Recently time domain averaging, referred to as “cyclic averaging”, was investigated in [6], where it was shown that time domain averaging in the frequency domain can be viewed as equivalent to a comb filter being applied to data (FRFs). However, the paper did not analyze the effect of time domain averaging on the noise suppression of input and output noise, as the focus of the paper was on the bias error due to the limited frequency resolution. In [7] it was demonstrated that using time domain averaging, the bias due to contaminating noise on both input and output signals can be suppressed when using periodic excitation signals in both single-input/single-output (SISO) and multipleinput/multiple-output (MIMO) cases. The present paper aims at developing some more fundamental understanding of this method. It will be shown that asymptotically unbiased estimates can be obtained by using the well-known pseudo random or periodic chirp excitation signal for SISO measurements, and periodic random excitation for MIMO measurements. SISO ESTIMATOR Using time domain averaging in the case of a single input is relatively straight-forward. The excitation signal can be any broadband periodic signal, typically pseudorandom or periodic chirp [7]. The procedure is to turn the excitation signal on, and then wait a number of periods until the linear system is achieving its steady-state response. For moderately damped systems 5 to 7 periods is often considered sufficient, whereas for very low damping, it can be necessary to wait much longer. After this waiting time, a number of periods are acquired for further processing. In Fig. 1 a model is shown with contaminating noise on both input and output. It is well known that in the case of input noise m(t) existing when using an H1 estimator, a biased estimate will result. The H1 estimator is 1 ˆ ˆ ˆ yx yx xx G f H f G f (1) where Gyx and Gxx are the cross- and autospectral densities, respectively, between the measured input signal, x(t), and the measured output signal, y(t). Fig 1 Illustration of linear system H( f ) with the input and output signals contaminated by extraneous noise. The normalized bias error in the magnitude of 1Hyx, when input noise is present [1], is ˆ mm b yx vv mm G H G G (2) where v(t) is the “true” input signal going through the linear system, see Fig. 1. 300
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