Using Time Deformation to Filter Nonstationary Time Series with Multiple Time-Frequency Structures

For nonstationary time series consisting of multiple time-varying frequency (TVF) components where the frequency of components overlaps in time, classical linear filters fail to extract components. The G-filter based on time deformation has been developed to extract components of multicomponent G-stationary processes. In this paper, we explore the wide application of the G-filter for filtering different types of nonstationary processes with multiple time-frequency structure. Simulation examples illustrate that the G-filter can be applied to filter a broad range of multicomponent nonstationary process where TVF components may in fact overlap in time. 1. Introduction and Background The traditional linear filter is defined as , where and and where and are the input and output processes. Papoulis [1] has shown that , where and denote the power spectra of the stationary input and output processes, and , and is the frequency response function. Based on this, certain filters (e.g., the Butterworth filter [2]) have been designed to filter or pass low frequency or high frequency components of the input process to the output process. Given a cutoff frequency, a low-pass (high-pass) Butterworth filter can extract components whose frequency content is below (above) this cutoff frequency. In general, traditional linear filters, such as the Butterworth filter, are time invariant. They are designed to extract components from stationary processes where the frequency behavior of the signal does not change with time. However, for time series data with time-varying frequency behavior (TVF), these filters can produce very poor results because the time-invariant nature of the filters does not properly account for the time-varying frequency behavior of the data. That is, these filters do not properly adjust the cutoff frequency with time according to the frequency behavior of the data. See discussion and examples in Xu et al. [3]. In order to address the filtering problem for nonstationary data with TVF, Xu et al. [3] developed the -filter utilizing the time-deformation method by deforming the index set (time axis) of the time series data. The use of time-deformation or time-warping methods to define or analyze nonstationary data has been investigated by several researchers. For example, self-similar processes are obtained by applying the Lamperti operator on the time scale of a stationary process [4]. Gray and Zhang [5] base time deformation on a log transformation of the time axis. They refer to processes that are stationary on the log scale as -stationary processes and