A state-space approach for obtaining spectral models from nonpositive covariance models

The problem considered in this paper is the following: given a state-space model for a symmetric sequence {rj} which is not positive, (i.e. its Fourier transform takes on negative values}, find a model for a positive sequence {r- j} which gives a good approximation to {rj}. The positive covariance model can then be used to define a spectrum, if desired. This problem arises, for example, when the original covariance model comes from an estimated covariance sequence which is not positive. A solution to the positivity problem is given which uses state-space models and a scaled algebraic Riccati equation. The procedure leaves the poles of the original model and the value of r0 unchanged. A simulation example is given to compare the proposed method with a different approach based on an ARMA parameterization of the spectrum. In this example, the squared error between the given sequence and the sequence obtained by the proposed method is within 5% of the optimal value.