Licentiate thesis 2002-003

Harmonic Signal Modeling Based on the Wiener Model Structure

Emad Abd-Elrady

May 2002

The estimation of frequencies and corresponding harmonic overtones is a problem of great importance in many situations. Applications can, for example, be found in supervision of electrical power transmission lines, in seismology and in acoustics. Generally, a periodic function with an unknown fundamental frequency in cascade with a parameterized and unknown nonlinear function can be used as a signal model for an arbitrary periodic signal. The main objective of the proposed modeling technique is to estimate the fundamental frequency of the periodic function in addition to the parameters of the nonlinear function. The thesis is divided into four parts. In the first part, a general introduction to the harmonic signal modeling problem and different approaches to solve the problem are given. Also, an outline of the thesis and future research topics are introduced. In the second part, a previously suggested recursive prediction error method (RPEM) for harmonic signal modeling is studied by numerical examples to explore the ability of the algorithm to converge to the true parameter vector. Also, the algorithm is modified to increase its ability to track the fundamental frequency variations. A modified algorithm is introduced in the third part to give the algorithm of the second part a more stable performance. The modifications in the RPEM are obtained by introducing an interval in the nonlinear block with fixed static gain. The modifications that result in the convergence analysis are, however, substantial and allows a complete treatment of the local convergence properties of the algorithm. Moreover, the Cramer-Rao bound (CRB) is derived for the modified algorithm and numerical simulations indicate that the method gives good results especially for moderate signal to noise ratios (SNR). In the fourth part, the idea is to give the algorithm of the third part the ability to estimate the driving frequency and the parameters of the nonlinear output function parameterized also in a number of adaptively estimated grid points. Allowing the algorithm to automatically adapt the grid points as well as the parameters of the nonlinear block, reduces the modeling errors and gives the algorithm more freedom to choose the suitable grid points. Numerical simulations indicate that the algorithm converges to the true parameter vector and gives better performance than the fixed grid point technique. Also, the CRB is derived for the adaptive grid point technique.

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