A guide on spectral methods applied to discrete data -- Part I: One-dimensional signals

Abstract: Spectral analysis in conjunction with discrete data in one and more dimensions can become a challenging task, because the methods are sometimes difficult to understand. This paper intends to provide an overview about the usage of the Fourier transform, 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 issue of frequency or band limitation of the signal. Or the source of artifacts might be of interest, when a Fourier transform is carried out. Another topic is the issue with fragmented data. Here, the Lomb-Scargle method will be explained with an illustrative example to deal with this special type of signal. Furthermore, a challenge encountered very often is the time-dependent spectral analysis, with which one can evaluate the point in time when a certain frequency appears in the signal. The information to solve such problems and to answer this questions is spread over many disciplines ranging from mathematics, electrical engineering, economic science to astrophysics. The goal of the first part 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 second part of this paper will then address the two- and more-dimensional data. The introduced methods are supported by the spectral package, which has been published for the statistical environment R prior this article.