## 4 Discussion  
Analyzing the results of the different frequency domain methods, the PSD, the CPSD and the MMSC, we can identify the frequencies corresponding to the first and second mode of the building. The CPSD and the MMSC lead to better results by taking advantage of simultaneous measurements. Table 3 shows a comparison of vibrating frequencies obtained with both on-site accelerometers and the non-invasive RAR technique. The first two fundamental frequencies obtained via RAR are virtually the same, with differences of less than 3% compared to those obtained with the accelerometer network. On the other hand, in this particular case study, it was possible to find the first two resonant frequencies with RAR sensing because the first and second modal shapes of the building have components in the LOS of the RAR. In order to achieve better understanding of the dynamic characteristics of buildings, measurements in both directions of the structures must be performed. It was not possible to find the third and further modes using the RAR, probably due to limitations in the displacement resolution of the device used in this study. An important requirement to obtain good dynamic displacement measurements with RAR is good reflectivity on the surface where the transmitted microwaves fall on. Usually, good SNRs in the Rbin \begin{table}
\begin{tabular}{c c c c c c} \hline \hline Mode & \multicolumn{2}{c}{ACC} & \multicolumn{2}{c}{RAR} & \multicolumn{2}{c}{Difference} \\ \hline  & f (Hz) & T (s) & f (Hz) & T (s) & (\%) \\ \hline
1 & 2.49 & 0.40 & 2.49 & 0.40 & 0.2 \\
2 & 2.79 & 0.36 & 2.71 & 0.37 & 2.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of the resonant frequencies and periods corresponding to the first and second mode, obtained with the contact method, using accelerometers, and the remote sensing technique, using RAR. Figure 20: (**a**) Displacement–time histories of the Rhins, from 60.25 to 60.65 s (0.4 s), corresponding to a period of the fundamental frequency of the structure (2.49 Hz). The colored circles represent five different time instants. (**b**) Displacement of the five time instants for each of the Rhins; the black lines correspond to the maximum and minimum displacements of the 0.4 s fragment. range profile indicate adequate quality of reflected signals. Surface roughness, geometry and shape of the structure and even the material of the buildings affect the quality of the back-scattered signal, but adequate signals correspond to the overall motion of the point where the signal back-scatters, whatever the quality of the materials of the wall. Before beginning to take the RAR profiles, it is important to analyze the SNR in the Rbin range profile, searching for adequate reflectors. In special cases, ad hoc artificial reflectors may be carefully installed at selected points of the structure to improve the SNR and the quality of the recordings [14,25]. That less clear results were obtained with the RAR technique may be due to the low amplitude of the displacement signals produced by environmental excitations in mid-rise buildings, compared to the acceleration signal amplitudes. The use of radar in structures with high fundamental frequencies may be limited due to the resolution of the RAR. This drawback, however, does not occur in tall or flexible structures where the radar has given excellent results [14,28]. More research into RAR limits and capabilities is necessary. The results confirm the capacity to find at least the fundamental frequency of the building using RAR in a straightforward way. However, uncertainty increases with the frequency complicating the detection of the higher resonant frequencies. Finally, the fundamental frequencies obtained herein, are consistent with the damage level of the structure. It is commonly known that structural damages cause changes in the dynamic response of the structure. Generally, the stiffness degradation induced by the damage lead to a decrease in the natural frequency.