THE PHASE-CORRECTED UNDECIMATED DISCRETE WAVELET PACKET TRANSFORM ANDITS APPLICATION TO INTERPRETING THE TIMING OF EVENTS

This paper is concerned with the development and application of the ph ase-corrected maximal overlap discrete wavelet packet transform (MODWP T). The discrete cyclic filtering steps of the MODWPT are fully explai ned. Energy preservation is proven. With filter coefficients chosen fr om Daubechies's least asymmetric class, the optimum time shifts to app ly to ensure approximate zero phase filtering at every level of the MO DWPT are studied, and applied to the wavelet packet coefficients to gi ve phase corrections which ensure alignment with the original time ser ies. Also, the time series values at each time are decomposed into det ails 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 MODWP T is applied to a non-stationary time series of hourly averaged Southe rn Hemisphere solar magnetic field magnitude data acquired by the Ulys ses spacecraft. The occurrence times of the shock waves previously det ermined 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 pickin g the location of complicated events. We demonstrate the superiority o f the MODWPT in interpreting timing information over two competing met hods, namely the cosine packet transform (or 'local cosine transform') , and the short-time Fourier transform.