Under some asymptotic conditions, modulated trajectories in the time-frequency plane can be directly extracted from the coefficients of the Gabor (1946) transform (or wavelets) of a signal. For simple modulated signals, it has been shown that these trajectories allow us to accurately characterize the frequency-modulation (FM) laws of the analyzed signals. For more modulated signals (for instance musical sounds), some interactions between the trajectories may occur and complex time-frequency structures appear. These interactions and the time-frequency diagrams thus obtained are mathematically and qualitatively described. In particular, some connections with the diagrams of the phase of the transform and with acoustical interaction phenomena (like beats phenomena) are discussed.

The behavior of the continuous wavelet and Gabor coefficients in the asymptotic limit using stationary phase approximations are investigated. In particular, it is shown how, under some additional assumptions, these coefficients allow the extraction of some characteristics of the analyzed signal, such as frequency and amplitude modulation laws. Applications to spectral line estimations and matched filtering are briefly discussed.>