Semiparametric estimation of the frequency of unknown periodic functions and its application to laser vibrometry signals

We propose a semiparametric approach to fundamental frequency estimation of an unknown periodic signal in additive white noise based on model selection. Our estimator maximizes a penalized version of the cumulated periodogram and is proved to be consistent and asymptotically efficient under very general conditions. When the number of observations is fixed, an implementation of this estimation method is proposed and illustrated on specific synthetic signals which arise in laser vibrometry. We extend this method for estimating the fundamental frequencies of two periodic functions having different fundamental frequencies when the data consist of their sum and additive white noise. We also compare the performances of our procedure with the so-called microdoppler technique, which is commonly used for laser vibrometry signals analysis. We show on simulated data that the penalized cumulated periodogram yields an accurate estimation of the frequencies at very low signal-to-noise ratios.