The phase–corrected undecimated discrete wavelet packet transform and its application to interpreting the timing of events

This paper is concerned with the development and application of the phase–corrected maximal overlap discrete wavelet packet transform (MODWPT). The discrete cyclic filtering steps of the MODWPT are fully explained. Energy preservation is proven. With filter coefficients chosen from Daubechie's least asymmetric class, the optimum time shifts to apply to ensure approximate zero phase filtering at every level of the MODWPT are studied, and applied to the wavelet packet coefficients to give phase corrections which ensure alignment with the original time series. Also, the time series values at each time are decomposed into details associated with each frequency band, and these line up perfectly with features in the original time series since the details are shown to arise through exact zero phase filtering. The phase–corrected MODWPT is applied to a non–stationary time series of hourly averaged Southern Hemisphere solar magnetic field magnitude data acquired by the Ulysses spacecraft. The occurrence times of the shock waves previously determined via manual pattern matching on the raw data match those times in the time–frequency plot where a broadband spectrum is obtained; in other words, the phase–corrected MODWPT provides an approach to picking the location of complicated events. We demonstrate the superiority of the MODWPT in interpreting timing information over two competing methods, namely the cosine packet transform (or ‘local cosine transform’), and the short–time Fourier transform.