The method

The method relies on the approximation of the signal by functions (time-frequency atoms) chosen from a very large and redundant set. Given a set of functions (dictionary) $G = \{g_1, g_2, \ldots, g_n\}$ such that $\vert\vert g_i\vert\vert=1$, we can define an optimal $\epsilon, M$-approximation as an expansion minimizing the error $\epsilon$ of the approximation of signal $f$ by $M$ atoms. Such an expansion is defined by the set of indices $\{\gamma_i \}_{i=1..M}$ of the chosen functions $g_{\gamma_i}$ and their weights $\omega_i$:

$$\epsilon=\vert\vert f(t)-\sum_{i=1}^{M} w_i g_{\gamma_i}(t) \vert\vert = \min$$ (3)

Finding such an optimal approximation is a NP-hard problem [Davis, 1994]. Another problem emerges from the fact that such an expansion would be unstable with respect to the number $M$ of used waveforms: changing $M$ even by one can completely change the set of waveforms chosen for the representation. These problems turn our attention to sub-optimal solutions. A sub-optimal expansion, stable with respect to the number of chosen waveforms, can be found by means of an iterative procedure, such as the Matching Pursuit algorithm proposed by Mallat and Zhang [Mallat and Zhang, 1993] (similar approach was discussed by Qian et al. [Qian et al., 1992]).