after the measurement takes place, so the content of only maps to a time interval of this process.