When was the time-frequency analysis invented?
According to the preface of Foundations of Time-Frequency Analysis, a rough timeline is as follows:


1930 - Early development of quantum mechanics by H. Weyl, E.Wigner, and J. von Neumann.
1946 - Theoretical foundation of information theory and signal analysis by D. Gabor (cf. "Theory of communication").
1980 - Time-frequency analysis established as an independent mathematical field (apart from engineering) by Guido Janssen.
1990 - Development of wavelet theory. Overview of the mutual influence given in "The wavelet transform, time-frequency localization and signal analysis" by Ingrid Daubechies.