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arXiv · 1703.10459

Random Discretization of the Finite Fourier Transform and Related Kernel Random Matrices

Aline Bonami·Abderrazek Karoui
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Abstract

This paper is centred on the spectral study of a Random Fourier matrix, that is an $n\times n$ matrix $A$ whose $(j, k)$ entries are $\exp(2i\pi m X_jY_k)$, with $X_j$ and $Y_k$ two i.i.d sequences of random variables and $1\leq m\leq n$ is a real number. When they are uniformly distributed on a symmetric interval, this may be seen as a random discretization of the Finite Fourier transform, whose spectrum has been extensively studied in relation with band-limited functions. Our study is two-fold. Firstly, by pushing forward concentration inequalities, we find an accurate comparison in $\ell^2$- norm between the spectrum of $A^*A$ and the one of an integral operator that can be defined in terms of the two probability laws chosen for the rows and the columns. Our study includes the one of stationary Hermitian kernel matrices and can be generalized to non stationary ones, for which the same kind of comparison with an integral operator is possible. Because of possible applications in the data science area, these last matrices have been largely studied in the literature and our results are compared with previous ones. Secondly we concentrate on uniform distributions for the laws of $X_j$'s and $Y_k$'s, for which the integral operator is the well-known Sinc-kernel operator with parameter $m.$ Our previous study allows to translate to random Fourier matrices the knowledge that we have on the spectrum of this operator. We have for them asymptotic results for $m, n$ and $n/m$ tending to $\infty$, as well as non asymptotic bounds in the spirit of recent work on the integral operators. As an application, we give fairly good approximations of the number of degrees of freedom and the capacity of a MIMO wireless communication network approximation model. Finally, we provide the reader with some numerical examples that illustrate the theoretical results of this paper.

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BibTeXRIS

Aline Bonami, Abderrazek Karoui. 2017-03-30. Random Discretization of the Finite Fourier Transform and Related Kernel Random Matrices. https://arxiv.org/abs/1703.10459

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