Frequency Shift in the Convolution of Non Stationary Time Series

Abstract

International Workshop in Recent Advances in Time Series Analysis (RATS), 9-12 June 2012, Protaras, CyprusThe Hilbert-Huang Transform (HHT, [1]) is a novel tool to analyze non stationary time series and describe them with their instantaneous, spectral information. HHT decomposes a time series into a number of zeromean, oscillatory modes (Intrinsic Mode Functions, IMF) in order to ensure existence of an interpretable analytic signal of each IMF. Then, it is possible to express the analytic signal in terms of time series of the instantaneous parameters: amplitude, phase and frequency. Therefore, each IMF resides in the timefrequency domain and is described by amplitude and phase as a function of time. In [2] we show that the convolution of IMFs with any spectral response function translates into a complex multiplication of the IMF with that response function. Since the convolution of a (non) stationary time series with a spectral response function in the time domain can be transformed into a basic algebraic formulation, in this work we focus on the repercussions of a convolution of non stationary signals by analyzing the instantaneous parameters and their time derivations of the resulting convolved signal. Most notably, we find that there can be a frequency shift in the resulting signal with respect to the original signal depending on the degree of non stationarity. This finding may be important for non stationary time series, which are filtered by a system response for technical reasons, as it is often the case for physical measurements. However, the work is in a preliminary state and only carried out in the theoryPeer Reviewe