The effects of phase on high-resolution frequency estimators

Several different ways in which the initial phase differences between two closely spaced sinusoids manifests itself, e.g., in the singular values of the data matrix, the angles between signal vectors, and the Cramer-Rao bound, are described. A closed-form expression for the singular values of the data matrices used in subspace-based algorithms and an expression for the angle between the signal components are presented. Based on these expressions, conjectures are made regarding the effect of the relative phase difference between signal components on the resolving ability of subspace-based techniques. The behavior of the Cramer-Rao bound for the two-complex-sinusoid case is examined as a function of phase difference. Predictions are made as to the effect of this behavior on expected algorithm performance as a function of the phase difference. Empirical results that validate the prediction are provided