Performance of Bearing Estimators Subject to Nonideal Inputs
R. Richter, R. D. Schalow · The Journal of the Acoustical Society of America · 1970
The practical case of computing the bearing θ to an object given N noisy measurements of the orthogonal components of the direction vector (X, Y) by using either θ̂ = arctan (〈Y/X〉) (1) or θ̂ = 〈arctan〉 (Y/X) (2) has been previously studied and reported for ideal descriptions of the noise associated with X and Y; namely, equal variance (σx2 = σy2) and zero correlation (ρxy = 0). Further analysis removes these restrictions. It is shown that both methods are asymptotically efficient but only Eq. 1 remains asymptotically unbiased for ρxy≠0 (in fact, Eq. 1 is a maximum likelihood estimator). For finite N, nonideal inputs cause bias with either method. Performance curves relating bias and estimation variance to signal-to-noise ratio, σx/σy, ρxy, N, and θ are developed which show that, as before, Eq. 1 is the preferable estimator in virtually all practical situations.