On even higher-order moments based on matching pursuit decomposition

In this paper, we derive a closed form formula for the spectral characteristic function based on the matching pursuit (MP) decomposition when using the chirplet dictionary. Thus the instantaneous frequency, the second-order and all higher-order conditional spectral moments can be obtained by a simple differentiation procedure. Although the second-order conditional spectral moment is a generalization of the notion of instantaneous bandwidth, this parameter is not positive for many bilinear time-frequency distributions. So this makes the usual interpretation of this quantity impossible. We prove that if we use the MP decomposition then not only the second conditional spectral moment be positive, but also all even higher-order moments always be positive, independent of the signal being analyzed. This feature is thus another advantage of using the MP decomposition in signal analysis.