# HERAS: A Modular Matlab Tool Using Physical Optics for the Analysis of Reflector Antennas
## 4 Applications of the Tool
### Efficient Estimation of an Array Fed Reflector Antenna with a Large Number of Feeds
\[\overline{f}_{b}=W\overline{f}_{f} \tag{4}\]  
The numerical error resulting from the use of the acceleration techniques is shown in Figure 10. An AFR with a reflector of 3 m aperture in offset configuration fed by an array with a magnification factor of 1.6 is used, leading to a similar peak gain of the SFPB configuration discussed in the previous sub-section. The same coverage definition of [56] is applied, so a direct comparison with the SFPB scenario can be obtained. This AFR architecture has higher side lobes, which are primarily contributed in this imaging configuration by the taper at feed array level. Hence, the threshold applied needs to be higher to reduce the error, and consequently the acceleration factor is slightly reduced. Almost 99.5% of points meet the requirements on the numerical error in terms of \(C/I\), which is statistically acceptable. When beam-forming is performed at a secondary level, the far-field of each defocused feed is computed independently by PO integration and the beam pattern is obtained as a weighted sum of the far-field patterns associated with individual feeds. This leads to a dif  
Figure 10: Numerical error resulting from the use of threshold and beam interpolation techniques in an AFR scenario. An acceleration factor in time of 60 is achieved. ferent problem which is described by Equation (5), where \(\vec{E}_{b}\) represents the beams far-field, \(\vec{E}_{f}\) the feeds far-field and \(W\) is the feeds excitation matrix describing the beam-forming. \[\vec{E}_{b}=W\vec{E}_{f} \tag{5}\]  
One advantage of beam-forming at secondary level is that, for a given antenna architecture, the far-field of each feed needs to be computed only once. Different beam-forming strategies and/or precoding processing can then be retrospectively applied to benchmark the antenna performance for different operating scenarios. However, in this case some of the acceleration techniques described previously cannot be directly used. In particular, the threshold method is generally not applicable to the far-field of each defocused feed. This is due to the very broad patterns of the beams associated with highly defocused feeds. Moreover, accurate information on the phase associated with the far-field response of each feed is essential in order to apply beam-forming and obtain the beam patterns. Hence, it is necessary to apply interpolation not only on the amplitude (which in general is sufficient in system studies involving passive SFPB/MFPB antennas) but also on the phase. This requirement, in turn, brings challenges associated to interpolating phase wrapped far-field characteristics. Consequently, exploiting interpolation becomes more challenging. Therefore, a new acceleration method has been studied which, rather than interpolating the far-field characteristics across different beams, exploits interpolation to reduce the number of far-field points that are computed within a given beam. Assuming that the amplitude and phase of the radiation at a subset of far-field points is known, the remaining points are estimated using interpolation performed in both the amplitude and phase domains. An example is illustrated in Figure 11. This method takes advantage of the low spatial variation of the far-field characteristics of defocused feeds and can lead to high acceleration factors, since the computational time highly depends on the number of far-field points to be computed using PO. Of particular significance in this case is the unwrapping of the phase in order to have a smooth variation that can be easily interpolated. This solution was introduced in [63] and proved useful for parametric studies using different beam-forming techniques and linear precoding in different multi-beam scenarios. A comparative study of different techniques to compute the elements excitation \(W\) was presented in [64]. The study is focused on direct evaluation techniques in order to provide a rapid estimation of the weights and reduce the computational cost.