On-line detection/estimation scheme in laser Doppler anemometry

Laser anemometers have become a promising technique for estimating velocities in a flow. In this paper, we study their use for onboard aircraft speed of flight estimation. More specifically, this paper addresses the problem of simultaneous detection of the arrival of aerosol particles in a laser anemometer and estimation of their velocity. A joint detection-estimation scheme is proposed. A Likelihood Ratio Test is presented and considerations about the specificities of the problem are used to calculate the threshold. Computationally efficient algorithms for estimating the parameters of in; terest are derived and on-line implementation issues are addressed. Numerical examples attest for the performance of the method, on both simulated and real data recorded during a flight test. 1. PROBLEM FORMULATION We consider the problem of estimating aircraft’s speed from an onboard laser anemometer. This system is basedon two coherent laser beams which are crossed and focused in the vicinity of the aircraft. Hence, a symmetric interference fringe pattern composed of bright and dark fringes is generated. When a particle of aerosol, with speed V, crosses the successivedark and bright fringes, it will scatter and not scatter light according to its velocity. It can be shown the signal recorded by a photodetector consists of a sinusoidal signal (whose frequency is representative of particle’s velocity, hence of aircraft’s speed) with a Gaussian shaped time-varying amplitude. In previous papers [ 1.21, we addressed the problem of estimating the frequency of such a signal. Here, we are concerned with the detection of arrival of an aerosol. As a matter of fact, since the flow is not continuous, particle appears randomly on the sides of the fringes pattern and then crosses the probe volume. A key point, prior to or in conjunction with estimation, is to detect whether a particle is currently present. Since we are looking for on-line detection, the problem can be adequately formulated as deciding, from a sliding window of N = PT + 1 data points, between the two following hypotheses: Ho : z(t) = b(t) or H, : s(t) = s (t tc) + b(t) where the parameter t, E ]-3T, 2T[ is representative of the fact that a particle has begun to appear in the probe volume and contributes to the recorded signal. The case t, = 0 corresponds to a particle at the center of the probe volume. b(t) is assumed to be a This work is supported by Sextant Avionique under contract number 1432411944164 sequence of independent and identically Gaussian distributed random variables N (0: cg). s(t) denotes the useful signal and is given by the following theoretical expression s(t) = Aexp {-2a2fit2 +j~dt} (1) where A denotes the amplitude of the signal, (rrd 4 2rfd is related to the particle’s velocity and a is an optical parameter of the system. Thus, the interesting information is the flow velocity V directly related to the frequency parameter fd of s(t) in (1). Once HI is assumed, an estimate of the Doppler frequency fd should be made available. However, [ I] showed that the Cramer-Rao Bound on the estimated signal’s parameter fd is minimum for t, = 0. Since the CRB does not significantly vary around t, = 0, a frequency estimate will be considered as valid when both HI is true and t, = 0 is small, classically -10 5 t, 5 10. In the sequel, we briefly present anon-line joint detection/estimation scheme based on Neyman-Pearson test. The reader is referred to [3] for detailed derivations that could be skipped here. 2. LIKELIHOOD RATIO TEST We begin with a Likelihood Ratio Test to tackle the detection problem. From the assumption made on b(t) N N (0, ui), one could prove that the likelihood ratio test can be written in the form [3,4,5] A(x) = P(X I-1 = P(xIHo) orl(x) = h (A(x)) = 11412 11;; A.412 I$ y (3) HO These forms suggest that we resort to a Neyman-Pearson test [4,5] using the probability of false alarm PFA and probability of detection PD to fix the threshold ‘1 of the test. From (2) and (3). it directly ensues that