Methodological developments of the wavelet transform using phase information

—Complex Meyer Matching Pursuit—

Abstract

Based on Fast Matching Pursuit (FMP), I have developed a new wavelet transform method “Complex Meyer Matching Pursuit (CMMP)” for overcoming the defects of current wavelet transform methods.
Recently, wavelet transform method was popularly applied to analysis of various time series data. However, some wavelet transform methods could even result different interprets during the analysis data. As an example, single wave signal was multi-used by the continuous wavelet transform (CWT). Moreover, only because the different start point, the different transform results were gotten from the discrete wavelet transform (DWT).
The Matching Pursuit (Mallat, 1993) and its extension FMP overcame the above mentioned defects. Their methods could get even results, respectively. However, the Matching Pursuit algorithms consumed computing loads, and caused biased estimation since it used the non-orthonormal wavelet function on the frequency domain. The wavelet function which was used in FMP was generally different shape to signal in time series. Therefore, single wave signal was transformed into many wavelets. Moreover there is some possibility that such signals are not transformed.
I proposed a new definition of the complex Meyer wavelet with the phase component in the frequency domain. Further, I used the algorithm to develop the CMMP. The wavelet is orthonormal wavelet function. Therefore, it overcomes the problem of bias on the frequency domain. Moreover it can be similar to shape of original signals, because it has argument. Next, I designed a new algorithm of transform priority decision and a matching method for overcoming the other defects of the Matching Pursuit, FMP, CWT and DWT.
The new method not only overcomes the defects but also expresses the signal shape as an argument of the complex Meyer wavelet. Therefore, the method can provide the new parameter “argument” of series data for interpretation in addition to the parameters of position on series data, frequency, and amplitude. I expect that the new parameter helps for interpretation of time series data.
For the applications of this new wavelet transform, I analyzed the long-period tremor at Aso volcano that was recorded by the time series data of broadband seismograph network (F-net) of the National Research Institute for Earth science and Disaster prevention (NIED). I was able to extract the waveform features of tremor by using the CMMP. Finally, I found the differences between the data recorded terms in the waveform shape of tremor which indicated that the tremors source was changed in these terms.