Fast and Exact Simulation of Complex-Valued Stationary Gaussian Processes Through Embedding Circulant Matrix

ABSTRACT

This article 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 article. Supplementary material for this article is available online.

KEYWORDS:

• Circularly symmetric processes
• Complex fractional Brownian motion
• Nonnegative definiteness

Acknowledgments

The authors are sincerely grateful to the reviewers and the associate editor for their careful reading, their stimulating comments and suggestions that considerably helped us in improving a previous version of the article. While revising this article, we have been aware of the report by Sykulski and Percival (2016) which extends the present article and considers the problem of simulating a stationary complex-valued Gaussian process with prescribed and arbitrary pseudo-covariance function h.

Additional information

Funding

Emilio Porcu is supported by Proyecto Fondecyt Regular Number 1130647 from Chilean Minestry of Science and Education. Jean-François Coeurjolly is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).