Abstract

This work addresses the problem of determining whether two multivariate random time series have the same power spectral density (PSD), which has applications, for instance, in physical-layer security and cognitive radio. Remarkably, existing detectors for this problem do not usually provide any kind of optimality. Thus, we study here the existence under the Gaussian assumption of optimal invariant detectors for this problem, proving that the uniformly most powerful invariant test (UMPIT) does not exist. Thus, focusing on close hypotheses, we show that the locally most powerful invariant test (LMPIT) only exists for univariate time series. In the multivariate case, we prove that the LMPIT does not exist. However, this proof suggests two LMPIT-inspired detectors, one of which outperforms previously proposed approaches, as computer simulations show.

Highlights

  • The problem of determining whether two multivariate time series possess the same power spectral density (PSD) at every frequency finds many diverse applications, ranging from the comparison of gas pipes [1], earthquake-explosion discrimination [2] or light-intensity emission stability determination [3], to physical-layer security [4] and spectrum sensing [5]

  • Similar to our previous works [10, 11], we use in this paper Wijsman’s theorem [12, 13], which allows us to derive the uniformly most powerful invariant test (UMPIT) or the locally most powerful invariant test (LMPIT), if they exist, without the aforementioned distributions, and even without identifying the maximal invariant statistic. Exploiting this powerful theorem and assuming Gaussianity, the main contribution of this paper is to formally show that the UMPIT does not exist for testing equality of PSD matrices, the LMPIT only exists for univariate time series, whereas it does not exist for multivariate processes

  • We compare the probability of missed detection of the generalized likelihood ratio test (GLRT) and the two LMPIT

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Summary

Introduction

The problem of determining whether two multivariate time series possess the same (matrix-valued) power spectral density (PSD) at every frequency finds many diverse applications, ranging from the comparison of gas pipes [1], earthquake-explosion discrimination [2] or light-intensity emission stability determination [3], to physical-layer security [4] and spectrum sensing [5]. This problem has originated an active field of research since it was first studied by Coates and Diggle [1]. We may apply the statistics for homogeneity at each frequency, and many ad-hoc detectors may be developed by appropriately choosing the function to fuse them

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