Multiple Constrained $\ell_2$ -Norm Minimization Algorithm for Adaptive Beamforming

This paper presents an effective adaptive beamforming method based on an <inline-formula> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula>-norm minimization problem with multiple constraints, named as MC-<inline-formula> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula>-M algorithm. In the proposed method, the output power minimization problem can be formulated as an <inline-formula> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula>-norm minimization problem of the weight vector, with multiple constraints including distortionless response constraint and zero-forcing response constraints toward the desired signal and the interference signals, respectively. Furthermore, via an interference subspace estimation, the presented <inline-formula> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula>-norm minimization problem can be rewritten as an equivalent constrained convex optimization problem of the weight vector, which is readily to be solved. Finally, the closed-form solution of the proposed optimization problem is given. Compared with the previous works, the proposed MC-<inline-formula> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula>-M approach provides better beamforming performance, for example: 1) it reduces the sensitivity to the variation of signal-to-noise ratio (SNR) so that it has better output array gain, even if in the large dynamic SNR changing circumstances; 2) the better sidelobe and nulling levels are maintained, since it weakens the component of the desired signal in the covariance matrix of the array snapshot vector by a new defined blocking matrix; and 3) it can not only effectively suppress the interference signals but also ensure that the response of the desired signal is distortionless. Simulation results demonstrate the efficiency of the presented approach.