The Carathéodory-Fejér-Pisarenko decomposition and its multivariable counterpart

When a covariance matrix with a Toeplitz structure is written as the sum of a singular one and a positive scalar multiple of the identity, the singular summand corresponds to the covariance of a purely deterministic component of a time- series whereas the identity corresponds to white noisethis is the CarathodoryFejrPisarenko (CFP) decomposition. In the present paper we study multivariable analogs for block-Toeplitz matrices as well as for matrices with the structure of state- covariances of finite-dimensional linear systems (which include block-Toeplitz ones). To this end, we develop theory which addresses questions of existence, uniqueness and realization of multivariable power spectra, possibly having deterministic components. We characterize state-covariances which admit only a deterministic input power spectrum, and we explain how to realize multivariable power spectra which are consistent with singular state covariances via decomposing the contribution of the singular part. We then show that multivariable decomposition of a state-covariance in accordance with a deterministic component $+$ white noise hypothesis for the input does not exist in general. We finally reinterpret the CFP-dictum and consider replacing the scalar multiple of the identity by a covariance of maximal trace which is admissible as a summand. The summand can be either (block-)diagonal corresponding to white noise or have a short- range correlation structure correponding to a moving average component. The trace represents the maximal variance/ energy that can be accounted for by a process at the input (e.g., noise) with the aforementioned structure, and this maximal solution can be computed via convex optimization. The decomposition of covariances and spectra according to the range of their time-domain correlations is an alternative to the CFP-dictum with potentially great practical significance.