Comparison of some simulation algorithms on basis of distribution

The statistical distribution of the time series generated by three simulation algorithms of Gaussian stochastic processes are derived and compared. In all three methods, the time histories are modeled as weighted linear combinations of terms of the form cos(ω_kt +φ_k) where the phases φ_k are independent random variables uniformly distributed in [0, 2π]. The frequencies ω_k, however, are either deterministic parameters (the spectral representation algorithm), or independent random variables either uniformly distributed in a very small interval (the randomized spectral representation scheme) or distributed according to the specified power spectral density (the random frequencies algorithm). The results of the present investigation show that, from the standpoint of normality of the generated time series and irrespectively of computational aspects, the random frequencies algorithm performs always (considering first-order distributions) and generally (considering second-order distributions) better than or as well as the spectral representation technique and its randomized version, which yield almost identical probability density functions. Finally, if the spectral representation algorithm or its randomized version is used, it is recommended that the frequencies ω_k be selected so that the energy associated with each term cos(ω_kt +φ_k) be the same.