Symmetric solutions and eigenvalue problems of Toeplitz systems

Algorithms and properties of symmetric solutions of a Toeplitz system are studied. It is shown that the numbers of positive and negative eigenvalues associated with symmetric (antisymmetric) eigenvectors are the same as the numbers of positive and negative predictor errors of symmetric (antisymmetric) filters. Based on the odd symmetric solutions and the property that all roots of symmetric and antisymmetric filters are on the unite circle, a method for Pisarenko´s decomposition is introduced. Compared with some other methods, it reduces the number of iterations and the computational cost in each iteration considerably