On the spectral coherence between two discrete time harmonizable simple processes

Simple processes (SPs) are a class of spectrally correlated (SC) processes that their Loeve bi-frequency spectrum is supported by a countable collection of special curves. While spectral coherence is a well-established tool for analyzing relationships between stationary processes, its extension to SC processes like discrete time harmonizable simple processes (DTHSPs) remains underexplored. This gap limits the ability to study the coherencies among many real-world signals. In this work, the spectral coherence for two DTHSPs is defined and robust estimator of the introduced spectral coherence for both known and unknown spectral structures is proposed. For known SP’s structures, we employ the periodogram covariances to establish the estimator and characterize its asymptotic properties. For unknown SP’s structures, a spectral ciphering-based approach is employed to estimate the coherence. Then, multiple testing procedures are developed to determine whether two DTHSPs are coherent or incoherent. Theoretical results are validated through simulations under two scenarios: (1) a time-lagged linear relationship between two DTHSPs and (2) independent DTHSPs. According to the numerical simulations, the proposed approach demonstrates superior performance compared to existing approaches in terms of key metrics such as precision, recall, F1-score and specificity. Finally, the proposed framework is applied to a real-world dataset, showcasing its practical utility in identifying spectral dependencies.