0.20 20.00 Hz -40.00 80.00 dB mm 2 /Hz F PSD FRLE:+Z F PSD FRRI:+Z F PSD RELE:+Z F PSD RERI:+Z (a) 0.20 20.00 Hz -40.00 80.00 dB mm 2 /Hz F PSD FRLE:+Z F PSD FRRI:+Z F PSD RELE:+Z F PSD RERI:+Z (b) Fig.4 The road profile roughness PSD of (a) the Blue-kay track (60 km/h) and of (b) the Florida track (80 km/h) 1 0 1 N T i m i m m R y y N − + = = ∑ (1) where N is the total number of samples and i the correlation sample index (also called time lag). A typical cross-correlation sequence is shown in Fig.5b. The fact that the correlations of a structure excited by white noise are similar to impulse responses is the basis for correlation-driven Stochastic Subspace Identification [8]. Frequency-domain methods require cross-spectra as primary data. As non-parametric spectrum estimate, the so-called weighted correlogram can be used. It is computed as the DFT of the weighted estimated correlation matrix (1): ( ) j j e L m t yy m m m L S w R ω ω − ∆ =− = ∑ (2) where L is the maximum number of time lags at which the correlations are estimated and mw denotes the time window. This number is typically much smaller than the number of data samples to avoid the greater statistical variance associated with the higher lags of the correlation estimates. As the correlation samples at negative time lags ( 0 m< ) contain redundant information, it suffices to consider only the positive time lags when computing the spectra. This lead to so-called half spectra of which even the auto spectra have a phase different from zero: ( ) j 0 0 1 j e 2 L m t yy m m m w R S w R ω ω + − ∆ = = +∑ (3) A more traditional non-parametric spectrum estimate is the so-called weighted averaged periodogram (also known as modified Welch’s periodogram). The advantage of the described correlogram approach (3) is that the use of a Hanning window can be avoided. A Hanning window introduces a bias on the damping estimates. Instead, just like in impact testing, an exponential window can be applied to the correlation functions before computing the DFT. 3.2 Pre-processing of the operational data In Fig.5a the vertical acceleration response signal at the driver seat rail location is shown in the time domain. Since most of the modal parameter estimation methods do not directly use the raw acceleration measurements, but rely on reduced data such as cross-correlations and cross-spectra between signals measured simultaneously at different locations, the correlation or covariance matrix l l iR × ∈ℝ between the measured output signals l my ∈ℝ , with l the number of outputs and m the sample index, has to be firstly estimated as 317
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