A state-space approach to positive sequences

The problem considered in this paper is the following: given a symmetric sequence{r_{j}}which is not positive, (i.e. its Fourier transform takes on negative values), find a positive sequence{\bar{r}_{j}}which is a good approximation to{r_{j}}. This problem arises, for example, when{r_{j}}is an estimated covariance sequence which is not positive, and a positive covariance sequence is desired. A solution to the positivity problem is given which uses state-space models and a scaled algebraic Riccati equation.