## 3 Results and Discussions  
Figure 3 shows the averaged waveform collected by the simulations performed with a 3 cm deep target at 1 GHz with a \(3\times 3\) antenna grid. This is the average that was used to calibrate the waveforms. Since the receiver was located at exactly the same positions as the voltage source, all the recordings contained the main excitation pulse which is the only waveform appreciable in the figure. Every backscattering would be orders of magnitude smaller than this signal. The calibrated waveforms for the simulation set of Figure 3 are reported in Figure 4. The waveforms are divided according to at which antenna they were recorded. A schematic of the antenna grid with the antenna position highlighted in gray is displayed at the top-right corner of every graph. Figure 4a shows the antenna in the corner positions, (b) and (c) those in symmetric position around the center, while (d) shows the calibrated waveform from the central antenna (see caption for details). It is noteworthy that the waveforms are not overlapping exactly inside the same group because of the finite space discretization level of the simulations. The triangle, the square, and circle in the middle of the graph indicate the expected time of some events. The triangle represents the central time of the excitation pulse. As it can be seen in Figure 3, at 1 ns the pulse had its zero-crossing middle point which can be used as reference time. In fact, this point is the same in all graphs of Figure 4. The second event, highlighted by a square, represents the return time of the echo from the first target interface. This time was calculated considering the average propagation speed of the electromagnetic wave in the homogeneous lung tissue and a straight line from the antennas to the target. This distance was the shortest for the central antenna (Figure 4d) and the longest for the antennas in the corner position (Figure 4a). In Figure 4b,c this return time is the same. The last event, highlighted by a circle, corresponds to the return time of the echo coming from the second interface of the target, which is when the electromagnetic wave traveled past through the target in a straight line. In this case also the target material was considered to calculate the propagation speed. Figure 3: Averaged waveform for a 3 cm deep target at 1 GHz central frequency and a \(3\times 3\) antenna grid. It can be seen that the waveforms are relatively flat at the positions where Figure 3 has the maximum intensity, showing that the calibration procedure of the DAS algorithm indeed helps in enhancing the backscattered signals. In all graphs, there are some modulations at the time corresponding to the two target echoes (square and circle). These modulations are the strongest for Figure 4a and the weakest for (**d**). In fact, the antennas at the corners had the largest distance from the target, while that in the center the shortest. As it can be seen from Figure 1b, the lung tissue shows a non-negligible attenuation in all the frequency range which sums up with the standard attenuation experienced by a radial propagating wave. Notice that there are other signals before the first target echo. These were produced by the bouncing of the electromagnetic wave in the half-space on top of the lung tissue. In fact, although PML boundaries were enforced, a fraction of the electromagnetic wave was not totally absorbed at the end of the computation domain and was reflected back toward the receiver. This is similar to a real case scenario where the materials surrounding or housing the antennas are not totally absorbing the radiation. These signals contribute to increasing the clutter, amount of undesired intensity, in the confocal maps. They are the strongest in Figure 4a). In this case, in fact, once hitting the target the electromagnetic pulse was, in addition, reflected vertically toward the lung-air interface which was considerably closer than the receiver.