Performance of Statistical Tests for Single-Source Detection Using Random Matrix Theory

Bianchi, Pascal; Debbah, Mérouane; Maïda, Mylène; Najim, Jamal

doi:10.1109/tit.2011.2111710

Cited by 187 publications

(223 citation statements)

References 49 publications

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“…For a detailed proof we refer the reader to [3]. Here we provide an intuitive explanation for this: In the joint limit p, n → ∞, p/n → c, we have that …”

Section: Introductionmentioning

confidence: 95%

“…Classical examples include testing for the presence of interactions in multi-way data [8] and testing for equality of the population covariance to a scaled identity matrix [11]. Some modern signal processing applications include testing for the presence of signals in cognitive radio as well as non-parametric signal detection in array processing [2,3]. Normalized Wishart matrices (e.g., with trace equal to one) are also a common model for random density matrices in quantum information channels, see for example [14,20].…”

Section: Introductionmentioning

confidence: 99%

“…(1.6) either to set the threshold corresponding to a given false alarm rate for various detection procedures, or to analyze them, see for example [3,10]. Since these thresholds depend on tail probabilities of U , an interesting question is how accurate is the approximation in Eq.…”

Section: Introductionmentioning

confidence: 99%

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On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

Nadler

2011

Journal of Multivariate Analysis

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The ratio of the largest eigenvalue divided by the trace of a p×p random Wishart matrix with n degrees of freedom and identity covariance matrix plays an important role in various hypothesis testing problems, both in statistics and in signal processing. In this paper we derive an approximate explicit expression for the distribution of this ratio, by considering the joint limit as both p, n → ∞ with p/n → c. Our analysis reveals that even though asymptotically in this limit the ratio follows a Tracy-Widom (TW) distribution, one of the leading error terms depends on the second derivative of the TW distribution, and is non-negligible for practical values of p, in particular for determining tail probabilities. We thus propose to explicitly include this term in the approximate distribution for the ratio. We illustrate empirically using simulations that adding this term to the TW distribution yields a quite accurate expression to the empirical distribution of the ratio, even for small values of p, n.

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“…For a detailed proof we refer the reader to [3]. Here we provide an intuitive explanation for this: In the joint limit p, n → ∞, p/n → c, we have that …”

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

Nadler

2011

Journal of Multivariate Analysis

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“…In [8] it is shown that, in the asymptotic regime N, K → ∞ with K/N fixed, the GLRT detection performance converges to that of RLRT. However, a natural question is: Simulated receiver operating characteristics (ROC) curves for detection methods with known and unknown noise variance.…”

Section: Detection Performance Of Rlrt and Glrtmentioning

confidence: 99%

Performance of Eigenvalue-Based Signal Detectors with Known and Unknown Noise Level

Nadler

Penna

Garello

2011

2011 IEEE International Conference on Communications (ICC)

134

109

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Abstract-In this paper we consider signal detection in cognitive radio networks, under a non-parametric, multi-sensor detection scenario, and compare the cases of known and unknown noise level. The analysis is focused on two eigenvalue-based methods, namely Roy's largest root test, which requires knowledge of the noise variance, and the generalized likelihood ratio test, which can be interpreted as a test of the largest eigenvalue vs. a maximum-likelihood estimate of the noise variance. The detection performance of the two considered methods is expressed by closed-form analytical formulas, shown to be accurate even for small number of sensors and samples. We then derive an expression of the gap between the two detectors in terms of the signal-to-noise ratio of the signal to be detected, and we identify critical settings where this gap is significant (e.g., small number of sensors and signal strength). Our results thus provide a measure of the impact of noise level knowledge and highlight the importance of accurate noise estimation.

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“…These methods do not require any prior information on the primary signal or on the noise power. In [30,31] two hypothesis tests allowing to detect the presence of an unknown transmitter using several sensors are proposed and random matrix theory is used to provide the error associated with both tests.…”

Section: Introductionmentioning

confidence: 99%

Non-commutative large entries for cognitive radio applications

Masucci

Debbah

2012

J Wireless Com Network

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Cognitive radio has been proposed as a solution for the problem of underutilization of the radio spectrum. Indeed, measurements have shown that large portions of frequency bands are not efficiently assigned since large pieces of bandwidth are unused or only partially used. In the last decade, studies in different areas, such as signal processing, random matrix theory, information theory, game theory, etc., have brought us to the current state of cognitive radio research. These theoretical advancements represents a solid base for practical applications and even further developments. However, still open questions need to be answered. In this study, free probability theory, through the free deconvolution technique, is used to attack the huge problem of retrieving useful information from the network with a finite number of observations. Free deconvolution, based on the moments method, has shown to be a helpful approach to this problem. After giving the general idea of free deconvolution for known models, we show how the moments method works in the case where scalar random variables are considered. Since, in general, we have a situation where more complex systems are involved, the parameters of interest are no longer scalar random variables but random vectors and random matrices. Random matrices are non-commutative operators with respect to the matrix product and they can be considered elements of what is called noncommutative probability space.Therefore, we focus on the case where random matrices are considered. Concepts from combinatorics, such as crossing and non-crossing partitions are useful tools to express the moments of Gaussian and Vandermonde matrices, respectively. Our analysis and simulation results show that free deconvolution framework can be used for studying relevant information in cognitive radio such as power detection, users detection, etc.

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Performance of Statistical Tests for Single-Source Detection Using Random Matrix Theory

Cited by 187 publications

References 49 publications

On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

Performance of Eigenvalue-Based Signal Detectors with Known and Unknown Noise Level

Non-commutative large entries for cognitive radio applications

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