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

TL;DR

This paper introduces an efficient and exact method for simulating complex-valued stationary Gaussian processes using embedding circulant matrices, with theoretical validation and practical examples including fractional Brownian motion.

Contribution

The paper develops simple conditions for the validity of the embedding circulant matrix method for complex Gaussian processes and demonstrates its effectiveness through simulations.

Findings

Method is well-suited for circularly-symmetric processes

Conditions for validity are satisfied by many examples

Simulation of fractional Brownian motion is demonstrated

Abstract

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.