  Consider a $N\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are
given by $Y_{ij}^{n}=\frac{\sigma(i/N,j/n)}{\sqrt{n}} X_{ij}^{n}$, the
$X_{ij}^{n}$ being centered i.i.d. and $\sigma:[0,1]^2 \to (0,\infty)$ being a
continuous function called a variance profile. Consider now a deterministic
$N\times n$ matrix $\Lambda_n=(\Lambda_{ij}^{n})$ whose non diagonal elements
are zero. Denote by $\Sigma_n$ the non-centered matrix $Y_n + \Lambda_n$. Then
under the assumption that $\lim_{n\to \infty} \frac Nn =c>0$ and $$ \frac{1}{N}
\sum_{i=1}^{N} \delta_{(\frac{i}{N}, (\Lambda_{ii}^n)^2)} \xrightarrow[n\to
\infty]{} H(dx,d\lambda), $$ where $H$ is a probability measure, it is proven
that the empirical distribution of the eigenvalues of $ \Sigma_n \Sigma_n^T$
converges almost surely in distribution to a non random probability measure.
This measure is characterized in terms of its Stieltjes transform, which is
obtained with the help of an auxiliary system of equations. This kind of
results is of interest in the field of wireless communication.
