An adaptive antenna array for broad-band signals using the constrained Kalman filtering

In order to overcome the slow convergence rate, in our previous paper (see IEEE Trans. Antennas and Propagation, p.1576, Nov. 1993) we developed a constrained Kalman algorithm for narrow-band beamforming by adding a constraint on the array response along the look direction to the measurement equation of the Kalman algorithm. We extend the constrained Kalman algorithm to broadband adaptive beamforming under linear constraints and show that the estimated weight vector of the proposed biased constrained Kalman algorithm may converge to a optimum solution with a small bias. To annihilate the bias, an unbiased constrained Kalman algorithm is presented. The error-correcting property of the constrained LMS algorithm is applied to separate the weight vector into two parts. One is a fixed part which is normal to and terminates on the constraint plane. The other is an adjustable part which is premultiplied by a projection matrix and lies in the constraint subspace. Hence the measurement and process equations of the unbiased constrained Kalman algorithm are different from those of the conventional Kalman algorithm. The iterative equations for adapting the estimated weight vector of the unbiased constrained Kalman algorithm are newly derived. Moreover, we show that the array system with the proposed algorithm can converge to an optimal broadband beamformer with probability 1 convergence.