无遍历性Wishart型随机矩阵在单通道实现中的渐近特征矩
Asymptotic Eigenvalue Moments of Wishart-Type Random Matrix Without Ergodicity in One Channel Realization
摘要：
考虑一个方差分布是随机的随机矩阵。如果对于每一列和行，其中元素的平方幅度的经验分布收敛于非随机分布，则该随机矩阵在一个信道实现中是遍历的。本文利用非交叉分割理论，导出了当$\bb H$具有随机方差分布且在单通道实现中非遍历时，大型Wishart型随机矩阵$\bb H\bb H^†$的几个渐近特征值矩（AEM）相关量的表达式.已知当$\bb H$在一个信道实现中是非遍历的（或遍历的）时，$\bb H\bb H^†$的经验特征值矩依赖于（或独立于）$\bb H$的方差分布的实现。对于非遍历$\bb H$，AEM可以通过i）根据$\bb H$的方差分布导出AEM的表达式，然后ii）在方差分布的集合上对导出的量求平均来获得。由于如果$\bb H$是遍历的，则AEM与方差分布无关，所以当方差分布的任何实现可用时，在i）中获得的表达式也可以用作遍历$\bb H$的AEM公式。
英文摘要：
Consider a random matrix whose variance profile is random. This random matrix is ergodic in one channel realization if, for each column and row, the empirical distribution of the squared magnitudes of elements therein converges to a nonrandom distribution. In this paper, noncrossing partition theory is employed to derive expressions for several asymptotic eigenvalue moments (AEM) related quantities of a large Wishart-type random matrix $\bb H\bb H^†$ when $\bb H$ has a random variance profile and is nonergodic in one channel realization. It is known the empirical eigenvalue moments of $\bb H\bb H^†$ are dependent (or independent) on realizations of the variance profile of $\bb H$ when $\bb H$ is nonergodic (or ergodic) in one channel realization. For nonergodic $\bb H$, the AEM can be obtained by i) deriving the expression of AEM in terms of the variance profile of $\bb H$, and then ii) averaging the derived quantity over the ensemble of variance profiles. Since the AEM are independent of the variance profile if $\bb H$ is ergodic, the expression obtained in i) can also serve as the AEM formula for ergodic $\bb H$ when any realization of variance profile is available.
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