Unbiased Estimation of Frequency Response in the Presence of Input and Output Noise Anders Brandt, Associate Professor Dept. of Industrial and Civil Engineering, University of Southern Denmark Niels Bohrs Allé 1, DK-5230, Odense M, Denmark ABSTRACT Many attempts have been made on finding a frequency response estimator which minimizes the bias error in cases where both the input and output signals of a linear system are contaminated by extraneous noise, for example, Hv and Hs. It is wellknown that these estimators only minimize the bias error if the input and output extraneous noise spectra are known, which they are normally not. This paper describes how time domain averaging (cyclic averaging) of periodic excitation signals can be used to eliminate the bias due to both input and output extraneous noise. It is demonstrated by simulation results that asymptotically unbiased estimates of frequency response functions can be obtained by using time domain averaging and periodic random noise. Examples are given of both single-input/single-output (SISO) and of multiple-input/multiple-output (MIMO) systems. The fact that periodic excitation signals in this way can be used to eliminate the bias error in FRF estimates does not seem to have been recognized previously. INTRODUCTION Estimating frequency response functions (FRFs) in mechanical measurements is an important tool. The main established theory for estimators commonly used was established in early editions of [1], most notably the H1 estimator which minimizes contaminating noise on the output. It is well-known that the H1 estimator is biased if contaminating noise exists in the measured input signal. This situation is not uncommon when using shaker excitation on weak structures. The H2 estimator, which minimizes noise on the input, on the other hand, is biased if contaminating noise exists in the measured output signal. The H2 estimator, furthermore, is only available for multiple-input/multiple-output system identification in the special case where the number of inputs equals the number of outputs, which is rarely the case. Early attempts to minimize noise on both input and output simultaneously led to the Hs estimator and the special case of this estimator called Hv, [2, 3], but these estimators have the limitation that knowledge of at least the ratio of the input and output contaminating noise is required to produce optimum estimates, which is rarely available in practice. Recently, [4], the Hv estimator has been shown to be a special case of a maximum likelihood estimator, but the requirement of knowing the relation between input and output contaminating noise remains. In [5], an unbiased estimator, the H estimator, based on special, cyclostationary excitation signals was presented. This is a promising estimator, particularly since the commonly used burst random excitation signal was shown to be useable as a cyclostationary excitation signal. T. Proulx (ed.), Modal Analysis Topics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 6, 299 DOI 10.1007/978-1-4419-9299-4_26, © The Society for Experimental Mechanics, Inc. 2011
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