Prior Knowledge Optimum Understanding by Means of Oblique Projectors and Their First Order Derivatives

Recently, an optimal Prior-knowledge method for DOA estimation has been proposed. This method solely estimates a subset of DOA's accounting known ones. The global idea is to maximize the orthogonality between an estimated signal subspace and noise subspace by constraining the orthogonal noise-made projector to only project onto the desired unknown signal subspace. As it could be surprising, no deflation process is used for. Understanding how it is made possible needs to derive the variance for the DOA estimates. During the derivation, oblique projection operators and their first order derivatives appear and are needed. Those operators show in consequence how the optimal Prior-knowledge criterion can focus only on DOA's of interest and how the optimality is reached.