基于样本广义特征值的噪声中信号检测的基本极限
Fundamental limit of sample generalized eigenvalue based detection of signals in noise using relatively few signal-bearing and noise-only samples
摘要：
统计信号处理中的检测问题可以简洁地表述为：给定m个（可能）信号方位，n维信号加噪声快照向量（样本）和N个统计独立的n维仅噪声快照向量，可以可靠地推断信号的存在吗？这个问题出现在雷达、声纳、无线通信、生物信息学和机器学习等多种应用的背景下，并且是随后的信号参数估计阶段中关键的第一步。信号检测问题可以很自然地用样本广义特征值来表示。样本广义特征值对应于通过用仅噪声样本协方差矩阵“白化”信号加噪声样本协方差矩阵而形成的矩阵的特征值。在本文中，我们证明了当信号方位和仅噪声样本相对较少时，基于样本广义特征值的任意有色噪声信号检测的基本渐进极限。数值模拟突出了我们的分析预测的准确性，并允许我们扩展我们的启发式定义的有效数量的有色噪声中可识别的信号。我们讨论了我们的研究结果的影响，在传感器阵列处理，传感器网络中的突变检测和聚类方法在机器学习中的微弱和/或密集信号的检测。
英文摘要：
The detection problem in statistical signal processing can be succinctly formulated: Given m (possibly) signal bearing, n-dimensional signal-plus-noise snapshot vectors (samples) and N statistically independent n-dimensional noise-only snapshot vectors, can one reliably infer the presence of a signal? This problem arises in the context of applications as diverse as radar, sonar, wireless communications, bioinformatics, and machine learning and is the critical first step in the subsequent signal parameter estimation phase. The signal detection problem can be naturally posed in terms of the sample generalized eigenvalues. The sample generalized eigenvalues correspond to the eigenvalues of the matrix formed by "whitening" the signal-plus-noise sample covariance matrix with the noise-only sample covariance matrix. In this article we prove a fundamental asymptotic limit of sample generalized eigenvalue based detection of signals in arbitrarily colored noise when there are relatively few signal bearing and noise-only samples. Numerical simulations highlight the accuracy of our analytical prediction and permit us to extend our heuristic definition of the effective number of identifiable signals in colored noise. We discuss implications of our result for the detection of weak and/or closely spaced signals in sensor array processing, abrupt change detection in sensor networks, and clustering methodologies in machine learning.
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