We study non-parametric frequency-domain system identification from a finite-sample perspective. We assume an open loop scenario where the excitation input is periodic and consider the Empirical Transfer Function Estimate (ETFE), where the goal is to estimate the frequency response at certain desired (evenly-spaced) frequencies, given input-output samples. We show that under sub-Gaussian colored noise (in time-domain) and stability assumptions, the ETFE estimates are concentrated around the true values. The error rate is of the order of $\mathcal{O}((d_{\mathrm{u}}+\sqrt{d_{\mathrm{u}}d_{\mathrm{y}}})\sqrt{M/N_{\mathrm{tot}}})$, where $N_{\mathrm{tot}}$ is the total number of samples, $M$ is the number of desired frequencies, and $d_{\mathrm{u}},\,d_{\mathrm{y}}$ are the dimensions of the input and output signals respectively. This rate remains valid for general irrational transfer functions and does not require a finite order state-space representation. By tuning $M$, we obtain a $N_{\mathrm{tot}}^{-1/3}$ finite-sample rate for learning the frequency response over all frequencies in the $ \mathcal{H}_{\infty}$ norm. Our result draws upon an extension of the Hanson-Wright inequality to semi-infinite matrices. We study the finite-sample behavior of ETFE in simulations.

翻译：我们从有限样本的角度研究非参数频域系统辨识。假设开环场景下激励输入是周期性的，并考虑经验传递函数估计（ETFE），目标是根据输入-输出样本估计某些期望（等间隔）频率下的频率响应。我们证明，在亚高斯有色噪声（时域）和稳定性假设下，ETFE估计值集中于真实值附近。误差率为 $\mathcal{O}((d_{\mathrm{u}}+\sqrt{d_{\mathrm{u}}d_{\mathrm{y}}})\sqrt{M/N_{\mathrm{tot}}})$，其中 $N_{\mathrm{tot}}$ 是总样本数，$M$ 是期望频率数，$d_{\mathrm{u}},\,d_{\mathrm{y}}$ 分别为输入和输出信号的维度。该率适用于一般无理传递函数，无需有限阶状态空间表示。通过调整 $M$，我们获得在 $\mathcal{H}_{\infty}$ 范数下所有频率上学习频率响应的 $N_{\mathrm{tot}}^{-1/3}$ 有限样本率。该结果基于将Hanson-Wright不等式推广到半无限矩阵。我们通过仿真研究了ETFE的有限样本行为。