Multidimensional ESPRIT for Damped and Undamped Signals: Algorithm, Computations, and Perturbation Analysis

In this paper, we present and analyze the performance of multidimensional ESPRIT (<inline-formula> <tex-math notation="LaTeX">$N$</tex-math></inline-formula>-D ESPRIT) method for estimating parameters of <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>-D superimposed damped and/or undamped exponentials. <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>-D ESPRIT algorithm is based on low-rank decomposition of multilevel Hankel matrices formed by the <inline-formula><tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-D data. In order to reduce the computational complexity for large signals, we propose a fast <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>-D ESPRIT using truncated singular value decomposition (SVD). Then, through a first-order perturbation analysis, we derive simple expressions of the variance of the estimates in <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>-D multiple-tones case. These expressions do not involve the factors of the SVD. We also derive closed-form expressions of the variances of the complex modes, frequencies, and damping factors estimates in the <inline-formula> <tex-math notation="LaTeX">$N$</tex-math></inline-formula>-D single-tone case. Computer results are presented to show effectiveness of the fast version of <inline-formula><tex-math notation="LaTeX">$N$</tex-math></inline-formula>-D ESPRIT and verify theoretical expressions.