Cramer-Rao bounds for estimating range, velocity, and direction with a sensor array

We derive Cramer-Rao bound (CRB) expressions for the range (time delay), velocity (Doppler shift), and direction of a point target using an active radar or sonar array. The CRB expressions are derived for an arbitrary signal waveform and a noise model that allows both spatial and temporal correlation. We discuss the relationship between the CRB and ambiguity function for this model. Then, we simplify the CRB for the cases of temporally or spatially uncorrelated noise. We compute the CRB for a 3-dimensional array with isotropic sensors in spatially uncorrelated noise and show that it is a function of the array geometry only through the "moments of inertia" of the array. The volume of the confidence region for the target's location is proposed as a measure of accuracy. For this measure, we show that the highest (and lowest) target location accuracy is achieved if the target lies along one of the principal axes of inertia of the array.