Robust approaches to remote calibration of a transmitting array

Abstract

We consider the problem of estimating the gains and phases of the RF channels of a M-element transmitting array, based on a calibration procedure where M orthogonal signals are sent through M orthogonal beams and received on a single antenna. The received data vector obeys a linear model of the type y ¼ AFg þ n where A is an unknown complex scalar accounting for propagation loss and g is the vector of unknown complex gains. In order to improve the performance of the least-squares (LS) estimator at low signal to noise ratio (SNR), we propose to exploit knowledge of the nominal value of g, viz g. Towards this end, two approaches are presented. First, a Bayesian approach is advocated where A and g are considered as random variables, with a non-informative prior distribution for A and a Gaussian prior distribution for g. The posterior distributions of the unknown random variables are derived and a Gibbs sampling strategy is presented that enables one to generate samples distributed according to these posterior distributions, leading to the minimum mean-square error (MMSE) estimator. A second approach consists in solving a constrained least-squares problem in which h ¼ Ag is constrained to be close to a scaled version of g. This second approach yields a closed-form solution, which amounts to a linear combination of g and the LS estimator. Numerical simulations show that the two new estimators significantly outperform the conventional LS estimator, especially at low SNR.