Mathematical Sciences: The Spectral Behavior of Large Dimensional Random Matrices Applied to Signal Processing

信号处理中的一般类问题是 研究的应用结果的限制行为 随机矩阵的特征值随维数的增加。 这些问题包括确定关于下列问题性质的信息： 在充满噪声的环境中从未知的 数据来源的数量，这些数据是从 传感器. 用于解决该问题的方法，例如MUSIC，依赖于 协方差矩阵的谱特性与 传感器值的向量。 该矩阵近似为 样本协方差矩阵由向量数据的样本形成 穿越时空 样本量通常需要相当大， 为了有效地近似协方差矩阵， 特别是当源的数量相当大时。 以帮助 后一种情况定理将在 本征值的极限经验分布函数 一类高维随机矩阵 施加到 信号处理问题当有很多信号源时， 定理可以用来确定源的数量和 噪声方差，样本量小得多 而不是近似协方差矩阵。 而且他们 在某些情况下，可以提供有关 信号的性质。 计算机模拟表明， 只要有20个来源就可以获得有用的信息。 研究还计划解决一些重要的数学问题， 关于极限定理的问题，为了完全 理解定理对信号的适用性 处理. 将进行计算机模拟， 的分析。

A general class of problems in signal processing is to be studied by the application of results on the limiting behavior of the eigenvalues of random matrices as the dimension increases. The problems consist of determining information on the nature of signals emitted in a noise-filled environment from an unknown number of sources from data gathered from a collection of sensors. Methods for solving the problem, such as MUSIC, rely on the spectral properties of the covariance matrix associated with the vector of sensor values. This matrix is approximated by the sample covariance matrix formed from samples of the vector data across time. The sample size usually needs to be quite large in order to effectively approximate the covariance matrix, especially when the number of sources is sizable. To aid in the latter situation theorems will be brought into play on the limiting empirical distribution function of the eigenvalues of a class of large dimensional random matrices. Applied to the signal processing problem when there are many sources, the theorems can be used to determine the number of sources and the variance of the noise, with a sample size considerably smaller than needed to approximate the covariance matrix. Moreover, they can, in certain situations, provide additional information on the nature of the signals. Computer simultations indicate that useful information can be obtained with as little as 20 sources. Research is also planned to solve some important mathematical questions concerning the limit theorems, in order to completely understand the applicability of the theorems to signal processing. Computer simulations will be performed to facilitate the analysis.