Widely-linear beamforming

In this paper we describe a beamforming algorithm based on widely-linear rather than linear data models. Initially, we develop this beamformer by generalizing the Capon (MVDR) optimization problem. That is, if the objective is to minimize output power while maintaining a specified directional gain, then we show that the output power of the widely-linear beamformer is less than or equal to the output power of the Capon (MVDR) beamformer. This result is valid regardless of the "true" distribution of the data. We also derive the widely-linear beamformer by considering beamforming to be an estimation problem. Linear models assume that the composite covariance matrix formed from the real and imaginary parts of the array-snapshot has a particular structure. This structure is often summarized by stating that the covariance formed from the array snapshot and its transpose (not Hermitian transpose) is zero. We could also call these data "proper" Gaussian vectors. The beamformers in this paper are appropriate for situations in which these implicit assumptions are violated.