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Estimation of quasipolynomials in noise: Theoretical, algorithmic and implementation aspects. (English) Zbl 0927.93004

Warwick, Kevin (ed.) et al., Computer-intensive methods in control and signal processing. The curse of dimensionality. Basel: Birkhäuser. 223-235 (1997).
Consider a quasipolynomial of degree \(K\) \[ h(t)=e^{\lambda t} \cdot \sum^k_{j=0} A_{j+1}t^j \sin(\omega t+\varphi_{j+1}), \] with the damping factor \(\lambda<0\), the frequency \(\omega>0\), the amplitudes \(A_{j+1}\neq 0\) and the phases \(\varphi_{j+1} \in[-\pi,\pi)\). The problem of parametric estimation of the sum of such quasipolynomials in white Gaussian noise is considered, where the parameters are supposed to be nonrandom. The paper focuses on the estimation of the damping factor and the frequency.
The Cramér-Rao lower bound (CRB) on the covariance matrix of the estimation error was derived by V. Slivinskas and V. Šimonyté [Acta Appl. Math. 38, No. 1, 55-78 (1995; Zbl 0826.62070)]. R. Kumaresan and Feng [IEEE Trans. Signal Process. 39, No. 3, 736-741 (1991)] proposed to use the Prony method of estimation with prefiltering, but the simulations show that the estimates are not very close to the CRB curve.
In the present paper, the nonlinear least squares estimator is considered, which is calculated by the Levenberg iterative procedure, see A. Sen and M. Srivastava [Regression analysis: Theory, methods and applications. New York etc. Springer-Verlag (1990; Zbl 0714.62057)]. Numerical results show that the Levenberg method outperforms the modified Prony method only in the case of closely-spaced frequencies and high signal-to-noise ratio, however at the expense of an increased computational time. A short description of a software developed by the authors is presented.
For the entire collection see [Zbl 0892.00024].

MSC:

93-04 Software, source code, etc. for problems pertaining to systems and control theory
93E10 Estimation and detection in stochastic control theory
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