Invariant Two-Parameter Family of Circulant Sparse Matrices and Its Application in Correlation Signal Theory

A new algebraic approach to approximation of correlation matrices of discrete signals based on a special family of square matrices with sparse circulant alternation of elements is developed. It is shown that the matrices of this family form a linear subspace that is invariant to a certain two-parameter system of shifts. An orthogonal matrix basis of this subspace is found and analytical expressions for the eigenvectors and eigenvalues of the matrices included in it are obtained. It is shown that the correlation matrices of discrete signals are well approximated by matrices from such an invariant family, and expressions are obtained for their optimal approximation by the criterion of minimum mean square error. It is shown that the obtained results make it possible to significantly simplify the computational implementation of algorithms for optimal correlation signal processing in practice without significant losses in their optimality.