This paper analyzes the detection of a MM–dimensional useful signal modeled as the output of a M×KM \times K MIMO filter driven by a KK–dimensional white Gaussian noise, and corrupted by a MM–dimensional Gaussian noise with mutually uncorrelated components. The study is focused on frequency domain test statistics based on the eigenvalues of an estimate of the spectral coherence matrix (SCM), obtained as a renormalization of the frequency-smoothed periodogram of the observed signal. If NN denotes the sample size and BB the smoothing span, it is proved that in the high-dimensional regime where M,B,NM,B,N converge to infinity while KK remains fixed, the SCM behaves as a certain correlated Wishart matrix. Exploiting well-known results on the behaviour of the eigenvalues of such matrices, it is deduced that the standard tests based on linear spectral statistics of the SCM fail to detect the presence of the useful signal in the high-dimensional regime. A new test based on the SCM, which is proved to be consistent, is also proposed, and its statistical performance is evaluated through numerical simulations.

中文翻译：

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存在空间不相关噪声时低秩信号的检测：频域方法


本文分析了 MM 维有用信号的检测，该信号被建模为由 KK 维高斯白噪声驱动的 M×KM × K MIMO 滤波器的输出，并被具有互不相关分量的 MM 维高斯噪声破坏。该研究的重点是基于频谱相干矩阵（SCM）估计的特征值的频域测试统计，该特征值是通过观测信号的频率平滑周期图的重整化而获得的。如果 NN 表示样本大小，BB 表示平滑跨度，则证明在 M、B、NM、B、N 收敛于无穷大而 KK 保持固定的高维状态下，SCM 表现为某个相关的 Wishart 矩阵。利用有关此类矩阵特征值行为的众所周知的结果，可以推断基于 SCM 线性谱统计的标准测试无法检测高维区域中有用信号的存在。还提出了一种基于单片机的新测试，该测试被证明是一致的，并通过数值模拟评估了其统计性能。