Introduction: Mathematical Aspects of Time-Frequency Analysis

Abstract

Time-frequency analysis of signals or images deals with mathematical transforms of continuous or discrete data with the aim of having more information accessible after than before the transform. The possible choice of the respective transform strongly depends on the mathematical model of the signal or image source. The modeling scheme affects which analyzing transforms can be applied in a mathematically sensible way, e.g., the mapping should be continuous, and also which transforms give access to new and, in particular, well-formulated interpretations of the data.

In this chapter, we present the ideas behind the optimal choice of an appropriate modeling scheme, the standard signal and image models, and the most important mathematical analysis transforms. This covers aspects of Fourier series and integrals, sampling or discretization problems, and various windowed transforms, such as the short-time Fourier transform, the Gabor transform, and wavelets.

The chapter gives an introduction to the main mathematical terms used in the subsequent four chapters of the book. It shows their relation and interplay and gives entrance points to the lectures presented in subsequent chapters. We provide citations to references for further and more in-depth reading and conclude with a list of exercises.