Simple and Accurate Closed-Form Approximation of the Standard Condition Number Distribution with Application in Spectrum
Standard condition number (SCN) detector is a promising detector that can work effectively in uncertain environments. In this paper, we consider a Cognitive Radio (CR) with large number of antennas (eg. Massive MIMO) and we provide an accurate and simple
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Abstract. Standard condition number (SCN) detector is a promising detector that can work effectively in uncertain environments. In this paper, we consider a Cognitive Radio (CR) with large number of antennas (eg. Massive MIMO) and we provide an accurate and simple closed form approximation for the SCN distribution using the generalized extreme value (GEV) distribution. The approximation framework is based on the moment-matching method and the expressions of the moments are approximated using bi-variate Taylor expansion and results from random matrix theory. In addition, the performance probabilities and decision threshold are also considered as they have a direct relation to the distribution. Simulation results show that the derived approximation is tightly matched to the condition number distribution. Keywords: Standard condition number matrix · Massive MIMO
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· Spectrum sensing · Wishart
Introduction
Cognitive Radio (CR) is being the technology that provides solution for the scarcity and inefficiency in using the spectrum resource. For the CR to operate effectively and to provide the required improvement in spectrum efficiency, it must be able to effectively detect the presence/absence of the Primary User (PU) to avoid interference if it exists and freely use the spectrum in the absence of the PU. Thus, Spectrum Sensing (SS), being responsible for the presence/absence detection process, is the key element in any CR guarantee. Several SS techniques were proposed in the last decade, however, Eigenvalue Based Detector (EBD) has been shown to overcome noise uncertainty challenges c ICST Institute for Computer Sciences, Social Informatics and Telecommunications Engineering 2016 D. Noguet et al. (Eds.): CROWNCOM 2016, LNICST 172, pp. 351–362, 2016. DOI: 10.1007/978-3-319-40352-6 29
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and performs adequately even in low SNR conditions. It presents an efficient way for multi-antenna SS in CR [1,2] as it does not need any prior knowledge about the noise power or signal to noise ratio. EBD is based on the eigenvalues of the received signal covariance matrix and it utilises results from random matrix theory. It detects the presence/absence of the PU by exploiting receiver diversity and includes the Largest Eigenvalue detector, the Scaled Largest Eigenvalue detector, and the Standard Condition Number (SCN) detector [1–6]. The SCN is defined as the ratio of maximum to minimum eigenvalues. The SCN detector compares the SCN of the sample covariance matrix with a certain threshold. This threshold was set according to Marchenco-Pastur law (MP) in [1], however, it is not related to any error constraints. In [2], the authors have provided an approximate relation between the threshold and the False-Alarm Probability (Pf a ) by exploiting the Tracy-Widom distribution (TW) for the maximum eigenvalue while maintaining the MP law for the minimum eigenvalue. This work was further improved in [3,4] by using the Curtiss formula for the distribution of the ratio of random variables. In these two cases, the threshold could
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