Locally Optimal Invariant Detector for Testing Equality of Two Power Spectral Densities

This work addresses the problem of determining whether two multivariate random time series have the same power spectral density (PSD), which has applications, for instance, in physical-layer security and cognitive radio. Remarkably, existing detectors for this problem do not usually provide any kind of optimality. Thus, we study here the existence under the Gaussian assumption of optimal invariant detectors for this problem, proving that the uniformly most powerful invariant test (UMPIT) does not exist. Thus, focusing on close hypotheses, we show that the locally most powerful invariant test (LMPIT) only exists for univariate time series. In the multivariate case, we prove that the LMPIT does not exist. However, this proof suggests two LMPIT-inspired detectors, one of which outperforms previously proposed approaches, as computer simulations show.