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

Abstract

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.