On consistency for the estimator of impulse response function
DOI:
https://doi.org/10.17721/2706-9699.2026.1.03Keywords:
impulse response function, cross-correlogram, consistency, Fourier seriesAbstract
The aim of the article is to construct and analyze a statistical estimator for the real-valued impulse response function (IRF) of a time-invariant continuous linear system, and to establish its properties of asymptotic unbiasedness and consistency. The estimation of the response function is performed using a sample inputoutput cross-correlogram approach. The input signal is modeled as a stationary zero-mean Gaussian stochastic process, represented as a trimmed Fourier series using a trigonometric orthonormal basis. The study employs functional analysis and probability theory to derive upper bounds for the mathematical expectation and variance of the proposed estimator. The study establishes that the proposed integral cross-correlogram estimator is asymptotically unbiased as the cutoff level N approaches infinity. Furthermore, by evaluating the upper bounds of the bias and variance, it is proven that the estimator is consistent in the mean square sense as both the cutoff level N and the averaging interval length T approach infinity. The proposed approach offers a rigorous mathematical framework for system identification applicable to signal processing, automatic control, econometrics, and oceanology. The use of a Fourier series representation makes this method particularly effective for analyzing linear systems driven by periodic or quasi-periodic input signals.
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