The state-covariance of a linear filter imposes a generalized interpolation constraint on the power spectrum of the input. We explore this observation by first characterizing minimal line-spectra which are consistent with given state-covariance data. In particular, we present a canonical decomposition for state-covariances which is analogous to the well-known decomposition of Toeplitz matrices due to Carathkodory-Fejkr and Pisarenko. Accordingly we develop subspace-based signal estimation techniques which apply to state covariances of linear filters and are analogous to MUSIC and ESPRIT. A method analogous to one due to Capon, but based on state-covariance data instead, is also presented. Finally, through a certain duality, subspace methods are shown to be useful in identifying absorption instead of emission lines in the power spectrum of the filter-input.