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

Abstract

Abstract— The finite Fourier transform operator, and in particular its singular values, have been extensively studied in relation with band-limited functions. We study here the sequence of singular values of a random discretization of the finite Fourier transform in relation with applications to wireless communication. We prove that, with high probability, this sequence is close to the sequence of singular values of the finite Fourier transform itself. This also leads us to develop 2 estimates for the spectrum of kernel random matrices. This seems to be new to our knowledge. As applications, we give fairly good approximations of the number of degrees of freedom and the capacity of an approximate model of a MIMO wireless communication network. We provide the reader with some numerical examples that illustrate the theoretical results of this paper.