Linear and Nonlinear Approximation of Power Density Spectra with Linear Dynamical Filter Systems

Abstract

Investigations on the behaviour of technical systems by means of computer simulations or laboratory tests require methods for the ‘construction’ of stochastic excition processes in order to imitate the natural effects as well as possible. In the stationary and gaussian case a process is completely described by its power density spectrum. W. Wedig suggested in 1985/6 the use of linear dynamical filter systems driven by White Noise for the artificial generation of the desired process. A linear approximation procedure is presented and discussed considering practical applications.

The direct approximation of the power density spectrum leads to a nonlinear and — in general — nonconvex optimization problem. It is shown that the direct adaptation can be performed in the time domain using the linear solutions as initial values. If the accuracy of these approximation solutions is not sufficiently high, evolution procedures may be applied to solve the resulting nonlinear optimization problem numerically. Some heuristic control concepts for evolution procedures based on statistical considerations are presented.

For piecewise constant excitation an equivalent discrete time system can be evaluated and transformed into an ARMA system. The generation of the process can be performed with a minimal amount of computation time using the ARMA system which is driven by a simple sequence of random numbers.