1Open Issues on the Statistical Spectrum Characterization of Random Vandermonde Matrices

Abstract — 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 ex-tend the combinatorial approach usually exploited to characterize the spectral behavior of large random matrices with indepen-dent 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 (de-convolution, 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 for