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Sensors 2016, 16(8), 1183; doi:10.3390/s16081183

Exact Distributions of Finite Random Matrices and Their Applications to Spectrum Sensing

1
School of Information Science and Engineering, Shandong University, Jinan 250100, China
2
Institute of Sensors, Signals and Systems, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, UK
3
Wireless Technology Innovation Institute, Beijing University of Posts and Telecommunications, Beijing 100876, China
*
Author to whom correspondence should be addressed.
Academic Editor: Vittorio M. N. Passaro
Received: 12 June 2016 / Revised: 22 July 2016 / Accepted: 25 July 2016 / Published: 29 July 2016
(This article belongs to the Section Physical Sensors)
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Abstract

The exact and simple distributions of finite random matrix theory (FRMT) are critically important for cognitive radio networks (CRNs). In this paper, we unify some existing distributions of the FRMT with the proposed coefficient matrices (vectors) and represent the distributions with the coefficient-based formulations. A coefficient reuse mechanism is studied, i.e., the same coefficient matrices (vectors) can be exploited to formulate different distributions. For instance, the same coefficient matrices can be used by the largest eigenvalue (LE) and the scaled largest eigenvalue (SLE); the same coefficient vectors can be used by the smallest eigenvalue (SE) and the Demmel condition number (DCN). A new and simple cumulative distribution function (CDF) of the DCN is also deduced. In particular, the dimension boundary between the infinite random matrix theory (IRMT) and the FRMT is initially defined. The dimension boundary provides a theoretical way to divide random matrices into infinite random matrices and finite random matrices. The FRMT-based spectrum sensing (SS) schemes are studied for CRNs. The SLE-based scheme can be considered as an asymptotically-optimal SS scheme when the dimension K is larger than two. Moreover, the standard condition number (SCN)-based scheme achieves the same sensing performance as the SLE-based scheme for dual covariance matrix K = 2 . The simulation results verify that the coefficient-based distributions can fit the empirical results very well, and the FRMT-based schemes outperform the IRMT-based schemes and the conventional SS schemes. View Full-Text
Keywords: finite random matrix theory; infinite random matrix theory; eigenvalue distributions; cognitive radio networks; spectrum sensing finite random matrix theory; infinite random matrix theory; eigenvalue distributions; cognitive radio networks; spectrum sensing
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Zhang, W.; Wang, C.-X.; Tao, X.; Patcharamaneepakorn, P. Exact Distributions of Finite Random Matrices and Their Applications to Spectrum Sensing. Sensors 2016, 16, 1183.

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