Asymptotic Eigenvalue Moments of Wishart-Type Random Matrix Without Ergodicity in One Channel Realization

Details Asymptotic Eigenvalue Moments of Wishart-Type Random Matrix Without Ergodicity in One Channel Realization Asymptotic Eigenvalue Moments of Wishart-Type Random Matrix Without Ergodicity in One Channel Realization

2008-10-06

arxiv.org/abs/0810.0882v1

Computer Science

Information Theory

36 pages, 6 figures, submitted to IEEE Transactions on Information Theory, Oct. 2008

Scientific paper

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 $\bb H\bb H^\dag$ when $\bb H$ has a random variance profile and is nonergodic in one channel realization. It is known the empirical eigenvalue moments of $\bb H\bb H^\dag$ 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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