On SVD for Estimating Generalized Eigenvalues of Singular Matrix Pencil in Noise

We review several algorithms for estimating generalized eigenvalues (GE's) of singular matrix pencils perturbed by noise. The singular value decomposition (SVD) is explored as the common structure in the three basic algorithms: direct matrix pencil algorithm, Pro-ESPRIT and TLS-ESPRIT. We show that several SVD-based steps inherent in those algorithms are equivalent to the first-order approximation. In particular, the Pro-ESPRIT and its variant TLS-Pro-ESPRIT are shown to be equivalent, and the TLS-ESPRIT and its earlier version LS-ESPRIT are shown to be asymptotically equivalent to the first-order approximation. For the problem of estimating superimposed complex exponential signals, the state space algorithm is shown to be also equivalent to the previous matrix pencil algorithms to the first-order approximation. The second-order perturbation and the threshold phenomenon are illustrated by simulation results based on a damped sinusoidal signal. An improved state space algorithm is found to be the most robust to noise.