Robust approaches to remote calibration of a transmitting array

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 1⁄4 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 1⁄4 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.