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.

[1]  Petre Stoica,et al.  Decentralized Control , 2018, The Control Systems Handbook.

[2]  Emad Abd-Elrady.  A nonlinear approach to harmonic signal modeling , 2004, Signal Process..

[3]  T. Söderström,et al.  A second order ODE is sufficient for modelling of many periodic signals , 2005 .

[4]  B. E. Ydstie,et al.  System identification using modulating functions and fast fourier transforms , 1990 .

[5]  Torsten Söderström,et al.  Second order ODEs are sufficient for modeling of many periodic signals , 2003 .

[6]  Petre Stoica,et al.  Introduction to spectral analysis , 1997 .

[7]  Peter Händel,et al.  Recursive estimation of frequencies and frequency rates of multiple cisoids in noise , 1997, Signal Process..

[8]  E. Abd-Elrady.  AN ADAPTIVE GRID POINT RPEM ALGORITHM FOR HARMONIC SIGNAL MODELING , 2002 .

[9]  Johan Schoukens,et al.  Least-squares periodic signal modeling using orbits of nonlinear ODEs and fully automated spectral analysis , 2005, Autom..

[10]  M. Farza,et al.  Simple nonlinear observers for on-line estimation of kinetic rates in bioreactors , 1998, Autom..

[11]  Petre Stoica,et al.  Statistical analysis of two nonlinear least-squares estimators of sine-wave parameters in the colored-noise case , 1989 .

[12]  B. Hofmann-Wellenhof,et al.  Introduction to spectral analysis , 1986 .

[13]  Masahiro Kaneda.  System Identification Using Fast Fourier Transform , 1980 .

[14]  Austin Blaquière,et al.  Nonlinear System Analysis , 1966 .

[15]  Torsten Söderström,et al.  Periodic signal modeling based on Lie/spl acute/nard's equation , 2004, IEEE Transactions on Automatic Control.

[16]  R. Kumaresan,et al.  Estimating the parameters of exponentially damped sinusoids and pole-zero modeling in noise , 1982 .

[17]  Sridhar Ungarala,et al.  Batch scheme recursive parameter estimation of continuous-time systems using the modulating functions method , 1997, Autom..

[18]  Peter Händel,et al.  Adaptive estimation for periodic signal enhancement and tracking , 1994 .

[19]  Emad Abd-Elrady,et al.  Least squares periodic signal modeling using orbits of nonlinear ode’s and fully automated spectral analysis* , 2004 .

[20]  Robert Boorstyn,et al.  Single tone parameter estimation from discrete-time observations , 1974, IEEE Trans. Inf. Theory.

[21]  Jian Li,et al.  An adaptive filtering approach to spectral estimation and SAR imaging , 1996, IEEE Trans. Signal Process..

[22]  E. Polak.  Introduction to linear and nonlinear programming , 1973 .

[23]  Arye Nehorai,et al.  Adaptive comb filtering for harmonic signal enhancement , 1986, IEEE Trans. Acoust. Speech Signal Process..

[24]  Torsten Söderström,et al.  Harmonic signal analysis with Kalman filters using periodic orbits of nonlinear ODEs , 2003, 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP '03)..

[25]  T Söderström,et al.  Bias analysis in least squares estimation of periodic signals using nonlinear ODEs , 2004 .

[26]  Heinz Unbehauen,et al.  Bilinear Continuous-Time Systems Identification via Hartley-Based Modulating Functions , 1998, Autom..

[27]  Michael P. Fitz,et al.  Further results in the fast estimation of a single frequency , 1994, IEEE Trans. Commun..

[28]  Torbjörn Wigren,et al.  Harmonic signal modeling using adaptive nonlinear function estimation , 1996, 1996 IEEE International Conference on Acoustics, Speech, and Signal Processing Conference Proceedings.

[29]  G. Larry Bretthorst,et al.  Estimating the Parameters , 1988 .

[30]  Torsten Söderström,et al.  Least Squares Harmonic Signal Analysis Using Periodic Orbits of ODEs , 2003 .

[31]  Torsten Söderström,et al.  Periodic signal analysis using orbits of nonlinear ODE's based on the Markov estimate , 2004 .

[32]  R. R. Boorstyn,et al.  Multiple tone parameter estimation from discrete-time observations , 1976, The Bell System Technical Journal.

[33]  Frédéric Rotella,et al.  Identification of non-stationary continuous systems using modulating functions , 1990 .

[34]  Benjamin Friedlander,et al.  On the accuracy of the Kumaresan-Tufts method for estimating complex damped exponentials , 1987, IEEE Trans. Acoust. Speech Signal Process..