Effect of unitary transformation on Bayesian information criterion for source numbering in array processing

Abstract

An approach based on unitary transformation for the problem of estimating the number of signals is proposed in this study. Among the information theoretic criteria, the authors focus on the conventional Bayesian information criterion (BIC) in the presence of a uniform linear array. The sample covariance matrix of this array is transformed into the real symmetric one by using a unitary transformation. This real symmetric matrix has real eigenvalues and eigenvectors. Therefore its eigenvalue decomposition needs only real computations. Since the eigenvalues of this real symmetric matrix are equal to the eigenvalues of the sample covariance matrix, by replacing them in BIC formula, the term log-likelihood of BIC does not change but it is obtained by fewer computations. Also by considering the resulting real eigenvectors instead of the complex eigenvectors as a part of free parameters in the parameter vector of the model, they have a reduction in the number of degrees of freedom in the penalty term of BIC. This reduction makes their proposed method outperform BIC. They refer to this approach as unitary BIC. A series of simulations are included to demonstrate the usefulness of this approach.