Multidimensional Nonlinear Schur Parametrization of NonGaussian Stochastic Signals, Part Two: Generalized Schur Algorithm

In the first part [1] of this paper we stated the multidimensional nonlinear Schur parametrization problem for higher-order (and non-Gaussian) stochastic sequences as a generalization of the linear prediction/innovations filter problem for second-order stochastic sequences. The subject of this paper is a multidimensional nonlinear generalization of the linear Schur parametrization procedure for second-order stochastic sequences. We state and solve the problem geometrically, in the space of generalized (multi-indexed) matrices. We derive multidimensional nonlinear Levinson and Schur recursions, resulting in a complete parametrization schema for higher-order nonstationary stochastic sequences and show its connection to solution the non-linear orthogonal approximate filter problem of the Volterra–Wiener class. In the third part [2] we propose a low-complexity nonlinear Schur parametrization procedure, following from a multidimensional nonlinear generalization of the staircase matrix extension problem.

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