In this chapter, we further study the spectra of the important random matrix models for wireless communications that are the sample covariance matrix and the information plus noise models. It has already been shown in Chapter 3 that, as the e.s.d. of the population covariance matrix (or of the information matrix) converges, the e.s.d. of the sample covariance matrix (or the information plus noise matrix) converges almost surely. The limiting d.f. can then be fully characterized as a function of the l.s.d. of the population covariance matrix (or of the information matrix). It is however not convenient to invert the problem and to describe the l.s.d. of the population covariance matrix (or of the information matrix) as a function of the l.s.d. of the observed matrices. The answer to this inverse problem is provided in Chapter 8, which however requires some effort to be fully accessible. The development of the tools necessary for the statistical eigen-inference methods of Chapter 8 is one of the motivations of the current chapter.

The starting motivation, initiated by the work of Silverstein and Choi [Silverstein and Choi, 1995], which resulted in the important Theorem 7.4 (accompanied later by an important corollary, due to Mestre [Mestre, 2008a], Theorem 7.5), was to characterize the l.s.d. of the sample covariance matrix in closed-form. Remember that, up to this point, we can only characterize the l.s.d. F of a sample covariance matrix through the expression of its Stieltjes transform, as the unique solution mF(z) of some fixed-point equation for all z ∈ ℂ\ℝ+.