  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
co-variance 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 algorithm. In particular, we present a simulation
study involving the circularly-symmetric fractional Brownian motion, a model
introduced in this paper.
