Consistency of subspace methods for signals with almost-periodic components

It is sometimes claimed in the literature that subspace methods provide consistent estimates, also when the underlying observed signal has purely oscillatory modes (or the generating system has uncontrollable eigenvalues on the unit circle) but a formal proof of this assertion does not seem to exist. In this paper, we prove consistency of subspace methods with purely oscillatory modes. A well-known subspace identification procedure based on canonical correlation analysis and approximate partial realization is shown to be consistent under certain conditions on the purely deterministic part of the generating system. The algorithm uses a fixed finite regression horizon and the proof of consistency does not require that the regression horizon goes to infinity at a certain rate with the sample size NN.