The empirical distribution of the eigenvalues of a Gram matrix with a given variance profile

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.Comment: 25 pages, revised version. Assumption (A2) has been relaxe

The empirical eigenvalue distribution of a Gram matrix: From independence to stationarity

Consider a $N\times n$ random matrix $Z_n=(Z^n_{j_1 j_2})$ where the individual entries are a realization of a properly rescaled stationary gaussian random field. The purpose of this article is to study the limiting empirical distribution of the eigenvalues of Gram random matrices such as $Z_n Z_n ^*$ and $(Z_n +A_n)(Z_n +A_n)^*$ where $A_n$ is a deterministic matrix with appropriate assumptions in the case where $n\to \infty$ and $\frac Nn \to c \in (0,\infty)$. The proof relies on related results for matrices with independent but not identically distributed entries and substantially differs from related works in the literature (Boutet de Monvel et al., Girko, etc.).Comment: 15 page

Methodes d'estimation spectrale 2D pour le traitement d'antenne bande etroite

eux méthodes d'estimation spectrale haute résolution 2D adaptées au problème de l'imagerie de sources bandes étroites non sinusoïdales en traitement d'antenne sont présentées dans cet article.La première basée sur les propriétés algébriques de la matrice de covariance spatio-temporelle est une extention 2D de Ia méthode du goniomètre.La deuxième: généralisation de la méthode TAM intoduite par Kung exploite la propriété que toute factorisation minimale de la matrice de covariance spatio-temporelle en l'absence de bruit coincide avec la matrice d'observabilité d'un systène linéaire 2D réalisable par un modèle d'Attasi.Des résultats de simulations comparent les performances de l'extension 2D du goniomètre à celles obtenues en exploitant la matrice de covariance spatiale et la matrice interspectrale

The Mutual Information of a MIMO Channel: A Survey

International audienc

On the fluctuations of the mutual information of large dimensional MIMO channels

International audienc