Linear statistical models for stationary sequences and related algorithms for Cholesky factorization of Toeplitz matrices

We review the Levinson-Durbin recursions for computing the Cholesky factors of the inverse of a Toeplitz correlation matrix R, and the Schur recursions for computing the Cholesky factors of R itself. We show that both algorithms may be imbedded in a single vector lattice recursion. Then, depending only on how the initial conditions are set, one gets either the Levinson-Durbin or the LeRoux-Gueguen recursions for computing reflection coefficients. We extend the analysis to treat the Toeplitz nonsymmetric case, and place the corresponding algorithms into the same framework. We also review the relation between the vector lattice recursions and their implementations in a serial lattice structure. Then we show how the factorizations may be run backwards to compute the Cholesky factors and correlations from the reflection coefficients. This generalizes a result usually attributed to Robinson and Treitel. One of our main purposes is to emphasize that the Levinson-Durbin recursions for going back-and-forth between correlations, reflection coefficients, and autoregressive filter parameters may be replaced with a dual set of recursions for going back-and-forth between correlations, reflection coefficients, and moving average filter parameters. When the correlation sequence is ARMA (p, p), then we show how the Schur recursions simplify to the Morf-Sidhu-Kailath algorithm for computing Kalman gains. Finally, the MSK algorithm may be specialized to the Rissanen algorithm for factoring a pure MA correlation sequence.