From Finite Covariance Windows to Modeling Filters: A Convex Optimization Approach

Thetrigonom5KO1 mrig tproblem is a classicalmass tproblem with numMO8S applications inm9M86VC5KS9 physics, and engineering. The rational covariance extensionproblem is a constrained version of thisproblem with the constraints arisingfrom the physical realizability of the corresponding solutions. Although themeCM um entropym ethod gives one well-known solution, in several applications a wider class of solutions is desired. In a semMVV paper, Georgiou derived an existence result for a broad class of m dels. In this paper, we review the history of thisproblem going back to Caratheodory, as well as applications to stochasticsystem and signal processing. In particular, we present a convex optimC5M87M problem for solving the rational covariance extensionproblem with degree constraint. Given a partial covariance sequence and the desired zeros of the shaping filter, the poles are uniquelydetermC98 from the uniquemiqu um of the corresponding optimi -CK6M problem In this way we obtain analgorithm for solving the covariance extension problem as well as a constructive proof of Georgiou's existence result and his conjecture, a generalized version of which we have recently resolved usinggeomCM6K mom ds. We also survey recent related results on constrained Nevanlinna--Pick interpolation in the context of a variationalform ulation of the generalmner tproblem Key words. rational covariance extension, interpolation, partial stochastic realization, trigonomon,C mrig tproblem spectralestim699SV speech processing, stochasticm deling, general mner tproblem AMS subject classifications. 30E05, 42A15, 49N15, 60G35, 62M15, 65K10, 93A30, 93E12 PII. S0036144501392194 1.