Estimation of the number of frequencies and frequencies of continuous-time sinusoidal signals: a nonlinear Luenberger observer approach

This paper addresses the online estimation of the number of frequencies and the frequency values for continuous-time multi-sinusoidal signals. Existing adaptive observer methods are limited by stringent prior requirements, such as full knowledge of the number of frequencies or a predefined upper bound on unknown frequencies. We extend the nonlinear Luenberger observer approach to overcome these restrictions. First, a nonlinear Luenberger observer is developed to identify the number of frequencies. With only a mild prior assumption on an upper bound on the number of frequencies, the proposed scheme precisely recovers the true number of frequencies. Second, based on the obtained number of frequencies, another nonlinear Luenberger observer is constructed to achieve accurate frequency estimation. Compared with existing adaptive observers, the proposed approaches possess a simpler linear filter structure. Numerical simulations on both noise-free and noisy signals validate the convergence of the developed estimators.