Exact Simulation of Noncircular or Improper Complex-Valued Stationary Gaussian Processes using Circulant Embedding
This work addresses a specific challenge in signal processing and statistics for researchers and practitioners needing efficient simulation of noncircular Gaussian processes, representing an incremental improvement over existing methods.
The paper tackles the problem of simulating improper complex-valued stationary Gaussian processes by developing an algorithm based on circulant embedding, achieving exact simulation in O(n log n) operations, except when eigenvalues are negative, and demonstrates its effectiveness with an improper fractional Gaussian noise process.
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.