Besides current new trends related to “big data”, there is still room for a detailed analysis of the fine structure of small size signals. Typical examples are briefly presented, ranging from bioacoustics to mathematics, via physics. Illustrations motivate the usefulness of the methods to be introduced – that are to be described in the rest of the book, the idea being to come back to the analysis of those signals in the last chapter. All of those signals are chirps, with instantaneous characteristics (amplitude and frequency) and are captured by the notion of analytic signal. The ubiquity of chirps is supported by the listing of many additional examples in science and engineering.

The different examples that had been used in Section 1 as motivations, are revisited at the light of what has been discussed later. The bat echolocation case is considered in a greater generality, with considerations about sequences of calls and the “why and how” of their structure in terms of optimality. Time-frequency formulations of matched filtering are proposed and used for, e.g., supporting in a geometrical way the solution to Doppler-tolerance. A similar analysis is provided for gravitational waves, with signal denoising complemented by parameter estimations and comparisons with theoretical models. Finally, Riemann’s zeta-function, as well as variations thereof and Weierstrass' functions, are given a time-frequency interpretation based on their disentanglement into chirp components.