Weighted Regression with Application to Array Antennas
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.
statistical estimation theory