Formula for Estimating the Frequency Response of LTI Systems From Noisy Finite-Length Datasets

In this letter, we revisit the classical problem of estimating the frequency response of an LTI system from noisy, non-periodic input-output data. Existing solutions fall into two categories: indirect methods, which compute the frequency response using identified models, and direct methods, which estimate the response directly from data. Direct methods bypass system identification, but have challenges when applied to noisy, finite-length datasets with unknown initial conditions. This letter proposes a new direct method that addresses these challenges, and provides an explicit formula for computing the frequency response. To develop this method, this letter leverages ideas from behavioral system theory and poses the problem as an optimization problem, whose objective is to minimize the projection of the solution onto the nullspace of an input-output data matrix. This letter also offers an alternative derivation of the formula, based on identifying an ARX model, thereby bridging the gap with classical indirect approaches. The proposed method is applied to experimental data collected from a DC motor, where we show that the proposed method outperforms other direct approaches based on Fourier transforms and low-rank approximations, and performs equally as good as direct subspace identification methods, even though no model class is prescribed.