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Cognitive Radio Communication and Networking: Principles and Practice by Robert Caiming Qiu, Michael C. Wicks, Zhen Hu, Husheng Li

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5.5 Case Studies and Applications

5.5.1 Fundamental Example of Using Large Random Matrix

We follow [279] for this development. Define an M × N complex matrix as

images

where (Xij)1≤i≤M, 1≤jN are (a number of MN) i.i.d. complex Gaussian variables images. x1, x2, …, xN are columns of X. The covariance matrix R is

images

The empirical covariance matrix is defined as

images

In practice, we are interested in the behavior of the empirical distribution of the eigenvalues of images for large M and N. For example, how do the histograms of the eigenvalues (λi)i=1, …, M of images behave when M and N increase? It is well known that when M is fixed, but N increases, that is, images is small, the large law of large numbers requires

images

In other words, ...

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