Bounding the Polynomial Approximation Errors of Frequency Response Functions

Abstract

Frequency response function (FRF) measurements take a central place in the instrumentation and measurement field because many measurement problems boil down to the characterization of a linear dynamic behavior. The major problems to be faced are leakage and noise errors. The local polynomial method (LPM) was recently presented as a superior method to reduce the leakage errors with several orders of magnitude while the noise sensitivity remained the same as that of the classical windowing methods. The kernel idea of the LPM is a local polynomial approximation of the FRF and the leakage errors in a small-frequency band around the frequency where the FRF is estimated. Polynomial approximation of FRFs is also present in other measurement and design problems. For that reason, it is important to have a good understanding of the factors that influence the polynomial approximation errors. This article presents a full analysis of this problem and delivers a rule of thumb that can be easily applied in practice to deliver an upper bound on the approximation error of FRFs. It is shown that the approximation error for lowly damped systems is bounded by ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LPM</sub> / <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3dB</sub> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R + 2</sup> with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LPM</sub> the local bandwidth of the LPM, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> the degree of the local polynomial that is selected to be even (user choices), and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dB</i> the 3 dB bandwidth of the resonance, which is a system property.