Generating data with prescribed power spectral density

Data generation is straightforward if the parameters of a time series model define the prescribed spectral density or covariance function. Otherwise, a time series model has to be determined. An arbitrary prescribed spectral density will be approximated by a finite number of equidistant samples in the frequency domain. This approximation becomes accurate by taking more and more samples. Those samples can be inversely Fourier transformed into a covariance function of finite length. The covariance in turn is used to compute a long autoregressive (AR) model with the Yule–Walker relations. Data can be generated with this long AR model. The long AR model can also be used to estimate time series models of different types to search for a parsimonious model that attains the required accuracy with less parameters. It is possible to derive objective rules to choose a preferred type with a minimal order for the generating time series model. That order will generally depend on the number of observations to be generated. The quality criterion for the generating time series model is that the spectrum estimated from the generated number of observations cannot be distinguished from the prescribed spectrum.