Instantaneous frequency tracking of a sinusoidally frequency-modulated signal with low modulation index: application to laser measurements in acoustics

In this paper, the problem of the instantaneous frequency (IF) estimation of sinusoidally frequency-modulated signals has been addressed, especially when small modulation indices are involved. Such a problem is encountered when laser-Doppler velocity measurements are carried out in acoustics. The amplitude and phase of the sine-wave IF law are then the searched parameters. A new time-frequency transform, specifically designed for tracking sine wave IF variations has been developed, using as a starting point the formalism of the Polynomial Wigner-Ville distributions. The computation of this transform, quick and simple, followed by an amplitude and phase estimation of the IF law, is the so-called Time-Frequency Synchronous Detector (TFSD). After a derivation of the Cramer-Rao bounds for the estimation of the searched parameters, the performances of the detector are studied and compared to those of the Cross Wigner-Ville-based estimator (CWV) and those of a parametric estimator based on the maximum likelihood (ML). For moderate SNR, the TFSD represents a better compromise in terms of bias, variance and computational time than the CWV. The ML performances are slightly better for synthesized signals than those of the time-frequency methods, but the latter provide significantly better estimates for experimental signals.