The principle of time domain averaging is that we make the excitation signal periodic in the time window of the DFT. We thus let the input signal x(t) be N-periodic, i.e. ( ) ( ) N N x t N x t , which means the measured input signal can be written as ( ) ( ) ( ) N x t x t m t (3) where xN(t) is the “pure” excitation signal with variance 2 Var ( ) N xN x t (4) and m(t) is a zero-mean random signal with variance 2 Var ( ) m m t . (5) Assuming K time averages are used to produce the time-averaged input signal, xa(t), the ensemble average of x(t) is ( ) E ( ) ( ) ( ) a N x t x t x t m t (6) where the variance of the average signal ( ) m t approaches zero at the rate 1/ K. The spectral density of xa, Gaa, is then 1 ( ) N N aa x x mm G f G G K . (7) The bias error of the FRF estimate in Eq. (2) is thus reduced by the time domain averaging using K averages to ˆ mm b yx mm vv G K H G G K (8) which means that the bias error due to the input noise m(t) is asymptotically eliminated by the averaging process. A similar treatment can be applied to the output signal in the case of output noise, n(t), see Fig. 1. To achieve maximum suppression of both input and output noise, all time data should be used for time domain averaging. In that case the FRF can be estimated as the simple ratio of the two spectra of the time averaged signals. We can denote this estimator ˆ ( ) t H f , or the Ht estimator, and it is computed by ˆ ˆ ( ) ˆ t a a Y H f X (9) where ˆ ( ) DFT ( ) a a Y f y t and ˆ ( ) DFT ( ) a a X f x t . When time domain averaging as proposed here is applied, the interpretation of the coherence function becomes somewhat more complicated than in the frequency domain averaging case. First of all, the coherence function has to be computed by using frequency domain averaging, on the raw data. In principle, the coherence function is a measure of the amount of contaminating noise, if we assume that the system is linear and no time delays exist. For measurements on linear mechanical systems, we can assume that around resonances and antiresonances, the dominating contaminating noise will be in the “small” signal (force signal at resonance, and response signal at antiresonance). At each of these frequencies, the coherence function then 301
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