# An Improved Circumferential Fourier Fit (CFF) Method for Blade Tip Timing Measurements
## 3 Method Evaluation Using Simulated Data
### Simulated Test
However, the relative error of amplitude identification was higher than that of Test 1. For the same natural frequency, the larger EO, the higher degree of under-sampling. That means the number of samples was less for a period of vibration of the blade. The amplitude that recovered was more sensitive to the number of samples than the EO identification. Although the relative error was increased for both methods, the quality of I  
Figure 7: Results of Test 3. (**a**) 95% confidence intervals of the EO identification and (**b**) 95% confidence intervals of the relative error of the amplitude identification. \(\bullet\) Test 5:  
Increasing the number of probes made the scatter of EO identification get larger for ICFF, which was shown in Figure 9a. Meanwhile, the relative error of amplitude identification was increased for both methods, which was shown in Figure 9b. The summated squared noise term in the **B\({}^{\text{T}}\)B** matrix would increase with the number of probes. Then the squared noise term was transferred to amplitude identification. Figure 9b still showed the relative error of ICFF was smaller than TCFF. Figure 8: Results of Test 4. (**a**) 95% confidence intervals of the EO identification and (**b**) 95% confidence intervals of the relative error of the amplitude identification. Figure 9: Results of Test 5. (**a**) 95% confidence intervals of the EO identification and (**b**) 95% confidence intervals of the relative error of the amplitude identification. The results of the EO identification that were less than 80 percent were marked in red in Table 3. Curve fits of lower PSR data are generally not conducive to parameter identification, especially for the EO identification, which was demonstrated by Test 2. Increasing the number of probes would have some negative effect on the EO identification for ICFF, and the degree of influence became significantly larger when the data noise up to 0.20 (Test 5). The higher PSR value and the larger amount of fitted data were conducive to EO identification for both methods. The higher EO value (higher degrees of under-sampling) and coupling between mistuning blades (unconsidering multi-frequency vibration) had little effect on the results of EO identification. Through the comparison from Test 1, Test 3, Test 4 and Test 6, ICFF performance was comparable to TCFF in terms of EO identification. The relative error for amplitude identification increased with the number of fitted data, which was verified by the results of Test 3. Higher EO value has a more negative effect on the amplitude identification of TCFF than that of ICFF (Test 4). Table 4 showed that the results of amplitude identification of ICFF were more accurate than TCFF, except that the low PSR (Test 2) resulted in the little difference between ICFF and TCFF. The larger number of fitted data and the number of probes would lead to a higher error of amplitude identification for both methods, which was caused by the squared noise term in the \(\mathbf{B}^{\text{T}}\mathbf{B}\) matrix that increased with the number of fitted data or probes. Compared with TCFF, ICFF considers the influence of blade vibration on the ToA of the blade tip. This improvement is more consistent with reality and leads to more accuracy of the amplitude identification. This was demonstrated through all of the simulated tests. Conclusions drawn above were based only on the numerical simulations but not based on the analysis of real signals measured during the experimental tests for a real object under loading conditions. In the experimental tests, different factors may affect the accuracy of the reconstruction method. So, the proposed method will be verified by experimental measurements and analysis in the next section.