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 theH1andH2-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.
Highlights
Estimation of Frequency Response Functions (FRFs) from experimental data is commonly carried out as a first step in the system identification process
Calculation of frequency-response-functions (FRFs) from finite data sequences often requires a segmentation of the data (Welch’s method) in order to obtain a consistent estimator when contaminating measurement noise is present
It is necessary to trade between the number of segments and the length on each segment in order to find an acceptable compromise between the bias error and the random error due to leakage noise and the random errors due to measurement noise in the estimated transfer function
Summary
Estimation of Frequency Response Functions (FRFs) from experimental data is commonly carried out as a first step in the system identification process. This paper is concerned with the frequency-domain approach known as Welch’s method, which uses the discrete Fourier transform and achieve averaging by segmenting the data as initially proposed in [1]. This is the most common estimator implemented in signal analysis software due to its speed and low memory requirement. It is well known that this method introduces leakage errors in the spectral densities since only a finite measurement time is studied. In [8,9] an approximate expression for the bias error at the resonance frequency was derived for a single-degree-of-freedom system with white noise input. The accuracy of the proposed expressions is examined with numerical simulations
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