Generalized Gerschgorin´s theorem for source number detection

A new family of source number estimators has appeared from the information provided by Gerschgorin radii and the centers of a unitary transformed covariance matrix. We suggest using a generalization of Gerschgorin´s theorem developed for the eigenvalue problem Ax = λBx. This generalization can be applied to the perturbation of multiple eigenvalues and the usual theorem of Gerschgorin appears only as a particular case. For this, we need defining regions that bound a distance called the chordal metric. The techniques of diagonalization based on unitary transformation are necessary to exploit the estimated covariance matrix too. With sinusoidal signals embedded in a colored noise, the used criterion GDE_dist with this generalization shows a better detection rate compared to that obtained by the simple Gerschgorin theorem.