( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) YY YY YX YX XY XY XX XX Z D Z D Z D Z D (9) and the Fourier transform of Eq. (9) defines the corresponding FRF estimates ( ) ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( ) ˆ ( ) ˆ 2 1 YX YY XX XY Z Z H Z Z H (10) COMPARING RD WITH TRADITIONAL FRF ESTIMATES In the following a 2 degree-of-freedom system is loaded by Gaussian white noise, the response is simulated using the theoretical FRF matrix and FFT, and finally the FRF’s are estimated by the traditional technique (in the following denoted “Welch”) and by RD functions. All simulations were made using 1,048,576 data points, and the damping, , and the data segment length, N, were varied. The data segment length is also used as the total size of the buffer window used for estimation of RD functions. An example of RD functions is shown in Figure 1. As it appears, the window buffer is not defined to be symmetrical around the triggering point (where the spike is located in ( ) XX D ). The reason is that the buffering window should be defined in order to include maximum information, and since the function ( ) YX D only has system information for negative times, the buffer is defined in order to include the RD functions for negative time lags. The corresponding FRF’s, estimated as ( ) ˆ 1 H according to Eqs. (2) and (10) for both techniques, are shown in Figure 2. It can be seen that at the resonance peak, the bias error in the RD estimate is significantly smaller than the bias error in the Welch estimate. The full band errors 0 are quantified by calculating the RMS value of the difference ( ) ˆ ( ) 1 H H , i.e. this error measure is the sum of the bias and random error. The narrow band error 1 is calculated as the maximum difference value, at the peak. Typical errors are shown in Tables 1 and 2. It appears that the random error of the estimates based on the RD technique is larger than for Welch estimates, which can be seen in larger full band errors in Table 1. The bias errors at the resonance peak are however considerably smaller in the RD estimates than in the Welch estimates. This is to be expected, as the trade-off between bias and random error is applicable also to RD functions. The noise properties of RD functions are, however, different from the noise properties of the more well-known correlation functions, and this will be further analyzed in future work. Table 1. Full band error, 0 0.2% 1% N 256 512 1024 2048 4096 256 512 1024 2048 4096 Welch 0,60 0,57 0,50 0,23 0,09 0,18 0,08 0,02 0,01 0,02 RD 0,85 1,06 1,76 1,77 1,82 0,77 0,81 0,82 0,81 0,82 500
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