## 2 Materials and Methods
### System Calibration  
Before using the received signals for image reconstruction, they must be properly calibrated. This operation is fundamental, since in this configuration the proximity between radiators provides very large mutual coupling, masking the contribution of all useful but very weak signals reflected by the illuminated scenario.  
This unwanted signal is usually referred to as \"crosstalk\" and it is easily removed by subtracting the measurement done without the target to the one acquired with the target. In this way, both the antenna coupling and the environment multiple reflections are largely suppressed. In addition, since the imaging procedure makes use of both amplitude and phase information to reconstruct the final image, it is fundamental to properly calibrate the  
Figure 4: Transmit/receive apodization matrix.  
Figure 3: Schema of data acquisition. In this simplified schema realized for explanation purposes, 15 antenna positions are considered (instead of the 28 positions of the real array). Red and blue dots represent transmitting and receiving antennas, respectively.  
phase of the received signal too. In fact, this is influenced by several factors (propagation in the waveguides, transitions, spurious backscattering, etc.). To this aim, a known calibration target is used, that is a metallic plate. This is a good standard, since its backscattering matrix is analytically known, and the corresponding phase delay is directly related to its distance from the antennas.  
Therefore, according to [30], the complete calibration procedure is as follows. A first measurement (\(M_{plate}\)) with the metallic plate at a given distance in front of the array is done and the corresponding scattering matrix is calibrated by the crosstalk contribution (i.e., measurement with the plate minus measurement without the plate (\(M_{match}\))). The delay matrix corresponding to an ideal metallic plate at the same distance is analytically computed (\(M_{analysis}\)) [31]; the elements of this matrix are complex numbers given by \(e^{-j2\pi f\tau}\), where \(f\) is the frequency and \(\tau\) is the time delay. After this, the calibration of the whole imaging systems, made by cables, transitions and antennas is achieved by means of the ratio between these two matrixes. Then, the measurement of the target (\(M_{target}\)) calibrated by the crosstalk is multiplied by this correction matrix and the result is used in the image reconstruction procedure. The complete calibration procedure (\(M_{obj}\)) can be summarized by the following formula:  
\[M_{obj}=\left(M_{target}-M_{match}\right)/\left(M_{plate}-M_{match}\right) \cdot M_{analytics} \tag{1}\]  
This approach is valid in general and no further steps are required for target detection in air. On the other hand, for target detection within realistic breast phantoms, an additional step must be carried out. In this case, the calibration procedure must also consider the artifacts due to the strong reflection at the air-phantom interface. This contribution is generally very high and masks the following useful signal contribution that contains information about tumor presence. The use of a proper coupling medium could reduce it. However, due to the difficult maintenance of the system submerged in oil and problems related to the sterilization between repeated applications on different patients (in a potential clinical environment), as well as the impossibility of using the abovementioned calibration procedure with a coupling liquid, in this work, no coupling medium was employed.  
To solve this problem, the time-gating approach is adopted; therefore, the image is reconstructed starting from the end of the interface reflection in time and space (therefore all the experimental images in phantoms do not start from z = 0 mm).