First class of solutions : the atomic decompositions As we have seen in the previous chapter, the Fourier transform is not adapted to the analysis of non-stationary signals since it projects the signal on infinite waves (sinusoids) which are completely delocalized in time. The concepts of instantaneous frequency and group delay are also inherently unadapted to a large number of non-stationary signals, those containing more than one elementary component, and in particular noisy signals. Thus mono-dimensional solutions seem not to be sufficient, and one has to consider bi-dimensional functions (functions of the variables time and frequency). A first class of such time-frequency representations is given by the atomic decompositions (also known as the linear time-frequency representations). To introduce this concept, we begin with the short-time Fourier transform which has a very intuitive interpretation.
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