Fast and exact simulation of complex-valued stationary Gaussian processes through embedding circulant matrix

This paper is concerned with the study of the embedding circulant matrix method to simulate stationary complex-valued Gaussian sequences. The method is, in particular, shown to be well-suited to generate circularly-symmetric stationary Gaussian processes. We provide simple conditions on the complex covariance function ensuring the theoretical validity of the minimal embedding circulant matrix method. We show that these conditions are satisfied by many examples and illustrate the simulation algorithm. In particular, we present a simulation study involving the circularly-symmetric fractional Gaussian noise, a model introduced in this paper.