Generating data with prescribed spectral density

Time series models are suitable for the generation of data with prescribed covariance or spectral characteristics. If the required covariance function or spectral density is defined by a time series model, data generation is straightforward. 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) process with the Yule-Walker relations. Data can be generated with this long AR process. Unfortunately, the most general prescribed spectral densities belong to infinitely wide covariance functions. Therefore, finite covariance representations are necessarily approximations. It is possible to derive objective rules to choose a minimal finite order for the generating AR process. This order depends on the number of observations to be generated. The criterion is that the spectrum of those observations cannot be distinguished from the prescribed spectrum.