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

被引:0
|
作者
Aline Bonami
Abderrazek Karoui
机构
[1] Institut Denis-Poisson,Department of Mathematics, Faculty of Sciences of Bizerte
[2] Université d’Orléans,undefined
[3] Collégium Sciences et Techniques,undefined
[4] University of Carthage,undefined
来源
Journal of Fourier Analysis and Applications | 2020年 / 26卷
关键词
Eigenvalues and singular values; Kernel random matrices; Random discretization; Finite Fourier transform; Sinc kernel operator; Number of degrees of freedom; Primary 42A38; 15B52; Secondary 60F10; 60B20;
D O I
暂无
中图分类号
学科分类号
摘要
This paper is centred on the spectral study of a random Fourier matrix, that is an n×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\times n$$\end{document} matrix A with (j, k) entries given by exp(2iπmYjZk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exp (2i\pi m Y_j Z_k)$$\end{document}, where Yj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_j$$\end{document} and Zk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_k$$\end{document} are two i.i.d sequences of random variables and 1≤m≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le m\le n$$\end{document} is a real number. When the Yj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_j$$\end{document} and Zk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_k$$\end{document} 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 ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^2$$\end{document}-norm, the spectrum of the matrix A∗A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^*A$$\end{document} 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 Yj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_j$$\end{document}’s and Zk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_k$$\end{document}’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.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^*A.$$\end{document} 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.
引用
收藏
相关论文
共 50 条
  • [31] Finite Random Matrices for Blind Spectrum Sensing
    de Abreu, Giuseppe Thadeu Freitas
    Zhang, Wensheng
    Sanada, Yukitoshi
    2010 CONFERENCE RECORD OF THE FORTY FOURTH ASILOMAR CONFERENCE ON SIGNALS, SYSTEMS AND COMPUTERS (ASILOMAR), 2010, : 116 - 120
  • [32] GOOD RANDOM MATRICES OVER FINITE FIELDS
    Yang, Shengtian
    Honold, Thomas
    ADVANCES IN MATHEMATICS OF COMMUNICATIONS, 2012, 6 (02) : 203 - 227
  • [33] Comparison of random number generators via Fourier transform
    Imai, Junichi
    MONTE CARLO METHODS AND APPLICATIONS, 2013, 19 (03): : 237 - 259
  • [34] Generalized Random Demodulator Associated with Fractional Fourier Transform
    Haoran Zhao
    Liyan Qiao
    Jingchao Zhang
    Ning Fu
    Circuits, Systems, and Signal Processing, 2018, 37 : 5161 - 5173
  • [35] Generalized Random Demodulator Associated with Fractional Fourier Transform
    Zhao, Haoran
    Qiao, Liyan
    Zhang, Jingchao
    Fu, Ning
    CIRCUITS SYSTEMS AND SIGNAL PROCESSING, 2018, 37 (11) : 5161 - 5173
  • [36] Random field discretization method for finite element reliability analysis
    Lin, Daojin
    Qin, Quan
    Qinghua Daxue Xuebao/Journal of Tsinghua University, 2002, 42 (06): : 839 - 842
  • [37] CONSISTENT FINITE-ELEMENT DISCRETIZATION OF DISTRIBUTED RANDOM LOADS
    LIU, Q
    ORISAMOLU, IR
    CHERNUKA, MW
    COMPUTERS & STRUCTURES, 1994, 51 (01) : 39 - 45
  • [38] Random matrices and random permutations
    Van Moerbeke, P
    PROCEEDINGS OF THE WORKSHOP ON NONLINEARITY, INTEGRABILITY AND ALL THAT: TWENTY YEARS AFTER NEEDS '79, 2000, : 207 - 218
  • [39] Kernel Regression Over Graphs Using Random Fourier Features
    Meireles Elias, Vitor Rosa
    Gogineni, Vinay Chakravarthi
    Martins, Wallace A.
    Werner, Stefan
    IEEE TRANSACTIONS ON SIGNAL PROCESSING, 2022, 70 : 936 - 949
  • [40] RANDOM FOURIER FEATURES MULTI-KERNEL LMS ALGORITHM
    Gao, Wei
    Song, Meiru
    Chen, Jie
    Zhang, Lingling
    2020 IEEE INTERNATIONAL CONFERENCE ON SIGNAL PROCESSING, COMMUNICATIONS AND COMPUTING (IEEE ICSPCC 2020), 2020,