This paper is centred on the spectral study of a random Fourier matrix, that is an \(n\times n\) matrix A with (j, k) entries given by \(\exp (2i\pi m Y_j Z_k)\), where \(Y_j\) and \(Z_k\) are two i.i.d sequences of random variables and \(1\le m\le n\) is a real number. When the \(Y_j\) and \(Z_k\) are uniformly distributed on a bounded 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. Moreover, this particular case of random finite Fourier matrix has been proposed in wireless telecommunication in order to approach the singular values of some channel fading matrices. We first compare in the \(\ell ^2\)-norm, the spectrum of the matrix \(A^*A\) with the spectrum of its associated integral operator. We show that the classical methods of concentration inequalities for kernel random matrices are well adapted for the spectral analysis of random Fourier matrices. We then concentrate on uniform distributions for the laws of \(Y_j\)’s and \(Z_k\)’s, for which the integral operator is the well-known Sinc-kernel operator with parameter m. We translate to random Fourier matrices the knowledge that we have on the spectrum of this operator. In particular, we study some asymptotic and non-asymptotic behaviours of the set of the eigenvalues of \(A^*A.\) This study is done in the spirit of recent work on the Sinc-kernel integral operator. As applications, we give fairly good approximations of the number of degrees of freedom, as well as an estimate of 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.

中文翻译：

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有限傅里叶变换的随机离散化及相关核随机矩阵

本文的重点是对随机傅立叶矩阵的频谱研究，该矩阵是一个\（n \ timesn \）矩阵A，其中具有（j，  k）项由\（\ exp（2i \ pi m Y_j Z_k）\）给出，其中\（Y_j \）和\（Z_k \）是两个随机变量的iid序列，而\（1 \ le m \ le n \）是一个实数。当\（Y_j \）和\（Z_k \）在有限的对称间隔上均匀分布，这可以看作是有限傅立叶变换的随机离散化，其频谱已与带限函数进行了广泛的研究。此外，在无线电信中已经提出了随机有限傅立叶矩阵的这种特殊情况，以便接近某些信道衰落矩阵的奇异值。我们首先在\（\ ell ^ 2 \）-范数中将矩阵\（A ^ * A \）的谱与其关联的积分算子的谱进行比较。我们表明，核随机矩阵浓度不等式的经典方法非常适合于随机傅里叶矩阵的频谱分析。然后，我们集中于\（Y_j \）定律的均匀分布和\（Z_k \），其积分运算符是著名的Sinc-kernel运算符，其参数为m。我们将这个算子的频谱知识转换为随机傅里叶矩阵。特别是，我们研究\（A ^ * A。\）特征值集的一些渐近和非渐近行为。本研究是根据Sinc-kernel积分算子的最新工作精神完成的。作为应用，我们给出了自由度的相当好的近似值，以及MIMO无线通信网络近似模型的容量估计。最后，我们为读者提供了一些数值示例，以说明本文的理论结果。