Fundamental Principles of Spectral Methods related to Discrete Data

Until the mid 20th century, the groundbreaking works of Claude Shannon regarding infor- mation theory laid the foundation for the vast field of digital signal processing and spectral data analysis, and thereby enabled almost all modern daily life information technologies we are now used to. Digital telecommunication, audio and video compression are just some ex- amples of advanced signal processing which are not possible without a theory about discrete data and its spectral representation. Signal processing starts with the process of sampling by means of a mathematical model, which results in a mapping between the continuous physical measure – like temperature, voltage fluctuations of a microphone or a camera picture – and its time and value discrete representation. According to that, information about the signal can be gained by filtering, data manipulation, pattern extraction and related procedures. Because of the vast variety of applications, many different spectral methods were developed which utilize different mathematical transforms, for example the Fourier transform or the Hilbert transform. Utilizing the appropriate transform for a problem also opens up shortcuts in calculations, or makes signal features visible by decomposing the signal into the associated spectral domain.