On estimating the amplitude of harmonic vibration from the Doppler spectrum of reflected signals

The Doppler spectrum of echoes from a sinusoidally vibrating scatterer has discrete spectral lines weighted by Bessel functions of the first kind. Because the signal and spectrum are complicated functions of the vibration amplitude, a number of different approaches have been tried in the past to estimate the vibration amplitude, given a received signal. Here, a new and simple relationship between the spread (or variance) of the Doppler spectrum and the vibration amplitude is derived. A method of estimating the vibration amplitude is proposed based on this relation and a noise compensation procedure is also demonstrated. The performance of the estimators is studied through simulations. High accuracy is predicted under proper sampling conditions even when the signal-to-noise ratio is poor. Slight deviations from single-frequency oscillation, as would be caused by nonlinear or nonideal medium or source effects, are found to have little contribution to the total estimation error.