Evolutionary spectral estimation from a single non-stationary sample using multiple generalized Morse wavelets

Abstract

Estimating the EPSD (evolutionary power spectral density) of a univariate non-stationary process from a single-sample time series is critical, given the often non-repeatable nature of in situ measurements in engineering applications. Existing methods usually exhibit substantial bias and random errors in EPSD estimation. To resolve this issue, an approach based on the multiple generalized Morse wavelet (GMW) is proposed in this study. The core of this method lies in multi-taper estimation based on the orthogonality of GMW functions to reduce errors, which is accompanied by a direct mapping between the wavelet transform and the EPSD, thereby avoiding matrix operations and reducing computational demands. In the numerical examples, the method is validated by comparing its performance with existing approaches under two scenarios: a limited number of sample time histories and a single sample time history. The results indicate that the ensemble average of the estimated EPSDs obtained from the proposed approach enhances the time–frequency representation by reducing both bias and random errors. Since the time–frequency resolution of the GMW function decreases as its order increases, using the first three to five orders generally yields a satisfactory EPSD estimation for the single non-stationary sample. Consequently, the proposed approach improves EPSD estimation by using fewer, more robust preset parameters than those commonly used.