Periodic signal analysis by maximum likelihood modeling of orbits of nonlinear ODEs

This paper treats a new approach to the problem of periodic signal estimation. The idea is to model the periodic signal as a function of the state of a second-order nonlinear ordinary differential equation (ODE). This is motivated by Poincare theory, which is useful for proving the existence of periodic orbits for second-order ODEs. The functions of the right-hand side of the nonlinear ODE are then parameterized by a multivariate polynomial in the states, where each term is multiplied by an unknown parameter. A maximum likelihood algorithm is developed for estimation of the unknown parameters, from the measured periodic signal. The approach is analyzed by derivation and solution of a system of ODEs that describes the evolution of the Cramer-Rao bound over time. This allows the theoretically achievable accuracy of the proposed method to be assessed in the ideal case where the signals can be exactly described by the imposed model. The proposed methodology reduces the number of estimated unknowns, at least in cases where the actual signal generation resembles that of the imposed model. This in turn is expected to result in an improved accuracy of the estimated parameters.