This paper investigates the signal detection problem in colored noise with an unknown covariance matrix. In particular, we focus on detecting an unknown non-random signal by capitalizing on the leading eigenvalue of the whitened sample covariance matrix as the test statistic (a.k.a. Roy's largest root test). Since the unknown signal is non-random, the whitened sample covariance matrix turns out to have a non-central $F$-distribution. This distribution assumes a singular or non-singular form depending on whether the number of observations $p\lessgtr$ the system dimensionality $m$. Therefore, we statistically characterize the leading eigenvalue of the singular and non-singular $F$-matrices by deriving their cumulative distribution functions (c.d.f.). Subsequently, they have been utilized in deriving the corresponding receiver operating characteristic (ROC) profiles. We also extend our analysis into the high dimensional domain. It turns out that, when the signal is sufficiently strong, the maximum eigenvalue can reliably detect it in this regime. Nevertheless, weak signals cannot be detected in the high dimensional regime with the leading eigenvalue.

翻译：本文研究了在协方差矩阵未知的有色噪声中的信号检测问题。具体而言，我们聚焦于利用白化样本协方差矩阵的主特征值作为检验统计量（即Roy最大根检验）来检测未知的非随机信号。由于未知信号是非随机的，白化样本协方差矩阵服从非中心$F$-分布。该分布根据观测数$p$与系统维度$m$的大小关系呈现奇异或非奇异形式。因此，我们通过推导奇异与非奇异$F$-矩阵主特征值的累积分布函数（c.d.f.），对其进行了统计刻画。随后，利用这些结果推导了相应的接收机工作特性（ROC）曲线。我们还将分析扩展至高维领域。结果表明，当信号足够强时，最大特征值能可靠地检测该场景下的信号。然而，在高维场景下，弱信号无法通过主特征值被检测到。