On Models of Gaussian Reciprocal Processes and the Reconstruction of Periodic Jacobi Matrices

Abstract

We consider algorithms to reconstruct models of scalar Gaussian reciprocal processes from covariance information. These methods exploit the fact that the covariance matrices of these processes have structured inverses either periodic Jacobi or Jacobi matrices. We develop these relationships and show how to pass back and forth between the covariance and its inverse both directly and by way of some knowledge of the eigenstructure. In particular we show how to use eigenpairs to reconstruct the matrices. This approach is markedly di erent from existing algorithms which use all of the eigenvalues of the full matrix and its largest principal submatrix.