General considerations are given about signal processing and its place within data science. It is argued that its specificity is rooted in a balanced implication of tools and concepts from physics, mathematics, and informatics. Examples (Fourier, wavelets) are given for supporting this claim, and arguments are detailed for justifying why time-frequency analysis, which is the topic of this book, can be viewed as a natural language for nonstationary signal processing. The Introduction is also the place where to present a roadmap for the way to read the book.

The simplest and most intuitive time-frequency transforms are the Short-Time Fourier Transform (STFT) and the associated spectrogram (squared STFT), which are in close connection with the Gabor transform when using a Gaussian window. Given a 1D signal, its STFT is a 2D function whose redundancy is controlled by a reproducing kernel. This kernel can be equivalently seen as an ambiguity function, i.e., a 2D time-frequency correlation function that is constrained by uncertainty. Building upon the classical duality which exists between “correlation functions” and “energy distributions,” we end up with the Wigner distribution as a central tool for time-frequency signal analysis. The Wigner distribution happens however to be a member of larger classes (such as Cohen's) to which the spectrogram belongs too.

Stationarity is a central concept in time-frequency analysis, but its formal definition in terms of global time-shift invariance is demanding, callingfor a more operational viewpoint that better matches intuition and common practice. A concept of relative stationarity is proposed, which not only makes the observation time-scale enter the picture but also permits to test for stationarity.

Rather than considering the disentanglement of multicomponent nonstationary signals as a time-frequency post-processing, a possibility is to first decompose the observation into modes that are amenable to some further demodulations. In this spirit, this chapter reviews a technique that has recently gained popularity, namely “Empirical Mode Decomposition” and the associated “Hilbert-Huang Transform.” The rationale of those data-driven methods is presented, as well as their actual implementation, with a brief discussion of pros and cons with respect to more conventional time-frequency analysis.

Thanks to its Bargmann representation, a Gaussian STFT can be factorized so as to be described by its zeros. This paves the way for a new approach that exploits the (usually ignored) zeros of the transform. Zeros can serve as centers for Voronoi cells whose statistics is investigated in terms of density, area, and shape. They can also be connected via a Delaunay triangulation, whose characterization in the noise-only situation permits, a contrario, to identify signals embedded in noise from “silent” points.

Web links are provided to free software tools and toolboxes (in Matlab and Python) that offer implementations of most of the methods described in the book and that have been used for producing the figures.