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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The non-centrality matrix is defined as \(\mathbf {\Omega }_K= \mathbf {\Sigma }_K^{-1}\varvec{M}\varvec{M}^\dagger \) where \(\mathbf {\Sigma }_K\) and \(\varvec{M}\) are respectively the covariance matrix and the mean of \(\varvec{Y}\) defined as \(\mathbf {\Sigma }_K = E[(\varvec{Y}-\varvec{M})(\varvec{Y}-\varvec{M})^\dagger ]\) and \(\varvec{M} = E[\varvec{Y}]\).
References
Cardoso, L., Debbah, M., Bianchi, P., Najim, J.: Cooperative spectrum sensing using random matrix theory. In: Proceedings of the IEEE International Symposium on Wireless Pervasive Computing (ISWPC), Greece, pp. 334–338, May 2008
Zeng, Y., Liang, Y.C.: Eigenvalue-based spectrum sensing algorithms for cognitive radio. IEEE Trans. Commun. 57(6), 1784–1793 (2009)
Penna, F., Garello, R., Spirito, M.: Cooperative spectrum sensing based on the limiting eigenvalue ratio distribution in wishart matrices. IEEE Commun. Lett. 13(7), 507–509 (2009)
Penna, F., Garello, R., Figlioli, D., Spirito, M.: Exact non-asymptotic threshold for eigenvalue-based spectrum sensing. In: Proceedings of the IEEE 4th International Conference CROWNCOM, Germany, pp. 1–5, June 2009
Zhang, W., Abreu, G., Inamori, M., Sanada, Y.: Spectrum sensing algorithms via finite random matrices. IEEE Trans. Commun. 60(1), 164–175 (2012)
Kobeissi, H., Nasser, Y., Bazzi, O., Louet, Y., Nafkha, A.: On the performance evaluation of eigenvalue-based spectrum sensing detector for mimo systems. In: XXXIth URSI General Assembly and Scientific Symposium (URSI GASS), pp. 1–4, August 2014
Tan, W.Y., Gupta, R.P.: On approximating the non-central wishart distribution with wishart distribution. Commun. Stat. Theory Method 12(22), 2589–2600 (1983)
Johansson, K.: Shape fluctuations and random matrices. Comm. Math. Phys. 209(2), 437–476 (2000)
Feldheim, O.N., Sodin, S.: A universality result for the smallest eigenvalues of certain sample covariance matrices. Geom. Funct. Anal. 20(1), 88–123 (2010)
Bornemann, F.: On the numerical evaluation of distributions in random matrix theory: A review with an invitation to experimental mathematics. Markov Proc. Relat. Fields 16, 803–866 (2009)
Baik, J., Ben Arous, G., Pch, S.: Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33(5), 1643–1697 (2005)
Baik, J., Silverstein, J.W.: Eigenvalues of large sample covariance matrices of spiked population models. J. Multivar. Anal. 97(6), 1382–1408 (2006)
Kritchman, S., Nadler, B.: Determining the number of components in a factor model from limited noisy data. Chemometr. Intell. Lab. Syst. 94, 19–32 (2008)
Bornemann, F.: Asymptotic independence of the extreme eigenvalues of gaussian unitary ensemble. J. Math. Phy. 51, 023514 (2009)
Hachem, W., Hardy, A., Najim, J.: A survey on the eigenvalues local behavior of large complex correlated wishart matrices. ARXIV to be published in the “Proceedings of the Journees MAS 2014” (September 2015)
Acknowledgment
This work was funded by a program of cooperation between the Lebanese University and the Azem & Saada social foundation (LU-AZM) and by CentraleSupélec (France).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 ICST Institute for Computer Sciences, Social Informatics and Telecommunications Engineering
About this paper
Cite this paper
Kobeissi, H., Nafkha, A., Nasser, Y., Bazzi, O., Louët, Y. (2016). Simple and Accurate Closed-Form Approximation of the Standard Condition Number Distribution with Application in Spectrum Sensing. In: Noguet, D., Moessner, K., Palicot, J. (eds) Cognitive Radio Oriented Wireless Networks. CrownCom 2016. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 172. Springer, Cham. https://doi.org/10.1007/978-3-319-40352-6_29
Download citation
DOI: https://doi.org/10.1007/978-3-319-40352-6_29
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-40351-9
Online ISBN: 978-3-319-40352-6
eBook Packages: Computer ScienceComputer Science (R0)