  Consider an $N\times n$ random matrix $Y_n=(Y^n_{ij})$ where the entries are
given by $Y^n_{ij}=\frac{\sigma_{ij}(n)}{\sqrt{n}}X^n_{ij}$, the $X^n_{ij}$
being independent and identically distributed, centered with unit variance and
satisfying some mild moment assumption. Consider now a deterministic $N\times
n$ matrix A_n whose columns and rows are uniformly bounded in the Euclidean
norm. Let $\Sigma_n=Y_n+A_n$. We prove in this article that there exists a
deterministic $N\times N$ matrix-valued function T_n(z) analytic in
$\mathbb{C}-\mathbb{R}^+$ such that, almost surely, \[\lim_{n\to+\infty,N/n\to
c}\biggl(\frac{1}{N}\operatorname
{Trace}(\Sigma_n\Sigma_n^T-zI_N)^{-1}-\frac{1}{N}\operatorname
{Trace}T_n(z)\biggr)=0.\] Otherwise stated, there exists a deterministic
equivalent to the empirical Stieltjes transform of the distribution of the
eigenvalues of $\Sigma_n\Sigma_n^T$. For each n, the entries of matrix T_n(z)
are defined as the unique solutions of a certain system of nonlinear functional
equations. It is also proved that $\frac{1}{N}\operatorname {Trace} T_n(z)$ is
the Stieltjes transform of a probability measure $\pi_n(d\lambda)$, and that
for every bounded continuous function f, the following convergence holds almost
surely \[\frac{1}{N}\sum_{k=1}^Nf(\lambda_k)-\int_0^{\infty}f(\lambda)\pi
_n(d\lambda)\mathop {\longrightarrow}_{n\to\infty}0,\] where the
$(\lambda_k)_{1\le k\le N}$ are the eigenvalues of $\Sigma_n\Sigma_n^T$. This
work is motivated by the context of performance evaluation of multiple
inputs/multiple output (MIMO) wireless digital communication channels. As an
application, we derive a deterministic equivalent to the mutual information:
\[C_n(\sigma^2)=\frac{1}{N}\mathbb{E}\log
\det\biggl(I_N+\frac{\Sigma_n\Sigma_n^T}{\sigma^2}\biggr),\] where $\sigma^2$
is a known parameter.
