摘要 将随机矩阵的非渐近谱理论应用到协作频谱感知中,对接收信号样本协方差矩阵的最大特征值和最小特征值进行分析,该文提出一种精确的最大最小特征值差(Exact Maximum Minimum Eigenvalue Difference, EMMED)的协作感知算法。对于任意给定的协作用户个数K和采样点数N,首先推导了最大最小特征值之差的精确概率密度函数(Probability Density Function, PDF)和累积分布函数(Cumulative Distribution Function, CDF),然后利用该分布函数设计了所提算法的判决阈值。理论分析表明,EMMED算法的判决阈值较已有的渐进最大最小特征值差(Asymptotic Maximum Minimum Eigenvalue Difference, AMMED)检测更为精确,算法无需主用户信号特征并且能够对抗噪声不确定度影响。仿真结果表明,存在噪声不确定度的感知环境下,EMMED算法较已有的精确最大特征值(Exact Maximum Eigenvalue, EME)和EMMER等频谱感知算法具有更好的检测性能。
Abstract:The non-asymptotic spectral theory of random matrix is applied to cooperative spectrum sensing, the maximum eigenvalue and the minimum eigenvalue of the sampled signal covariance matrix are analyzed and an Exact Maximum Minimum Eigenvalues Difference (EMMED) algorithm is proposed. For any given numbers of cooperative users K and sampling points N, the exact Probability Density Function (PDF) and Cumulative Distribution Function (CDF) of the difference between the maximum and minimum eigenvalues are derived. Then, an accurate decision threshold is designed by using the distribution function. Theoretical analysis shows, the EMMED algorithm has the characteristics that the decision threshold is more accurate than the existing Asymptotic Maximum Minimum Eigenvalue Difference (AMMED) algorithm, without the characteristics of the main user signal and not affected by noise uncertainty. In addition, the simulation results show that the EMMED algorithm has better detection performance than the existing Exact Maximum Eigenvalue (EME) and EMMER algorithms in the real sensing environment with noisy uncertainty.
OH S W, MA Y, PEH E, et al. Introduction to Cognitive Radio and Television White Space[M]. Hoboken, NJ,USA, John Wiley & Sons, Inc., 2016: 1-22. doi: 10.1002/ 9781119110491.ch1.
[2]
KHATTAB A and BAYOUMI M A. An overview of IEEE standardization efforts for cognitive radio networks[C]. 2015 IEEE International Symposium on Circuits and Systems, Lisbon, 2015: 982-985. doi: 10.1109/ ISCAS.2015.7168800.
[3]
SHARMA S K, BOGALE T E, CHATZINOTAS S, et al. Cognitive radio techniques under practical imperfections: A survey[J]. IEEE Communications Surveys & Tutorials, 2015, 17(4): 1858-1884. doi: 10.1109/COMST.2015.2452414.
MI Yin and LU Guangyue. Cooperative spectrum sensing algorithm based on limiting eigenvalue distribution[J]. Journal on Communications, 2015, 36(1): 84-89. doi: 10.11959/j.issn.1000-436x.2015010.
[5]
TAHERPOUR A, NASIRI-KENARI M, and GAZOR S. Multiple antenna spectrum sensing in cognitive radios[J]. IEEE Transactions on Wireless Communications, 2010, 9(2): 814-823. doi: 10.1109/TWC.2009.02.090385.
[6]
CARDOSO L S, DEBBAH M, BIANCHI P, et al. Cooperative spectrum sensing using random matrix theory [C]. 2008 3rd International Symposium on Wireless Pervasive Computing, Santorini, 2008: 334-338. doi: 10.1109/ISWPC. 2008.4556225.
[7]
ZENG Y, KOH C L, and LIANG Y C. Maximum eigenvalue detection: Theory and application[C]. 2008 IEEE International Conference on Communications, Beijing, 2008: 4160-4164. doi: 10.1109/ICC.2008.781.
[8]
ZENG Y and LIANG Y C. Eigenvalue based spectrum sensing algorithms for cognitive radio[J]. IEEE Transactions on Communications, 2009, 57(6): 1784-1793. doi: 10.1109/ TCOMM.2009.06.070402.
WANG Yingxi and LU Guangyue. DMM based spectrum sensing method for cognitive radio systems[J]. Journal of Electronics & Information Technology, 2010, 32(11): 2571-2575. doi: 10.3724/SP.J.1146.2009.01434.
[10]
ZANELLA A, CHIANI M, and WIN M Z. On the marginal distribution of the eigenvalues of Wishart matrices[J]. IEEE Transactions on Communications, 2009, 57(4): 1050-1060. doi: 10.1109/TCOMM.2009.04.070143.
[11]
PENNA F, GARELLO R, FIGLIOLI D, et al. Exact non- asymptotic threshold for eigenvalue-based spectrum sensing[C]. 2009 4th International Conference on Cognitive Radio Oriented Wireless Networks and Communications, Hannover, 2009: 1-5. doi: 10.1109/CROWNCOM.2009. 5189008.
[12]
RATNARAJAH T, ZHONG C, KORTUN A, et al. Complex random matrices and multiple-antenna spectrum sensing[C]. 2011 IEEE International Conference on Acoustics, Speech and Signal Processing, Prague, 2011: 3848-3851. doi: 10.1109/ICASSP.2011.5947191.
[13]
CHATZINOTAS S, SHARMA S K, and OTTERSTEN B. Asymptotic analysis of eigenvalue-based blind Spectrum Sensing techniques[C]. 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, Vancouver, 2013: 4464-4468. doi: 10.1109/ICASSP.2013.6638504.
[14]
RATNARAJAH T, VAILLANCOURT R, and ALVO M. Eigenvalues and condition numbers of complex random matrices[J]. SIAM Journal on Matrix Analysis & Applications, 2005, 26(2): 441-456. doi: 10.1137/S089547980342204X.
[15]
CHIANI M. On the probability that all eigenvalues of Gaussian and Wishart random matrices lie within an interval[J]. IEEE Transactions on Information Theory, 2017, 63(7): 4521-4531. doi: 10.1109/TIT.2017.2694846.
[16]
TRACY C A and WIDOM H. On orthogonal and symplectic matrix ensembles[J]. Communications in Mathematical Physics, 1996, 177(3): 727-754. doi: 10.1007/BF02099545.
LIU Ning, SHI Haoshan, LIU Liping, et al. A novel blind spectrum sensing algorithm based on random matrix[J]. Journal of Northwestern Polytechnical University, 2016, 34(2): 262-267. doi: 10.3969/j.issn.1000-2758.2016.02.013.
[18]
CHIANI M. Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution[J]. Journal of Multivariate Analysis, 2014, 129: 69-81. doi: 10. /j.1016jmva.2014.04.002.