Finite Sample Identification: From Frequency to Time Domain

We review frequency domain system identification under finite samples and study how to recover time domain models. We focus on an open loop setting where the excitation input is periodic and consider the Empirical Transfer Function Estimate (ETFE), i.e., we estimate the frequency response at certain desired evenly-spaced frequencies from input-output data. Under sub-Gaussian colored noise and certain stability assumptions, we can establish finite-sample guarantees for the ETFE. The estimates are concentrated around the true values with an error rate of the order of $\mathcal{\tilde O}(\sqrt{M/N_{\mathrm{tot}}})$, where $N_{\mathrm{tot}}$ is the total number of samples and $M$ is the number of desired frequencies. Given the ETFE, we show that by tuning the number of frequencies $M$, we can recover the impulse responses of the system at a rate of $\mathcal{\tilde O}(N_{\mathrm{tot}}^{-1/3})$ for general irrational systems and $\mathcal{\tilde O}(N_{\mathrm{tot}}^{-1/2})$ for state-space systems.