# Terahertz Frequency-Scaled Differential Imaging for Sub-6 GHz Vehicular Antenna Signature Analysis
## 2 The Imaging Process
### Formulation of the Total Image Reconstruction Process  
The ultrawide-band (UWB) near-field imaging radar system used for the measurement is a flat SAR system composed of a transmitter antenna (_Tx_) and a receiver antenna (_Rx_) with fixed spatial separation, named as \(X\) offset distance, being a monostatic configuration but with a bi-static angle. The _Tx_-_Rx_ system is fixed and explores the environment by moving the object in an _XY_ plane using a positioning table. The system points toward the target located at a distance \(r_{t}\) in the _Z_-axis, as shown in Figure 1.  
Figure 1 shows the measurement arrangements, depicting the _Tx_-_Rx_ and the object. The objective is to get the spatial distribution image of the electric contrast associated with the object under test. The electric contrast is written as \(c\!\left(\overset{\rightarrow}{r}\right)=\frac{\left(\varepsilon_{t}\!\left( \overset{\rightarrow}{r}\right)-\varepsilon_{b}\right)}{\varepsilon_{b}}\), with \(\varepsilon_{t}\!\left(\overset{\rightarrow}{r}\right)\) being the object's complex dielectric permittivity relative to its environment (background) constant value \(\varepsilon_{b}\!\left(\overset{\rightarrow}{r}\right)\). The imaging system generates an illuminating field with polarization in the _y_-axis (transversal) that is incident on the object under investigation, represented by a metallic car in Figure 1. \(\overset{\rightarrow}{E}\!\left(\overset{\rightarrow}{r},f;\overset{ \rightarrow}{r}_{t_{i}}\right)\) denotes the electric field generated in point \(\overset{\rightarrow}{r}_{t_{i}}\) by the _Tx_ antenna placed at \(\overset{\rightarrow}{r}_{t}\) and operating at frequency \(f\). \[\overset{\rightarrow}{J}_{eq}\!\left(\overset{\rightarrow}{r},f;\overset{ \rightarrow}{r}_{t_{i}}\right)=j\omega\varepsilon_{b}\,c\!\left(\overset{ \rightarrow}{r}\right)\overset{\rightarrow}{E}\!\left(\overset{ \rightarrow}{r},f;\overset{\rightarrow}{r}_{t_{i}}\right) \tag{1}\]  
The equivalent current \(\overset{\rightarrow}{J}_{eq}\) is the scattering field source generated by the object. Then, \(\overset{\rightarrow}{J}_{eq}\) is understood as a \"trace\" of the original object (image). Figure 1: Measurement arrangements to explore the object in an _XY_ plane using a positioning table. Figure 1 shows the \(Tx\) and \(Rx\) antennas fixed in position, and the object is moved using \(XY\) plane geometry located at a distance of \(\overset{\rightarrow}{r}_{t}\), on a set of positions \(\left(N_{x},N_{y}\right)\) across \(x\)-axis and \(y\)-axis, to gather a 2D array that includes monostatic measurements. For targets with highly conductive behavior (\(\sigma\gg\omega\varepsilon_{0}\varepsilon_{r}^{\prime}\)), like the metallic car used in this paper, the equivalent currents tend to produce the conduction current \(\overset{\rightarrow}{f}_{\alpha q}=\sigma\overset{\rightarrow}{E}\)[9]. The equivalent current \(\overset{\rightarrow}{f}_{\alpha q}\left(\overset{\rightarrow}{r}_{t},f; \overset{\rightarrow}{r}_{t_{t}}\right)\) generates a scattered field (\(\overset{\rightarrow}{E}_{s}\)) in the point \(\overset{\rightarrow}{r}_{t_{t}}\) of the object, which is measured by the \(Rx\) antenna at each point of measurement. \(\overset{\rightarrow}{E}_{s}\) has the following equation:  
\[\overset{\rightarrow}{E}_{s}\left(\overset{\rightarrow}{r}_{R_{i}},f; \overset{\rightarrow}{r}_{t_{i}}\right)=-j\omega\mu_{0}\int\limits_{V_{0}} \overset{\rightarrow}{f}_{\alpha q}\left(\overset{\rightarrow}{r}_{t},f; \overset{\rightarrow}{r}_{t_{i}}\right)\mathcal{G}\left(\left|\overset{ \rightarrow}{r}_{R_{i}}-\overset{\rightarrow}{r}_{t}\right|,\ f\right)dV \tag{2}\]  
where \(\omega\) is the angular frequency, \(\mu_{0}\) is the magnetic permeability in vacuum, \(\overset{\rightarrow}{f}_{\alpha q}\) is the equivalent current, and \(\mathcal{G}\left(\left|\overset{\rightarrow}{r}_{R_{i}}-\overset{\rightarrow} {r}_{t}\right|,\ f\right)\) is the Green's function that matches the geometry of the problem.