Exact simulation of complex-valued Gaussian stationary processes via circulant embedding

Abstract

Circulant embedding is a technique that has been used to generate realizations from certain real-valued Gaussian stationary processes. This technique has two potential advantages over competing methods for simulating time series. First, the statistical properties of the generating procedure are exactly the same as those of the target stationary process. Second, the technique is based upon the discrete Fourier transform and hence is computationally attractive when this transform is computed via a fast Fourier transform (FFT) algorithm. In this paper we show how, when used with a standard 'powers of two' FFT algorithm, circulant embedding can be readily adapted to handle complex-valued Gaussian stationary processes.

Fractional Brownian motion, the Matérn process, and stochastic modeling of turbulent dispersion

“…For simulation of stationary processes, the Toeplitz matrix structure can in principle be used to accelerate the Cholesky decomposition to O(N 2 ) or even O (N log N ), see Yagle and Levy (1985) and Dietrich and Newsam (1997) respectively. The latter method, termed circulant embedding, while O (N log N ), involves embedding the covariance matrix of interest within a larger matrix, and may lead to somewhat unpredictable tradeoffs between minimizing error and increasing the matrix size (Percival, 2006). The method presented here has the advantages that it is very straightforward to implement, and that the error terms are well understood provided the Green's function is known.…”

Tapering Promotes Propriety for Fourier Transforms of Real-Valued Time Series

“…So the DFT of a proper complex-valued vector gives rise to proper complex variables at each Fourier frequency. This property was exploited in [13] for the simulation of proper scalar complex-valued Gaussian proceses. However, if X is real-valued, the case of interest to us, no such assurance is forthcoming.…”

Simulation of Improper Complex-Valued Sequences

“…Recently, Percival [5] presented a scheme for simulation of a sample of size N from such a Gaussian process. In this correspondence, we provide a method for simulation from a general improper complex-valued SOS process, i.e., it is not assumed that r = 0, 2 .…”