# Finite Sample Frequency Domain Identification

arXiv: https://arxiv.org/abs/2404.01100 · HTML: https://mlsanity.com/paper/2404.01100/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 32 references that are in the map.
- First posted on arXiv 1 Apr 2024.

## Builds on (most cited first, 11 of 11)

- [Learning Without Mixing: Towards A Sharp Analysis of Linear System Identification](https://mlsanity.com/paper/1802.08334/) (2018)
- [Finite Sample Analysis of Stochastic System Identification](https://mlsanity.com/paper/1903.09122/) (2019)
- [Finite Time Identification in Unstable Linear Systems](https://mlsanity.com/paper/1710.01852/) (2017)
- [Statistical Learning Theory for Control: A Finite Sample Perspective](https://mlsanity.com/paper/2209.05423/) (2022)
- [Active Learning for Identification of Linear Dynamical Systems](https://mlsanity.com/paper/2002.00495/) (2020)
- [Learning with little mixing](https://mlsanity.com/paper/2206.08269/) (2022)
- [Single Trajectory Nonparametric Learning of Nonlinear Dynamics](https://mlsanity.com/paper/2202.08311/) (2022)
- [A Tutorial on the Non-Asymptotic Theory of System Identification](https://mlsanity.com/paper/2309.03873/) (2023)
- [Learning from many trajectories](https://mlsanity.com/paper/2203.17193/) (2022)
- [Minimax Lower Bounds for H_∞-Norm Estimation](https://mlsanity.com/paper/1809.10855/) (2018)
- [Frequency Domain Gaussian Process Models for H^∞ Uncertainties](https://mlsanity.com/paper/2211.15923/) (2022)

## Built on by (most cited first, 1 of 1)

- [Frequency Response Identification of Low-Order Systems: Finite-Sample Analysis](https://mlsanity.com/paper/2508.17142/) (2025)

## Abstract (arXiv)

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 O((du+√dudy)√M/Ntot), where Ntot is the total number of samples, M is the number of desired frequencies, and du,,dy 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 Ntot-1/3 finite-sample rate for learning the frequency response over all frequencies in the  H∞ 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.