## 3 Numerical Simulation
### Computation Time  
Time consumption for computation is another issue in the field application; actually, it is even more crucial if SHM systems are designed for a real-time or online application. To ensure the computation time, the frame responses are successively down-sampled to several sampling rates, such as 500, 250, 200, 100, 50, and 40 Hz, as the measurement. Two scenarios, fixed \(i\) and fixed \(C_{E}\), are studied and RSI is individually conducted using the same input, output, and parameters in each scenario. Moreover, all the analyses are carried out in the same computer environment for comparison. For the hardware environment, the CPU is Intel(R) Core(TM) i7-6700K and the RAM is 32.0 GB. For the software environment, OS is 64 bit Windows and the platform is MATLAB R2021a [42]. The detailed parameters and the results of computation time are listed in Tables 2 and 3.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Measurement** & \multicolumn{5}{c}{**2 nd, 4 th, 6 th, and 8 th Floor**} \\ \hline \(i\) & \multicolumn{5}{c}{20} \\ \hline \(C_{MAC}\) & \multicolumn{5}{c}{0.98} \\ \hline Sampling Rate (Hz) & 500 & 250 & 200 & 100 & 50 & 40 \\ \hline Total Computation Time (seconds) & 214.77 & 100.89 & 80.58 & 36.23 & 15.97 & 12.03 \\ \hline Averaged Computation Time (ms) & 10.96 & 10.51 & 10.6 & 10.06 & 9.97 & 10.01 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The computation time over various sampling rates and a fixed \(i\).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Measurement** & \multicolumn{5}{c}{**2 nd, 4 th, 6 th, and 8 th Floor**} \\ \hline \(C_{E}\) & \multicolumn{5}{c}{around 4} \\ \hline \(C_{MAC}\) & \multicolumn{5}{c}{0.98} \\ \hline \(i\) & 100 & 50 & 40 & 20 & 10 & 8 \\ \hline Sampling Rate (Hz) & 500 & 250 & 200 & 100 & 50 & 40 \\ \hline Total Computation Time (seconds) & 5763.64 & 574.62 & 268.68 & 36.23 & 7.79 & 4.98 \\ \hline Averaged Computation Time (ms) & 320.17 & 63.83 & 37.31 & 10.06 & 4.32 & 3.45 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The computation time over various sampling rates and a fixed \(\alpha\).  
Figure 6: The identified system poles with and without MAC criterion under high noise level. (**a**) without MAC criterion; (**b**) with MAC criterion.  
In Table 2, as expected, the averaged computation time, \(t_{c}\), per step is quite similar; however, it exponentially increases if \(C_{E}\) is considered in Table 3. Doubtless, too frequent identification may bring no advantage to SHM systems for infrastructures. A ratio is introduced to represent the percentage spent on computation in each step as  
\[r_{c}=\frac{t_{c}}{\Delta t}\times 100\% \tag{31}\]  
The ratios, \(r_{c}\), are shown in Figure 7 and must be smaller than 100% to guarantee real-time or online processing. Figure 7a clearly reveals the exponential increase in the second scenario and the averaged computation time surpasses the result from the fixed \(i\) around 112 Hz. Furthermore, according to Figure 7b, the sampling ratio should be smaller than 92 Hz so that the proposed recursive formulation can be finished within each sampling interval. Although the computation time varies with different computer environments, one can conclude that down-sampling processing is generally needed for a real-time or online application. Otherwise, the proposed method may not complete the computation in each step because SHM systems commonly have a sampling rate from 100 Hz to 200 Hz [43; 44; 45]. One can note that on the other hand, the sampling rate needs to be large enough to see those significant frequencies. In some literature, 50 Hz or 60 Hz is a good suggestion for the sampling rate [46; 47; 48; 49].