Cramer-rao lower bounds for the estimation of an incident signal's direction-of-arrival upon rectangular/circular/spherical arrays of isotropic sensors

Abstract

Direction-of-arrival (DOA) estimation accuracy relies on both the estimation algorithm and the sensor array geometry. The sensor array geometry has several parameters, such as the position of sensors and the spatial aperture that could significantly affect the direction-of-arrival estimation accuracy. In order to improve the DOA estimation accuracy, it is therefore desirable to optimize the sensor array geometry. However, in practice, manufacturing and field deployment faults could distort the sensor array geometry. For instance, the sensors could be perturbed from their nominal positions. This thesis attempts to devise rules/ways on how to optimize the direction-of-arrival estimation accuracy by considering different sensor array geometries, and investigates the effect of sensor position perturbations on the DOA estimation accuracy. In particular, rectangular, spherical, and circular arrays are considered. A common approach to assess the accuracy of parameter estimation methods is to derive a theoretical lower bound and compare it to the variance of the devised estimators. The common lower bound is the classical or hybrid (in the case of both random and deterministic parameters) Cram´er-Rao bound (CRB), which lower bounds the variance of any statistically unbiased parameter. Theoretically, the closer the estimation variance is to the lower bound, the better the performance of the parameter estimation method. Therefore, in order to investigate the DOA estimation accuracy obtainable from the considered different sensor array geometries, this thesis analytically derives the direction finding Cram´er-Rao bound for the three array geometries, and provides qualitative insights into how the array design parameters affect the DOA estimation accuracy. Two configurations of the uniform rectangular array (URA) are considered. The first configuration has its sensors placed only on the rectangle's perimeter (uniform rectangular "rim" array), whereas the second configuration places its sensors on the rectangle's perimeter and interior (uniform rectangular "filled" array), as well. The above two URA configurations are compared and experiments found that the "rim" URA outperforms the "filled" URA in terms of the direction-of-arrival estimation accuracy. For sensor position perturbations' effects on the DOA estimation accuracy, the thesis considers a uniform circular array, whose constituent sensors are allowed to be perturbed three-dimensionally in space. Suprisingly, this investigation, discovers that the sensor position perturbations not only degrade the DOA estimation accuracy via the nuisance effects on Fisher information, but could also improve the DOA estimation accuracy via the resulting enlarged array spatial aperture. Lastly, the thesis determines the DOA estimation accuracy obtainable from an open uniform spherical array. This study develops rules-of-thumb that provide insights into how a uniform spherical array's direction-of-arrival estimation accuracy depends on the total number of sensors. In all the mentioned investigations, the analytically derived direction finding Cramer-Rao lower bounds are validated by maximum a posteriori/maximum likelihood estimation Monte Carlo simulations.