*2.2. GB-RAR Basic Principles*

A ground-based real aperture radar (RAR) system is a stepped-frequency continuous-wave (SF-CW) radar that emits a continuous wave with different progressive frequencies within a given frequency band. The frequency bandwidth B provides the range resolution ΔR of the radar according to the relationship, Equation (2):

$$
\Delta \mathcal{R} = \frac{c}{2B} \tag{2}
$$

where c is the speed of electromagnetic waves in vacuum. The corresponding echoes, backscattered by the scene, give rise to the raw data. Table 2 summarizes the main technical specifications of the IBIS-F/L® GB-RAR, IDS S.p.a, Pisa, Italy, used in this experiment.

The inverse Fourier transform of the raw signal acquired by the radar, normalized and converted to a logarithmic scale, provides the normalized radar cross-section (NRCS) profile, which shows the amplitude of the radar signal backscattered by targets located within the scene, discriminating them with the range resolution (2).

The interferometric phase is computed as follows:

$$
\Delta\!\!\!\!\/\!\/\/\_2 = \text{atan}\!\!\!\/\text{S}\_2 \cdot \text{con}\!\!\/(\text{S}\_1)\!\!\/\text{(}\!\!\/^\text{(}\text{S}\_1)\!\!\/)\!\!\/^\text{(}\text{con}\!\!\/^\text{(}\text{S}\_1)\!\!\/^\text{(}\text{con}\!\!\/^\text{(}\text{)}\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\!\/^\text{(}\text{)}\!\/^\text{(}\text{)}\!\/^\text{(}\text{)}\!\/^\text{($$

where S1 and S2 are two coherent complex-values radar data acquired at times t1 and t2, respectively, during the data acquisition. The line-of-sight (LoS) displacement D1,2 of a point P, occurring in the time interval [t1, t2], is related to the interferometric phase <sup>Δ</sup>ϕ1,2, computed in Equation (3), by the relationship:

$$D\_{1,2} = \frac{\lambda}{4\pi} \Delta \varphi\_{1,2} \tag{4}$$

where λ is the radar wavelength. The precision of displacement measurements depends on the precision of phase measurements and it is a fraction of millimeter.

In the case of RAR interferometric applications, the radar acquires data with a sampling time in the order of a fraction of second, usually of a few milliseconds. This means that the radar can accurately track in time the deformation profile of the target. The vibration frequency spectrum is obtained by a spectral analysis of the displacement profile, computed in Equation (4). The vibration frequency spectra can be visualized as 2D maps. For each target, discriminated in range with a range resolution ΔR, the frequency spectra and displacement profiles are displayed vs. frequency and time, respectively.

**Table 2.** Technical Specifications of IBIS-F/L GB-RAR.

