  Analytical methods for finding moments of random Vandermonde matrices with
entries on the unit circle are developed. Vandermonde Matrices play an
important role in signal processing and wireless applications such as direction
of arrival estimation, precoding, and sparse sampling theory, just to name a
few. Within this framework, we extend classical freeness results on random
matrices with independent, identically distributed (i.i.d.) entries and show
that Vandermonde structured matrices can be treated in the same vein with
different tools. We focus on various types of matrices, such as Vandermonde
matrices with and without uniform phase distributions, as well as generalized
Vandermonde matrices. In each case, we provide explicit expressions of the
moments of the associated Gram matrix, as well as more advanced models
involving the Vandermonde matrix. Comparisons with classical i.i.d. random
matrix theory are provided, and deconvolution results are discussed. We review
some applications of the results to the fields of signal processing and
wireless communications.
