An Improved Multi-Taper S-Transform Method to Estimate Evolutionary Spectrum and Time-Varying Coherence of Nonstationary Processes

Abstract

Obtaining the accurate evolutionary spectrum and time-varying coherence of multivariate nonstationary processes is essential in engineering applications. However, estimating the time-varying coherence is challenging and has been rarely studied for a single realization of a nonstationary process. To address this issue, an improved multi-taper S-transform (MTST) method is developed to reduce the variance of the evolutionary spectrum and time-varying coherence estimates with a sufficient time-frequency resolution. This new method uses the different number of tapers (NoT) at each frequency to achieve the trade-off between the resolution and variance of the estimated spectrum and coherence. The nonstationary process defined by Wold-Cramer decomposition with a time-varying coherence is considered in this study. First, the analytical formulae of the bias and variance of the evolutionary spectrum of the improved MTST are deduced. The NoT is then analytically derived when time-frequency sine and Hermite tapers are adopted. Subsequently, an iterative procedure is proposed to determine the optimal NoT at each frequency without manual tuning. The proposed technique is finally applied to a nonstationary wind speed process and an earthquake ground motion with different spectral characteristics. Results demonstrate that the improved MTST technique outperforms Priestley’s method, the wavelet-based method, and the MTST method with fixed NoT by estimating a more accurate evolutionary spectrum and time-varying coherence.

Keywords: Multi-taper S-transform method, Nonstationary process, Evolutionary spectrum, Time-varying coherence