Damped-amplitude & polynomial frequency model for short-time ambient vibrations of a building

The modelling of signal in the context of multiple components, few samples, strong non-stationarity and non-linearity is a difficult problem. For signals in this category, classical methods such as Fourier-based techniques, time-frequency representation, and high order ambiguity function suffer from limitations of various kinds. For this concern, some other techniques which avoid these limitations should be applied. A model based on polynomial approximation of both the instantaneous amplitude and frequency had been proved to be a favorable choice. With consideration of adapting to ambient vibration signals where the amplitude is damped, we propose to use a new model of which the amplitude is approximated by damped exponentials. Meanwhile, the instantaneous frequency is represented by low-order orthonormal polynomials in order to track the strong local variations. Then the parameters estimation is carried out via a maximum likelihood procedure followed by a stochastic optimization method. For the purpose of faster convergence, the adaptive simulated annealing is employed which permits a more flexible temperature tuning. In order to study the behavior of the proposed algorithm, we then detail its performance based on simulated multi-component signals. Cramer-Rao bounds are recalculated and compared with those obtained by the model previously proposed, where the amplitude is based on polynomial approximation. Analysis of the frequency resolution between two closely spaced components is also shown as another important evidence of performance. In order to achieve a direct physical interpretation in real world applications, the proposed amplitude-frequency model is transformed to the physical damping model, under which the estimated signal is decomposed into time-varying resonance frequencies and damping ratios. The algorithm is further applied on ambient vibrations of buildings as a local analysis. Results are discussed in agreement with the model of a dissipative dynamic system, which corresponds to the ambient vibrations of a building.