Distribution of the Ratio of the Largest Eigenvalue to the Trace of Complex Wishart Matrices
Distribution of the Ratio of the Largest Eigenvalue to the Trace of Complex Wishart Matrices
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DOI：
10.1109/tsp.2012.2205922
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发表时间：
2012-10-01
影响因子：
5.4
通讯作者：
Zhong, Caijun
中科院分区：
文献类型：
--
作者：
Kortun, Ayse;Sellathurai, Mathini;Zhong, Caijun
这种对应关系研究了比率T = lambda（1）/Sigma（m）（i=1）的统计特性，其中lambda（1）>= lambda（2）>=. >= lambda（m）是具有n个自由度的m × m复中心Wishart矩阵W的m个特征值。我们推导出新的精确的解析表达式的概率密度函数（PDF）和累积分布函数（CDF）的T为复杂的中心Wishart矩阵与任意尺寸。我们还制定了简化的统计T的特殊情况下的对偶不相关和对偶相关的复中心Wishart矩阵（m = 2）。研究的比值T是盲频谱感知中最重要的比值，因为它代表了广义似然比检验（GLRT）的充分统计量。因此，推导出的分析结果被用来找到准确的判决阈值为所需的概率的盲GLRT（B-GLRT）检测器的虚警。结果表明，基于精确判决门限的B-GLRT检测器的性能优于文献中提出的渐近判决门限检测器，从而有效地利用了认知无线电中的频谱资源。
This correspondence investigates the statistical properties of the ratio, T = lambda(1)/Sigma(m)(i=1) where lambda(1) >= lambda(2) >= ... >= lambda(m) are the m eigenvalues of an m x m complex central Wishart matrix W with n degrees of freedom. We derive new exact analytical expressions for the probability density function (PDF) and cumulative distribution function (CDF) of T for complex central Wishart matrices with arbitrary dimensions. We also formulate simplified statistics of T for the special case of dual uncorrelated and dual correlated complex central Wishart matrices (m = 2). The investigated ratio T is the most important ratio in blind spectrum sensing, since it represents a sufficient statistics for the generalized likelihood ratio test (GLRT). Thus, the derived analytical results are used to find the exact decision threshold for the desired probability of false alarm for Blind-GLRT (B-GLRT) detector. It is shown that the exact decision threshold based B-GLRT detector gives superior performance over the asymptotic decision threshold schemes proposed in the literature, which leads to efficient spectrum usage in cognitive radio.