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Abstract
An estimation method is presented for signals described by linear differential equations whose coefficients are functions of the unknown parameters. Various types of identifiability are defined according to these functions. In the linear case, an estimation algorithm is derived directly from the identifiability conditions, by use of elementary rules of operational calculus. The main steps of the algorithm are first presented through an introductory example of a damped sinusoid estimation. Unlike the modified Prony's method (used as reference), the presented one (1) provides explicit closed-form expressions and (2) remains valid for linear differential equations with non-constant coefficients. These expressions depend only on iterated integrals of the observed signals. Some basic features of the method are analysed and, in particular, we show that a least squares interpretation can be attached to it. Application to the estimation of a chirp signal is also presented with numerical simulations.
Notes
Notes
1. See also Citation8 for an algebraic frequency estimation.
2. The invertibility of the system in the operational domain is not necessary. For example, although x and are proportional, their respective images in the time domain, x and
, are not.
3. Note that a fully discrete-time counterpart of the presented method can be found in Citation15–17.
4. In all the simulations, the integrals are numerically computed using the ‘work horse’ trapezoidal method.
5. The Matlab code used for simulating this method was found at http://www.statsci.org
6. In general either d of D is zero.
7. Compare with (Equation24) which describes the true parameter vector Θ. Indeed, since the term Q′ in the second member of (Equation24) depends on Θ, that equation is not computable.