A new concept for the distributions of wavelet packet decomposition coefficients in detail subbands

This paper presents a new concept that the distributions of wavelet packet decomposition (WPD) coefficients in the detail subbands are not always zero-mean. In general opinion, the detail coefficients of WPD, which contain high-frequency sharp transitions and noise components of the observed signal, are supposed to be distributed as zero-mean if the boundary effect is not considered. However, by the research of the application of WPD for the periodic signals, we find that the WPD coefficients in the detail subbands are a series of non-zero constants rather than distributed as zero-mean when the sampling frequency, the fundamental frequency of periodic signals and the decomposed level of WPD meet a constraint condition. In this paper, the constraint condition is discussed and presented. It can be concluded that the distributions of WPD coefficients in the detail subbands are regarded as zero-mean only when the constraint condition is dissatisfied. This new concept has been validated by using a harmonic signal and a piecewise smooth signal corrupted with harmonics and white noise. The research of the new concept makes the whole theory of WPD coefficients distribution more systematic, complete and scientific.