Emad Abd-Elrady

Least squares harmonic signal analysis using periodic orbits of ODEs

Abstract The idea of the paper is to model harmonic signals by means of second order nonlinear ordinary differential equations (ODEs). This is motivated by the well known theoretical results on the existence of periodic orbits for nonlinear second order ODEs. The right hand side functions of the ODE are the estimated quantities, this being accomplished by the use of a polynomial parameterisation. A least squares estimation algorithm is then derived. The methodology reduces the number of parameters needed in cases where the signal generation can be accurately described by the suggested model, thereby enhancing estimation performance.

A nonlinear approach to harmonic signal modeling

Periodic signals can be estimated recursively by exploiting the fact that a sine wave passing through a static nonlinear function generates a spectrum of overtones. A real wave with unknown period in cascade with a piecewise linear function is thefefore used as a parameterization for the estimated signal model. In this paper the driving periodic wave can be chosen depending on any prior knowledge. A recursive Gauss-Newton prediction error identification algorithm for joint estimation of the driving frequency and the parameters of the nonlinear output function is introduced. Local convergence properties as well as the Cramer-Rao bound (CRB) are derived for the suggested algorithm.

Harmonic signal analysis with Kalman filters using periodic orbits of nonlinear ODEs

The paper suggests a new approach to the problem of harmonic signal estimation. The idea is to model the harmonic signal as a function of the state of a second order nonlinear ordinary differential equation (ODE). The function of the right hand side of the nonlinear ODE is parameterized with a polynomial model. A Kalman filter and an extended Kalman filter are then developed. The proposed methodology reduces the number of estimated unknowns in cases where the actual signal generation resembles that of the imposed model. This is expected to result in an improved accuracy of the estimated parameters, as compared to existing methods.

Periodic signal modeling based on Lie/spl acute/nard's equation

The problem of modeling periodic signals is considered. The approach taken here is motivated by the well known theoretical results on the existence of periodic orbits for Lie/spl acute/nard systems and previous results on modeling periodic signals by means of second-order nonlinear ordinary differential equations. The approach makes use of the appropriate conditions imposed on the polynomials of the Lie/spl acute/nards system to guarantee the existence of a unique and stable limit cycle. These conditions reduce the number of parameters required to generate accurate models.

Least-squares periodic signal modeling using orbits of nonlinear ODEs and fully automated spectral analysis

Periodic signals can be modeled by means of second-order nonlinear ordinary differential equations (ODEs). The right-hand side function of the ODE is parameterized in terms of known basis functions. The least-squares algorithm developed for estimating the coefficients of these basis functions gives biased estimates, especially at low signal-to-noise ratios. This is due to noise contributions to the periodic signal and its derivatives evaluated using finite difference approximations. In this paper a fully automated spectral analysis (ASA) technique is used to eliminate these noise contributions. A simulation study shows that using the ASA technique significantly improves the performance of the least-squares estimator.

Periodic signal analysis using orbits of nonlinear ODE's based on the Markov estimate*

Abstract Periodic signals can be modeled by means of second order nonlinear ordinary differential equations (ODEs). This approach is motivated by the well known theoretical results on the existence of periodic orbits for nonlinear second order ODEs. The right hand side functions of the ODE are parameterized in terms of known basis functions. A Markov estimation algorithm is then derived. A simulation study shows that the Markov estimator has significantly better performance than the least squares estimator.

AN ADAPTIVE GRID POINT RPEM ALGORITHM FOR HARMONIC SIGNAL MODELING

Abstract Periodic signals can be modeled as a real wave with unknown period in cascade with a piecewise linear function. A recursive Gauss-Newton prediction error method (RPEM) for joint estimation of the driving frequency and the parameters of the nonlinear output function parameterized in a number of adaptively estimated grid points is introduced. The Cramer-Rao bound (CRB) is derived for the suggested algorithm. Numerical examples indicate that the suggested algorithm gives better performance than using fixed grid point algorithms.

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

This report 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, and a maximum likelihood algorithm is developed for estimation of the parameters of these unknown functions 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. 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.

Least squares periodic signal modeling using orbits of nonlinear ode’s and fully automated spectral analysis

Abstract Periodic signals can be modeled by means of second order nonlinear ordinary differential equations (ODEs). The right hand side function of the ODE is parameterized in terms of known basis functions. The least squares algorithm developed for estimating the coefficients of these basis functions gives biased estimates, especially at low signal to noise ratios. This is due to noise contributions to the periodic signal and its derivatives evaluated using finite difference approximations. In this paper a fully automated spectral analysis (ASA) technique is used to eliminate these noise contributions. A simulation study shows that using the ASA technique significantly improves the performance of the least squares estimator.

BIAS ANALYSIS IN PERIODIC SIGNALS MODELING USING NONLINEAR ODE'S

Abstract Second-order nonlinear ordinary differential equations (ODEs) can be used for modeling periodic signals. The right hand side function of the ODE model is parameterized in terms of polynomial basis functions. The least squares (LS) algorithm for estimating the coefficients of the polynomial basis gives biased estimates at low signal to noise ratios (SNRs). This is due to approximating the states of the ODE model using finite difference approximations from the noisy measurements. An analysis for this bias is given in this paper.