Generating Stationary Gaussian Processes Using The Spectral Representation Theorem

The paper presents the Spectral Representation Theorem of a stationary process and two simulation methods derived from it. In order to test the accuracy of the two simulation methods two numerical applications are employed. First, using a theoretical power spectral density (PSD), two sets of sample functions, corresponding to each method, are generated and a comparison of the obtained numerical results with the analytical PSD is carried out. The second example is more complex and consists in using the stationary zone of the strong motion of the recorded NS acceleration registered at INCERC during the 1977 Vrancea earthquake. The corresponding Fourier Spectrum is calculated. In order to obtain a smoother PSD representation for the real Fourier spectrum, a specific barrier model spectrum (SBM) is fitted to it and the corresponding PSD calculated. This PSD is used to generate two sets of samples. The mean PSD obtained using both methods of simulation is compared with that characterizing the registered acceleration. The paper shows that the generated time series possess all the theoretical probabilistic characteristics discussed below, when the number of terms used in the simulation formulas is large. Three types of estimators are employed in the numerical evaluation of both simulation methods