Conditional spectral moments in matching pursuit based on the chirplet elementary function

Abstract

In this paper, we derive closed form formulae for the first- and the second-order conditional spectral moments, which are generalizations of the concept of instantaneous frequency (IF) and instantaneous bandwidth (IB). We prove that for the matching pursuit (MP) distribution, based on the chirplet elementary function, the first-order conditional spectral moment or the IF is guaranteed to be real valued and the IB is always positive. In addition, not only is the second-order conditional spectral moment positive but so also are all even higher-order conditional spectral moments. These new properties are shown to be additional reasons for using the MP decomposition. We also compare the chirplet and Gaussian atoms, because both always give positive values for the Wigner-Ville distribution (WVD), and therefore the MP distribution is also always positive. We show that when the chirplet atom is used, the MP distribution has superior resolution and the convergence is faster when compared with the Gaussian atom. Finally, as the MP algorithm is computationally demanding, we study how to overcome this limitation when using the chirplet atom.