FAST ALGORITHMS FOR CLOSE-TO-TOEPLITZ-PLUS-HANKEL SYSTEMS AND 2-SIDED LINEAR PREDICTION

We extend the low-displacement-rank definition of close-to-Toeplitz (CT) matrices to close-to-Toeplitz-plus-Hankel (CTPH) matrices, and develop new fast algorithms for solving CTPH systems of equations. A matrix is defined as CTPH if it is the sum of a CT matrix and a second CT matrix post-multiplied by an exchange matrix; an equivalent definition in terms of UV rank is also given. This definition is motivated by our application of the new algorithms to two-sided linear prediction (TSP), which differs from one-sided linear prediction (OSP) in that both past and future time series values are used in a symmetric manner to estimate the present value. We define autocorrelation and covariance forms of TSP analogous to those for OSP; the covariance form of TSP is solved using the new CTPH fast algorithms, just as the covariance form of OSP is solved using CT fast algorithms. Numerical examples show that: 1) TSP produces smaller residuals than OSP (we prove this); 2) TSP resolves sharp spectral peaks better than OSP; and 3) covariance TSP produces smaller residuals than autocorrelation TSP.