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 noise—this is the Carathéodory-Fejér-Pisarenko (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). We characterize state-covariances which admit only a deterministic input power spectrum. We 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, and develop formulae for spectra corresponding to singular covariances via decomposing the contribution of the singular part. We consider replacing the “scalar multiple of the identity” in the CFP decomposition 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 (e.g., noise) with the aforementioned structure at the input, and the optimal 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. Index Terms Multivariable time-series, spectral analysis, spectral estimation, central solution, Pisarenko harmonic decomposition, short-range correlation structure, moving average noise, convex optimization.