Data-domain penalty function algorithm for stabilised adaptive beamforming

A data-domain formulation of the penalty function method is derived. It is based on the application of QR decomposition by Givens rotations. This formulation involves a Cholesky square-root decomposition of the integrated penalty function matrix which is, theoretically, nonnegative definite. However, this important condition may be violated in practice owing to numerical inaccuracy in which case the Cholesky factorisation is bound to fail. It is shown how this problem can be avoided by defining the penalty function in terms of a finite number of soft constraints in the data domain. The resulting algorithm is compared with the original penalty function method and found to give comparable results in the context of full precision arithmetic