Two Problems of Wavelet Packet Transform

Wavelet packet transform (WPT) is a useful tool for time-frequency analysis. The WPT based on discrete wavelet transform (DWT) has a well-known problem. It is called shift-variance behavior. It varies the energy of DWT result and fails to detect a changing-point even when an original signal shifts only by one sample. Although the solution of multi-resolution analysis (MRA) by using the complex DWT has been released, many reports about the WPT do not accomplish shift-invariance. In this paper, we describe the reason not to complete the shift-invariance for the WPT. In the complex MRA, it constructs a theoretical structure of a parallel DWT with one sample delayed. However the WPT partly achieves it. In addition, the WPT evenly divides a frequency band into 2n parts where n is the decomposition level. Each band is placed in the wrong frequency order. Whenever high-pass filtered signals are down sampled, the origin of the frequency axis is translated to. We show how to make the correct order. The two problems of the shift-invariance and the wrong frequency order are solved.