7.1 Heuristic considerations

A time-frequency analysis of a signal is a representation of the signal as a linear combination of time-frequency atoms. These time-frequency atoms are essentially characterized by an arbitrary duration t ₂ ? t ₁ and an arbitrary frequency ?. The instant t ₁ is the moment when the signal is first heard (if it is a speech signal, for example), and t ₂ is the instant when it ceases to be heard. The frequency ? is an average frequency; this is the frequency of the emitted tone in the case of a musical signal, while the frequency spectrum given by Fourier analysis takes into consideration the parasitic frequencies created by the note's attack and decay.

We also think of a time-frequency atom as occupying a symbolic region in the time-frequency plane (Figure 7.1). This symbolic region is a rectangle R with area 2 ?, which expresses the Heisenberg uncertainty principle.

Figure 7.1: A Heisenberg box in the time-frequency plane.

The most famous example of time-frequency atoms is that of the Gabor wavelets. For these we have f _R( t) = e ^{i ?0 t} g _h( t ? t ₀), where is the center of the time-frequency atom and , g ( t) = ? ^{? 1/4} e ^{? t2/2}. In this case, the "size" of the time-frequency atom is approximately h, and h is approximately equal...