SPECTRAL FACTORIZATION OF LINEAR PERIODIC-SYSTEMS WITH APPLICATION TO THE OPTIMAL PREDICTION OF PERIODIC ARMA MODELS

A cyclostationary process is a stochastic process whose statistical parameters, such as mean and autocorrelation, exhibit suitable periodicity. In this paper, we consider the cyclospectral factorization problem which consists of finding a Markovian (state-space) realization of a given cyclostationary process. It is shown that a significant class of periodic state-space representations is in a one-to-one correspondence with the periodic solutions of a difference periodic Riccati equation. This result is applied to the solution of the prediction problem for ARMA models with periodically varying coefficients. If the periodic ARMA model is minimum-phase, the optimal predictor is given a simple input-output expression that generalizes the well-known one for time-invariant ARMAs. Otherwise, the computation of the optimal predictor calls for the solution of a cyclospectral factorization problem.