New inverse wavelet transform method with broad application in dynamics

Extracting multi-scale models from system identification of stationary or nonstationary measured signals (e.g., time series) is of great importance in engineering and the applied sciences. We propose a new computational method for harmonic analysis and decomposition of signals based on the inverse wavelet transform and demonstrate its efficacy in diverse areas in dynamics. The wavelet transform is a linear transformation of a signal measured in the temporal/spatial domain to the time–frequency/space-wavenumber domain and applies to stationary and nonstationary measurements. The new method is based on a numerical inverse wavelet transform and yields decomposition of the measured signal in terms of its dominant harmonic components. First, we formulate the analytical continuous inverse wavelet transform in a way that is suitable for computational implementation. Then, taking as example a general measured signal in the time domain, (i) we numerically compute its numerical wavelet transform spectrum, (ii) define a set of “harmonic regions” in the wavelet spectrum containing the dominant harmonics to be inverted and studied, and (iii) by numerically inverse wavelet transforming each of the harmonic regions separately, obtain the respective decomposed harmonics in the time domain. Note that, by construction, the superposition of all decomposed harmonics reconstructs the original signal. Next, we demonstrate the efficacy of the method with some examples. We start with an artificial signal with prescribed harmonic components to highlight the method and its accuracy. Then, we show applicability of the method to system identification, by applying it to the modal analysis of a system of linearly coupled oscillators with closely spaced modes. Lastly, we show how the new method enables quantification of the energy captured by each of the decomposed components (harmonics) in the response of a strongly nonlinear system. To this end, a single degree of freedom geometrically nonlinear oscillator is considered, and the method is used to quantify nonlinear energy “scattering” in its frequency domain. These examples hint at the broad applicability of the new method to diverse areas of signal processing and dynamics, including discrete and continuous dynamical systems with strongly (and even non-smooth) nonlinearities.