## 2 Materials and Methods
### The Real-Aperture Radar (RAR)
#### 2.3.1 RAR Interferometry  
The RAR measures displacements by interferometry. The principle entails the measurement of the phase difference between the transmitted microwave signals and signals back reflected from the moving target building. Then, the radial displacement in the line of sight (LOS) of the RAR is calculated according to Equation (1).  
\[d=\frac{\lambda}{4\pi}(\varphi_{2}-\varphi_{1}) \tag{1}\]  
where \(\lambda\) is the wavelength and \(\varphi_{1}\) and \(\varphi_{2}\) are the phases acquired at the different times. The expected precision is in the order of tens of micrometers, when good phase stability is achieved with high SNR. A detailed description of the working principle of the RAR can be found in [13; 14].  
One main feature of the radar is that it can simultaneously capture the movement of different objects that are in the field of view (FOV) of the RAR antenna. Objects separated from each other by more than 0.5 m can be captured in different so-called Range bins (Rbin) [14]. In the case of buildings, the radar allows recording the displacement at different heights of the building, depending on the height of the building and on the measuring distance. However, the vibration of non-structural objects on the facades and any other objects in the FOV of the RAR can also be captured. Therefore, the interpretation of the time histories corresponding to each Rbin must be performed carefully. It is necessary to bear in mind that: (a) the structure must be spatially located within the range profile in order to select and study only the bins containing the structure displacement and, (b) the building displacement signal must be distinguished from the noise produced by non-structural elements and other objects within the RAR LOS.  
In order to help in this careful interpretation, the range profile expresses the quality of the received signals, in terms of SNR, as a function of the Rbin number. In this case, high SNR peaks, within the bins corresponding to the structure, indicate that the time history recorded corresponds well to the displacement of the building. Thus, the range profile is very useful in locating Rbins that more likely correspond to the true displacement of the structure.  
Figure 8 depicts a scheme of a typical RAR measurement: (**a**) the radar sends and back receives the signal from different parts of the building in different Rbins located at a distance \(D_{bin}\); (**b**) the amplitude of the radar echoes according to the different Rbins is plotted into a range profile in terms of SNR, allowing Rbin selection according to the range of the structure and the SNR; and (**c**) the interferometric phase of the selected Rbins is converted to a displacement-time recording using Equation (1) [25].  
Figure 8: Procedure to record the displacement recordings of a building: (**a**) RAR sensing; (**b**) Rbins selection; (**c**) displacement–time recording of the selected Rbins (see explanation above).  
It is required to record a displacement history long enough to obtain sufficient frequency resolution that allows investigating the buildings harmonics. In addition, filtering the signal also serves to set aside the vibration frequencies of objects that are not within the range of interest. Although the above precautions may be sufficient for high-rise buildings with long periods/very low frequencies, in the case of mid- and low-rise buildings with short periods/high frequencies, as discussed later in this article, the resonant frequency estimation by these traditional frequency domain techniques can be a difficult task. Therefore, it is necessary to explore different techniques for signal analysis. Several techniques such as multiple coherence are discussed below.