Exact Simulation of Noncircular or Improper Complex-Valued Stationary Gaussian Processes using Circulant Embedding

TL;DR

This paper introduces an efficient, exact algorithm for simulating improper complex-valued stationary Gaussian processes using circulant embedding, applicable to various process lengths with practical guarantees.

Contribution

It presents a novel circulant embedding-based method for exact simulation of improper Gaussian processes, extending existing techniques to complex-valued cases.

Findings

Algorithm operates in O(n log n) time.

Method is exact except for negative eigenvalues.

Empirical evaluation confirms efficiency and accuracy.

Abstract

This paper provides an algorithm for simulating improper (or noncircular) complex-valued stationary Gaussian processes. The technique utilizes recently developed methods for multivariate Gaussian processes from the circulant embedding literature. The method can be performed in $\mathcal{O}(n\log_2 n)$ operations, where $n$ is the length of the desired sequence. The method is exact, except when eigenvalues of prescribed circulant matrices are negative. We evaluate the performance of the algorithm empirically, and provide a practical example where the method is guaranteed to be exact for all $n$, with an improper fractional Gaussian noise process.