Exact and Asymptotic Analysis of Largest Eigenvalue Based Spectrum Sensing
Olav Tirkkonen, Lu Wei · InTech eBooks · 2012
Will-be-set-by-IN-TECHlargest eigenvalue distribution is utilized to set a decision threshold, considering only the false alarm probability.This result characterizes the LE detector performance in the asymptotical region where the sample sizes and the number of cooperating sensors are huge.In (Kritchman & Nadler, 2009) a more general problem of estimating the number of signals using the largest eigenvalue is studied, where the estimation probability is obtained using the Tracy-Widom distribution as well.Finally we note that the LE detector is similar to the energy detector in that the test statistics are functions of the noise variance.Therefore the LE detector is pestered by the noise uncertainty problem as well.In this chapter, the analysis of eigenvalue detector is carried out in a setting where there is only one primary user transmitting.The detection problem is a hypothesis test between two possible hypotheses; either there is a primary user, or there is none.The covariance matrices under these hypotheses can be formulated as central and non-central Wishart matrices, respectively.Empirically we found that the largest eigenvalue calculated from the received covariance matrix is an efficient quantity to discriminate between the two hypotheses, which motivates the investigation of the LE detection.The contribution of this chapter is two-fold.Firstly we derive the exact largest eigenvalue distributions for central and non-central Wishart matrices.We modify the results on the largest eigenvalue distributions from (Dighe et al., 2003;Kang & Alouini, 2003;Khatri, 1964) in order to derive distribution functions suitable for performance analysis.As a result we obtain exact characterizations for both the false alarm probability and the probability of missed detection.Secondly, we investigate the detection performance in the asymptotical region where both the number of sensors and the sample size are large.Specifically we derive closed-form asymptotic largest eigenvalue distributions for central and non-central Wishart matrices.These results are possible due to recent breakthrough in random matrix theory.Moreover a simple closed-form formula for the receiver operating characteristics (ROC) can also be derived.Besides gaining more insights into the detection performance, the low complexity asymptotic results can be used for the implementation of the LE detector.The accuracy of the asymptotic approximations is investigated by comparing to the exact distributions through various realistic spectrum sensing scenarios.The results confirm the usefulness of the asymptotic distributions in analyzing the detection performance in practice.We also compare the detection performance of the LE detection with other well-known detection schemes.It turns out that in the case of perfectly estimated noise power the LE detector performs best among the detectors considered.In order to see the whole picture, we extend the analysis to the case where the noise power is not perfectly known.With worst case noise uncertainty, the LE detector performs worse than the ER detector, but is by far more robust against noise uncertainty the energy detector.The rest of this chapter is organized as follows.In Section 2, we formulate the primary user detection problem in a multi-antenna spectrum sensing setting.We then motivate the choice of largest eigenvalue as the test statistics.Section 3 is devoted to deriving the exact as well as the asymptotical largest eigenvalue distributions.In Section 4, we first study the impact of approximation accuracy of the asymptotic distributions on the detection performance.We then compare the detection performance of the LE detector with that of other detection methods.Lastly, we investigate the impact of the noise uncertainty on the detection performance.Finally in Section 5 we conclude the main results of this chapter.