Locally Most Powerful Invariant Tests for Correlation and Sphericity of Gaussian Vectors

In this paper we study the existence of locally most powerful invariant tests (LMPIT) for the problem of testing the covariance structure of a set of Gaussian random vectors. The LMPIT is the optimal test for the case of close hypotheses, among those satisfying the invariances of the problem, and in practical scenarios can provide better performance than the typically used generalized likelihood ratio test (GLRT). The derivation of the LMPIT usually requires one to find the maximal invariant statistic for the detection problem and then derive its distribution under both hypotheses, which in general is a rather involved procedure. As an alternative, Wijsman's theorem provides the ratio of the maximal invariant densities without even finding an explicit expression for the maximal invariant. We first consider the problem of testing whether a set of $N$-dimensional Gaussian random vectors are uncorrelated or not, and show that the LMPIT is given by the Frobenius norm of the sample coherence matrix. Second, we study the case in which the vectors under the null hypothesis are uncorrelated and identically distributed, that is, the sphericity test for Gaussian vectors, for which we show that the LMPIT is given by the Frobenius norm of a normalized version of the sample covariance matrix. Finally, some numerical examples illustrate the performance of the proposed tests, which provide better results than their GLRT counterparts

Detection of multivariate cyclostationarity

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 Loe`ve 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.The work of P. Schreier was supported by the Alfried Krupp von Bohlen und Halbach Foundation, under its program “Return of German scientists from abroad”. The work of I. Santamaría and J. Vía was supported by the Spanish Government, Ministerio de Ciencia e Innovación (MICINN), under project RACHEL (TEC2013-47141-C4-3-R). The work of L. Scharf was supported by the Airforce Office of Scientific Research under contract FA9550-10-1-0241

An asymptotic LMPI test for cyclostationarity detection with application to cognitive radio

We propose a new detector of primary users in cognitive radio networks. The main novelty of the proposed detector in comparison to most known detectors is that it is based on sound statistical principles for detecting cyclostationary signals. In particular, the proposed detector is (asymptotically) the locally most powerful invariant test, i.e. the best invariant detector for low signal-to-noise ratios. The derivation is based on two main ideas: the relationship between a scalar-valued cyclostationary signal and a vector-valued wide-sense stationary signal, and Wijsman's theorem. Moreover, using the spectral representation for the cyclostationary time series, the detector has an insightful interpretation, and implementation, as the broadband coherence between frequencies that are separated by multiples of the cycle frequency. Finally, simulations confirm that the proposed detector performs better than previous approaches.The work of P. Schreier was supported by the Alfried Krupp von Bohlen und Halbach Foundation, under its program “Return of German scientists from abroad”. The work of I. Santamaría and J. Vía was supported by the Spanish Government, Ministerio de Ciencia e Innovación (MICINN), under project RACHEL (TEC2013-47141-C4-3-R). The work of L. Scharf was supported by the Airforce Office of Scientific Research under contract FA9550-10-1-0241

LMPIT-inspired Tests for Detecting a Cyclostationary Signal in Noise with Spatio-Temporal Structure

In spectrum sensing for cognitive radio, the presence of a primary user can be detected by making use of the cyclostationarity property of digital communication signals. For the general scenario of a cyclostationary signal in temporally colored and spatially correlated noise, it has previously been shown that an asymptotic generalized likelihood ratio test (GLRT) and locally most powerful invariant test (LMPIT) exist. In this paper, we derive detectors for the presence of a cyclostationary signal in various scenarios with structured noise. In particular, we consider noise that is temporally white and/or spatially uncorrelated. Detectors that make use of this additional information about the noise process have enhanced performance. We have previously derived GLRTs for these specific scenarios; here, we examine the existence of LMPITs. We show that these exist only for detecting the presence of a cyclostationary signal in spatially uncorrelated noise. For white noise, an LMPIT does not exist. Instead, we propose tests that approximate the LMPIT, and they are shown to perform well in simulations. Finally, if the noise structure is not known in advance, we also present hypothesis tests using our framework

Towards Robust Spectrum Sensing in Cognitive Radio Networks

This thesis focuses on multi-antenna assisted energy based spectrum sensing. The studies leading to this thesis have been motivated by some practical issues with energy based detection. These include the noise uncertainty problem at the secondary receiver, the presence of multiple active primary users in cognitive cellular networks, the existence of unknown noise correlations and detection in the low signal-to-noise ratio regime. In this thesis, the aim is to incorporating these practical concerns into the design of spectrum sensing algorithms. To this end, we propose the use of various detectors that are suitable for different scenarios. We consider detectors derived from decision-theoretical criteriors as well as heuristic detectors. We analyze the performance of the proposed detectors by deriving their false alarm probability, detection probability and receiver operating characteristic. The main contribution of this thesis consists of the derived closed-form performance metrics. These results are obtained by utilizing tools from multivariate analysis, moment based approximations, Mellin transforms, and random matrix theory. Numerical results show that the proposed detectors have indeed resolved the concerns raised by the above practical issues. Some detectors could meet the needs of one of the practical challenges, while others are shown to be robust when several practical issues are taken into account. The use of detectors constructed with decision-theoretical considerations over the heuristically proposed ones is justified as well

Robust spectrum sensing techniques for cognitive radio networks

Cognitive radio is a promising technology that improves the spectral utilisation by allowing unlicensed secondary users to access underutilised frequency bands in an opportunistic manner. This task can be carried out through spectrum sensing: the secondary user monitors the presence of primary users over the radio spectrum periodically to avoid harmful interference to the licensed service. Traditional energy based sensing methods assume the value of noise power as prior knowledge. They suffer from the noise uncertainty problem as even a mild noise level mismatch will lead to significant performance loss. Hence, developing an efficient robust detection method is important. In this thesis, a novel sensing technique using the F-test is proposed. By assuming a multiple antenna assisted receiver, this detector uses the F-statistic as the test statistic which offers absolute robustness against the noise variance uncertainty. In addition, since the channel state information (CSI) is required to be known, the impact of CSI uncertainty is also discussed. Results show the F-test based sensing method performs better than the energy detector and has a constant false alarm probability, independent of the accuracy of the CSI estimate. Another main topic of this thesis is to address the sensing problem for non-Gaussian noise. Most of the current sensing techniques consider Gaussian noise as implied by the central limit theorem (CLT) and it offers mathematical tractability. However, it sometimes fails to model the noise in practical wireless communication systems, which often shows a non-Gaussian heavy-tailed behaviour. In this thesis, several sensing algorithms are proposed for non-Gaussian noise. Firstly, a non-parametric eigenvalue based detector is developed by exploiting the eigenstructure of the sample covariance matrix. This detector is blind as no information about the noise, signal and channel is required. In addition, the conventional energy detector and the aforementioned F-test based detector are generalised to non-Gaussian noise, which require the noise power and CSI to be known, respectively. A major concern of these detection methods is to control the false alarm probability. Although the test statistics are easy to evaluate, the corresponding null distributions are difficult to obtain as they depend on the noise type which may be unknown and non-Gaussian. In this thesis, we apply the powerful bootstrap technique to overcome this difficulty. The key idea is to reuse the data through resampling instead of repeating the experiment a large number of times. By using the nonparametric bootstrap approach to estimate the null distribution of the test statistic, the assumptions on the data model are minimised and no large sample assumption is invoked. In addition, for the F-statistic based method, we also propose a degrees-of-freedom modification approach for null distribution approximation. This method assumes a known noise kurtosis and yields closed form solutions. Simulation results show that in non-Gaussian noise, all the three detectors maintain the desired false alarm probability by using the proposed algorithms. The F-statistic based detector performs the best, e.g., to obtain a 90% detection probability in Laplacian noise, it provides a 2.5 dB and 4 dB signal-to-noise ratio (SNR) gain compared with the eigenvalue based detector and the energy based detector, respectively

Relativistic Gravitational Experiments in Space

The results are summarized of a workshop on future gravitational physics space missions. The purpose of the workshop was to define generic technological requirements for such missions. NASA will use the results to direct its program of advanced technology development