Open Issues on the Statistical Spectrum Characterization of Random Vandermonde Matrices

Recently, analytical methods for finding moments of random Vandermonde matrices with entries on the unit circle have been proposed in the literature. Vandermonde matrices play an important role in signal processing and wireless applications, among which the multiple-antenna channel modeling, precoding or sparse sampling theory. Recent investigations allowed to extend the combinatorial approach usually exploited to characterize the spectral behavior of large random matrices with independent and identically distributed (i.i.d.) entries to Vandermonde structured matrices, under fairly broad assumptions on the entries distributions. While in several cases explicit expressions of the moments of the associated Gram matrix, as well as more advanced models involving the Vandermonde matrix could be provided, several issues are still open in the spectral behavior characterization, with applications either in signal processing (deconvolution, compressed sensing) and/or wireless communications (capacity estimation, topology information retrieving, etc). I. PROBLEM DESCRIPTION A Vandermonde matrix [6] with entries on the unit circle has the following form V = 1 √ N   1 · · · 1 e−jω1 · · · e−jωL .. . . . .. e−j(N−1)ω1 · · · e−j(N−1)ωL   ; (1) we will mainly focus on the case where ω1,..., ωL are i.i.d. random variables taking values on [0, 2π). Throughout the paper, the ωi will be called phase distributions, V will denote any Vandermonde matrix, of dimensions N ×L, with a given phase distribution. Let c denote the aspect ratio of the abovedefined matrix, i.e. limL,N→+∞ L N → c. Other models of particular interest are the generalized Vandermonde matrices, whose columns do not consist of uniformly distributed powers, namely V = 1 √ N   e−jbNf(0)cω1 · · · e−jbNf(0)cωL e−jbNf( 1 N )cω1 · · · e−jbNf( 1 N )cωL .. . . . .. e−jbNf( N−1 N )cω1 · · · e−jbNf( N−1 N )cωL   , (2) where f is called the power distribution, and is a map from [0, 1) to itself. More general cases can also be considered, for SUPELEC, Alcatel-Lucent Chair on Flexible Radio, Plateau de Moulon, 3 rue Joliot-Curie, 91192 GIF SUR YVETTE CEDEX, France and Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway instance by replacing f with a random variable, i.e. V = 1 √ N   e−jNλ1ω1 · · · e−jNλ1ωL e−jNλ2ω1 · · · e−jNλ2ωL .. . . . .. e−jNλN ω1 · · · e−jNλN ωL   , (3) with λi’s, i.i.d. on [0, 1), and also independent from the ωj’s. The basic quantities of interest are defined in the following: Definition 1: Let us consider an N ×N Hermitian matrix A. The averaged empirical cumulative distribution function of the eigenvalues (also referred to as the averaged empirical spectral distribution (ESD)) of A is defined as