Affiliations: School of Engineering, Blekinge Institute of
Technology, Karlskrona, Sweden | Department of Functional Product Development,
Luleå University of Technology, Luleå, Sweden
Note: [] Corresponding author: Andreas Josefsson, School of Engineering,
Blekinge Institute of Technology, SE-371 79 Karlskrona, Sweden. Tel.: +46 730
336306; E-mail: andreas.josefsson@bth.se
Abstract: Frequency response functions are often utilized to characterize a
system's dynamic response. For a wide range of engineering applications, it is
desirable to determine frequency response functions for a system under
stochastic excitation. In practice, the measurement data is contaminated by
noise and some form of averaging is needed in order to obtain a consistent
estimator. With Welch's method, the discrete Fourier transform is used and the
data is segmented into smaller blocks so that averaging can be performed when
estimating the spectrum. However, this segmentation introduces leakage effects.
As a result, the estimated frequency response function suffers from both
systematic (bias) and random errors due to leakage. In this paper the bias
error in the H_1 and H_2-estimate is
studied and a new method is proposed to derive an approximate expression for
the relative bias error at the resonance frequency with different window
functions. The method is based on using a sum of real exponentials to describe
the window's deterministic autocorrelation function. Simple expressions are
derived for a rectangular window and a Hanning window. The theoretical
expressions are verified with numerical simulations and a very good agreement
is found between the results from the proposed bias expressions and the
empirical results.
Keywords: Frequency response functions, bias error, leakage effects, Welch's method
