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Extracting information from a signal exhibiting damped resonances is a challenging task in many practical cases due to the presence of noise and high attenuation. The interpretation of the signal relies on a model whose order (i.e., the number of resonances) is in general unknown. In this study, the signal is modeled as a sum of Lorentzian lineshapes, and a Bayesian framework is designed to simultaneously remove the baseline distortion, select the number of resonances, and recover the parameters of each lineshape including frequency, damping factor, resonance amplitude, and noise magnitude. The Bayesian problem is solved resorting to a reversible jump Markov chain Monte Carlo (RJ-MCMC) sampling scheme. The algorithm is tested on synthetic signals as well as experimental data from a resonant ultrasound spectroscopy experiment aiming to measure elastic properties. The results show that, compared to the well-known linear prediction singular value decomposition method, the RJ-MCMC method achieves a better performance with the advantages of joint model selection, high accuracy estimation, and uncertainty evaluation. We found that when the signal-to-noise-ratio is larger than 20 dB, the average relative error for frequency extraction is smaller than
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