Time-frequency analysis. Wavelets - Theory

Fourier analysis is a basic tool in signal processing, indispensable in many fields of research, but it quickly shows its limitations, which are justified as soon as we leave the rigorous framework of its definition: the domain of stationary signals of finite energy. In Fourier analysis, all temporal aspects (beginning, end, duration of an event), although present in the phase, become illegible in the spectrum. In particular, the Fourier transform (FT) of a piece of music does not reveal the rhythm played, but simply the notes present. The spectrum alone cannot be used to separate two different scores with the same notes. And yet, we would sometimes like to carry out both time and frequency analysis, to find the "musical range" associated with these non-stationary signals.

The study of non-stationary signals therefore requires either an extension of TF (or stationary methods), by introducing a temporal aspect, or the development of specific methods.

The first solution, introduced intuitively in the middle of the century, corresponds to the sliding-window Fourier (FFG) or short-term Fourier analyses introduced as early as 1945 by D. Gabor [1] with the idea of a time-frequency plane in which frequency modulations would thus be expressed, and in which time would become a complementary parameter to frequency. These methods show that exact joint localization in time and frequency is impossible, and introduce the idea of a discrete, minimal basis, translating into a few coefficients the energy distribution of the signal in the time-frequency plane thus revealed. These approaches were joined by the wavelet transform, which existed in a latent state in both mathematics and signal processing, but whose real boom began in the early 1980s.

A second possible approach is to consider the signal energy density as a distribution of the two variables time and frequency. This joint, bilinear decomposition, introduced by J. Ville [2], highlighted the central role of the Wigner-Ville distribution, and then led to the general affine Cohen classes, encompassing time-frequency representations and previously cited [3][4].