In this paper we consider the problem of obtaining a state space realization of a zero mean gaussian vector process. A new algorithm is presented for the case in which the process is given in terms of its spectral density function. Contrary to the usual procedure followed, which requires a partial fraction expansion, the algorithm presented starts with a (deterministic) realization of the spectral density function itself.
van der Schaft, A. J., & Willems, J. C. (1984). A new procedure for stochastic realization of spectral density matrices. SIAM journal on control and optimization, 22(6), 845-855. https://doi.org/10.1137/0322053