  Consider the random matrix $\Sigma = D^{1/2} X \widetilde D^{1/2}$ where $D$
and $\widetilde D$ are deterministic Hermitian nonnegative matrices with
respective dimensions $N \times N$ and $n \times n$, and where $X$ is a random
matrix with independent and identically distributed centered elements with
variance $1/n$. Assume that the dimensions $N$ and $n$ grow to infinity at the
same pace, and that the spectral measures of $D$ and $\widetilde D$ converge as
$N,n \to\infty$ towards two probability measures. Then it is known that the
spectral measure of $\Sigma\Sigma^*$ converges towards a probability measure
$\mu$ characterized by its Stieltjes Transform.
  In this paper, it is shown that $\mu$ has a density away from zero, this
density is analytical wherever it is positive, and it behaves in most cases as
$\sqrt{|x - a|}$ near an edge $a$ of its support. A complete characterization
of the support of $\mu$ is also provided. \\ Beside its mathematical interest,
this analysis finds applications in a certain class of statistical estimation
problems.
