Weighted Regression with Application to Array Antennas

Doctoral thesis, 2007

A nonlinear system can be modelled with a simple linear model if the model is only valid locally. This can be done by assigning weights to the estimation data, as a function of the distance to the modelled point. The weighting is here used to develop a direction-dependent calibration method for array antennas. The aim of the calibration method is to be able to handle imperfection, such as position errors, which give rise to errors which is dependent of the direction of the incoming signal. The calibration method is non-parametric which makes the calibration method general, i.e. not dependent on knowledge about the nature of the imperfections in the antenna. The direction-dependence is achieved by including a weight matrix which is a function of the angle of the incoming signal and the direction of the calibration measurements. To improve the performance the model for the ideal steering vector is included as nominal model. The calibration method is shown to improve the performance of direction of arrival estimation methods, and the impact of the density of the calibration grid and the magnitude of the position errors are investigated. Most results are based on simulated data, but also data from a laboratory ultra-sound array, known to have large direction-dependent errors, is included. The impact of the calibration method on the statistical properties of the estimate is also investigated. The Cramer-Rao lower bound is derived for a data model including direction dependent noise in the calibration data to model the loss of information due to the direction-dependent errors in the array. This result is compared to one from a more conventional data model including known errors. The maximum likelihood estimates for the data models are also derived, and a way to choose the weighting matrix by an approximative maximum likelihood result. It is also shown how the calibration method can be incorporated in a convex optimization algorithm to achieve a low sidelobe level for a receiving antenna using classical beamforming. Lastly, the calibration method is generalized by showing how a nominal model can be introduced in a weighted regression, and how an angular distance measure can be used instead of the conventional Euclidian norm.