A Novel Approach to Stationarize Nonstationary Stochastic Signals by Wavelet Transforms

Because the observations of physical phenomena are nonstationary in most cases, if there exists a bijective transformation for stationarization, it becomes realistic to process nonstationary signals in the laboratory. In 1994, Houdré proposed an approach that can stationarize the nonstationary processes with wide-sense stationary increments (W.S.S.I.) by the continuous wavelet transform (CVVT). However, some tough assumptions are necessary for the wavelet functions. Unfortunately, most of the well-known wavelet functions do not qualify for all of the assumptions at the same time. The novelty of our work is to provide loose constraints for wavelet functions such that the most famous wavelet functions are qualified to stationarizc nonstationary stochastic processes with W.S.S.I. or wide-sense stationary jumps (W.S.S.J.). Moreover, the CWT of a second-order process with W.S.S.I/W.S.S.J. or the W.S.S. property is also shown to be W.S.S. Because physical data is observed as the form of a discrete sequence, we extend the work to the discrete-time wavelet transform so that a nonstationary sequence with W.S.S.I./W.S.S..I. can be stationarized by an easily realizable perfect reconstructive-quadrature mirror filter structure of the discrete wavelet transform. Six examples for fractional Brownian motion signals and nonstationary signals generated by autoregressive integrated moving average models are provided to show the stationarization both theorectically and numerically.