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.

我们从有限样本的角度研究非参频域系统辨识，利用经验传递函数估计（ETFE）来估计特定频率下的频率响应，证明了在次高斯有色噪声和稳定性假设条件下，ETFE 估计会集中在真值附近，并且得到学习全部频率下频率响应的有限样本速率。