On SVD for estimating generalized eigenvalues of singular matrix pencil in noise

Several algorithms for estimating generalized eigenvalues (GEs) of singular matrix pencils perturbed by noise are reviewed. The singular value decomposition (SVD) is explored as the common structure in three basic algorithms: direct matrix pencil algorithm, Pro-ESPRIT, and TLS-ESPRIT. It is shown that several SVD-based steps inherent in those algorithms are equivalent to the first-order approximation. Also, Pro-ESPRIT and TLS-Pro-ESPRIT are shown to be equivalent, and TLS-ESPRIT and 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 threshold phenomenon is illustrated by a simulation result based on a damped sinusoidal signal. An improved state space algorithm is found to be the most robust to noise.<>