The Euclid algorithm and the fast computation of cross-covariance and autocovariance sequences

A simple linear procedure is given to compute the cross-covariance sequence associated with the outputs of two rational digital transfer functions driven by the same white noise sequence. Such a computation often appears in the study of digital filters, in Wiener filtering, in noise variance estimation, in the study of low-order approximations, and in the study of multichannel systems. A fast algorithm based on the Euclid algorithm is introduced to solve the linear system of equations involved in the computation, and a detailed analysis of the matrix is given. The special case of the autocovariance computation is reviewed, and the same study is performed. Alternate polynomial presentations are given and are shown to involve the same matrices and similar fast algorithms.<>