## 7 Simulation Results  
An example of the spatial-average power density (sPD) over \(1\,\mathrm{cm}^{2}\) resulting from a FT to \(50\,\mathrm{mm}\) distance in the case without noise (Figure 5a,e) is depicted in Figure 6. It shows that the field distribution is reconstructed with only minimal distortions. When noise is added to the emulated measurements-in this example, at a level of \(-24\,\mathrm{dB}\) as described in Section 6 (Figure 5c,g)--the field gets distorted visibly, as can be observed in Figure 7; the error in the psPD\({}_{tot}\) in this example is \(-0.08\,\mathrm{dB}\) without noise and \(-0.53\,\mathrm{dB}\) with \(-24\,\mathrm{dB}\) noise. The quantitative results for all antennas are shown in Figure 8. For all antennas operating at \(30\,\mathrm{GHz}\) and above, up to a noise level of \(-24\,\mathrm{dB}\), the mean absolute error in the psPD flux crossing surface A, psPD\({}_{n}\), is below \(0.61\,\mathrm{dB}\) for all distances, ranging from very close (\(2.1\,\mathrm{mm}\), corresponding to \(100\,\mathrm{\SIUnitSymbolMicro m}\) or \(\lambda/100\) at \(30\,\mathrm{GHz}\) distant from the plane of equivalent currents) to far (\(150\,\mathrm{mm}\), or \(45\,\lambda\) for \(90\,\mathrm{GHz}\)). The average absolute error over all antennas is shown in Figure 9. It can be seen that, for small distances (\(2.1\), \(2.5\) and \(5\,\mathrm{mm}\)), the results are similar for the different noise levels up to \(-21\,\mathrm{dB}\). For high noise levels, the error increases, especially for larger distances (\(50\,\mathrm{mm}\) and \(150\,\mathrm{mm}\)). This may be attributed to phase reconstruction errors, which become more apparent at larger distances. For distances extremely close to the plane of equivalent currents, as seen for \(2.1\,\mathrm{mm}\), the error in the spatial-averaged norm of the Poynting vector psPD\({}_{tot}\) is larger than psPD\({}_{n}\), up to approximately \(0.8\,\mathrm{dB}\) (for \(30\,\mathrm{GHz}\) and above), even for very low noise levels. This discrepancy (also visible in Figure 8) is  
Figure 5: Example of first (**a**–**d**) and third (**e**–**h**) planes used as inputs of the algorithm at different noise levels for the \(30\,\mathrm{GHz}\) dipole array. Figure 6: Example simulation and FT of the spatial-average power density (sPD, over a \(1\,\mathrm{cm}^{2}\) circle) at \(z=50\,\mathrm{mm}\) without noise for the dipole array at \(30\,\mathrm{GHz}\). likely due to the reactive components that contribute to psPD\({}_{tot}\); they are more difficult to reconstruct than the mostly propagating components contained in psPD\({}_{n}\). From 5 mm and larger distances, no substantial difference can be observed between the two error metrics. The marginally higher error for the no noise case compared with the \(-\)30 dB noise case shows that there are other error sources, such as errors in the phase reconstruction algorithm and numerical errors originating from the expansion of magnetic current in terms of step functions. Figure 8: Mean absolute error in psPD\({}_{n}\) (**a**) and psPD\({}_{tot}\) (**b**) for different noise levels (\(y\)-axis of each figure) and for different distances. The plane of equivalent currents was placed at 2 mm for all cases. Mean values and standard deviations over 20 simulations per noise level are shown. Figure 7: Example simulation and FT of the sPD (over a 1 cm\({}^{2}\) circle) at \(z=50\) mm with a noise level of \(-\)24 dB for the dipole array at 30 GHz. It is apparent that the errors for the antennas operating at 10 GHz are relatively high. This can be explained by the very close distance of the plane of equivalent currents \(z_{1}\), 2 mm from the AUT, which corresponds to \(\lambda/30\). To investigate this in more detail and to illustrate how the problem is resolved at greater distances, different distances of \(z_{1}\), i.e., 2 mm, 4 mm and 6 mm, were investigated. These correspond to the same electrical distance as the measurement of a 10 GHz, 20 GHz or 30 GHz antenna at 2 mm distance.