Detection of Multivariate Cyclostationarity

David Ramirez,Peter J Schreier,Ignacio Santamaria,Javier Via,Louis L Scharf

doi:10.1109/tsp.2015.2450201

David Ramirez, Peter J Schreier + Show 3 more

Open Access

https://doi.org/10.1109/tsp.2015.2450201

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Abstract

This paper derives an asymptotic generalized likelihood ratio test (GLRT) and an asymptotic locally most powerful invariant test (LMPIT) for two hypothesis testing problems: 1) Is a vector-valued random process cyclostationary (CS) or is it wide-sense stationary (WSS)? 2) Is a vector-valued random process CS or is it nonstationary? Our approach uses the relationship between a scalar-valued CS time series and a vector-valued WSS time series for which the knowledge of the cycle period is required. This relationship allows us to formulate the problem as a test for the covariance structure of the observations. The covariance matrix of the observations has a block-Toeplitz structure for CS and WSS processes. By considering the asymptotic case where the covariance matrix becomes block-circulant we are able to derive its maximum likelihood (ML) estimate and thus an asymptotic GLRT. Moreover, using Wijsman's theorem, we also obtain an asymptotic LMPIT. These detectors may be expressed in terms of the Loeve spectrum, the cyclic spectrum, and the power spectral density, establishing how to fuse the information in these spectra for an asymptotic GLRT and LMPIT. This goes beyond the state-of-the-art, where it is common practice to build detectors of cyclostationarity from ad-hoc functions of these spectra.

Highlights

A zero-mean, discrete-time, complex-valued random process u[n] is said to be cyclostationary (CS) if its covariance function is periodic with period P [1], [2]: ruu[n, m] = E[u[n]u∗[n − m]] = ruu[n + P, m]The period P is a natural number greater than 1 because P = 1 corresponds to a wide-sense stationary (WSS) process
DERIVATION OF THE GLRT In the previous section, we showed that the three covariance matrices are block-diagonal without further structure but different block sizes
We present the asymptotic LMPIT for testing cyclostationarity vs. wide-sense stationarity

Summary

INTRODUCTION

This has been done in the context of CR to detect the presence of primary users [31] These two papers test the correlation in the temporal domain, it is possible to do so in the frequency domain [23], where the frequency coherence between u[n] and v[n] is used as the detector statistic. We instead use Wijsman’s theorem [38]–[41], which allows us to obtain the ratio of the distributions of the maximal invariant statistic without deriving the distributions or even the maximal invariant statistic Both GLRT and LMPIT are functions of coherence matrices, as are the detectors for spatial correlation in [42]–[44].

PROBLEM FORMULATION

REWRITING THE HYPOTHESES

DERIVATION OF THE GLRT

DERIVATION OF THE LMPIT

INTERPRETATION OF THE DETECTORS

VIII. CONCLUSIONS