Cramer-Rao bounds for estimating range, velocity, and direction with an active array

We derive Cramer-Rao bound (CRB) expressions for the range (time delay), ve locity (Doppler shift), and direction of a point target using an active rad ar or sonar array, First, general CRB expressions are derived for a narrowb and signal and array model and a space-time separable noise model that allo ws both spatial and temporal correlation. We discuss the relationship betwe en the CRB and ambiguity function for this model, Then, we specialize our C RB results to the case of temporally white noise and the practically import ant signal shape of a linear frequency modulated (chirp) pulse sequence. We compute the CRB for a three-dimensional (3-D) array with isotropic sensors in spatially white noise and show that it is a function of the array geome try only through the "moments of inertia" of the array. The volume of the c onfidence region for the target's location is proposed as a measure of accu racy, For this measure, we show that the highest (and lowest) target locati on accuracy is achieved if the target lies along one of the principal axes of inertia of the array, Finally, we compare the location accuracies of sev eral array geometries.