Estimating particles velocity from laser measurements: maximum likelihood and Cramer-Rao bounds

We address the problem of estimating particles velocity in the vicinity of an aircraft by means of a laser Doppler system. When a particle passes through the region of interference fringes generated by two coherent laser beams, the signal backscattered is of the form Aexp{-2α²·f_d²t² }cos{2πf_dt}, where the Doppler frequency f_d is related to the aircraft speed. This paper is concerned with the most precise estimation of the parameters A and f_d in the model considered. Cramer-Rao bounds (CRBs) on the accuracy of estimates of A and f_d are derived, and closed-formed expressions are given. Approximated formulas provide quantitative insights into the influence of α and f_d. Additionally, a maximum likelihood estimator (MLE) is presented. Numerical examples illustrate the performance of the MLE and compare it with the CRB. The influence of the SNR, the sample size, the optical parameter α, and the frequency f_d on the estimation performance is emphasized. Finally, an application to real data is presented