## 4 Objective Evaluation
### Beam and Stop Area Constraints  
The results of the loop design mentioned in Section 2.5 are shown in Figure 8. This loop design finds a compromise between a good fit in the beam area and low directivity pattern values in the stop area.  
For low frequencies \(f_{p}<187.5\,\mathrm{Hz}\), the directivity pattern is quite omnidirectional (\(\|\mathbf{error}_{stop}\|_{\infty}>0.2\) and \(\|\mathbf{error}_{beam}\|_{\infty}>0.1\)), so that Loop 3 has to increase \(\epsilon_{beam}\) to \(\|\epsilon_{beam}^{opt}\|_{\infty}>0.1\). For higher frequencies \(f_{p}\geq 187.5\,\mathrm{Hz}\), there is a good fit in the beam area \(\|\mathbf{error}_{beam}\|_{\infty}\leq 0.1\) so that Loop 1 and Loop 2 find the ideal upper bound for the beam and the stop area. Overall, it can be said that the best result is found in the frequency range of \(281.3\,\mathrm{Hz}\leq f_{p}\leq 1969\,\mathrm{Hz}\): a good fit in the beam area combined with low directivity pattern values in the stop area \(\|error_{stop}\|_{\infty}\leq 0.2\). At high frequencies (\(f_{p}\geq 16,690\,\mathrm{Hz}\)), Figure 8b shows aliasing effects (\(\|\mathbf{error}_{stop}\|_{\infty}=1\)), which are expected, since the aliasing frequency of the logarithmically-spaced array is \(f_{al}(\Delta x=s)\approx 17,000\,\mathrm{Hz}\).  
Figure 9 shows the polar plot of the desired directivity pattern in addition to the absolute value of the directivity patterns of the frequencies \(f_{p}=250\,\mathrm{Hz}\), \(f_{p}=1000\,\mathrm{Hz}\), \(f_{p}=4000\,\mathrm{Hz}\) and \(f_{p}=8000\,\mathrm{Hz}\). For all frequencies, there is a good fit (a small difference between desired and obtained directivity pattern) in the beam area, as already quantified by Figure 8a. Comparing the side-lobe-levels of the different frequencies, the following can be stated: the side-lobe-level decreases from \(f_{p}=250\,\mathrm{Hz}\) to \(1000\,\mathrm{Hz}\); there is no big difference in side-lobe-level between \(f_{p}=1000\,\mathrm{Hz}\) and \(f_{p}=4000\,\mathrm{Hz}\)  
Figure 8: The difference between the simulated directivity pattern and the desired one (\(error\)) in the beam (**a**) and the stop (**b**) area, as well as the corresponding upper bounds of both areas as function of the frequency.  
the side-lobe-level increases from \(f_{p}=4000\) Hz to \(f_{p}=8000\) Hz. This analysis is described in a quantified matter in Figure 8b.  
Figure 10a allows for a more detailed analysis, as it shows the absolute value of the difference between the directivity pattern and the desired one in the whole angular range \(|error(\phi_{m},f_{p})|\). The omnidirectional behavior of the directivity pattern up to \(f_{p}=187.5\) Hz can be also seen there. For higher frequencies, side lobes appear at \(\phi_{m}=\pm 180\)\({}^{\circ}\) and move with increasing frequency into the direction of the beam \(-60\)\({}^{\circ}\leq\phi_{m}\leq 60\)\({}^{\circ}\). Aliasing effects can be seen in Figure 10a, like in Figure 8b.  
In addition to the absolute value of the directivity pattern, the phase \(\arg(b(f_{p},\phi_{m}))\) is represented in Figure 10b.  
The directivity pattern is purely real: the phase shows only three possible values \(\arg(b)=\{-\pi,0,\pi\}\) as mentioned in Section 2.3. In the beam area, the phase has, in fact, only values \(\arg(b)=0\), which leads to no phase differences between the two arrays in the recording configuration mentioned in Section 2.3.  
Figure 10: The difference between the directivity pattern and the desired one \(|error(f_{p},\phi_{m})|\) (**a**), as well as the phase of the directivity pattern \(\arg(b(f_{p},\phi_{m}))\) (**b**).  
Figure 9: Polar plot of the desired directivity pattern (grey markers) and the absolute value of the obtained directivity patterns of the frequencies \(f_{p}=250\) Hz (solid line), \(f_{p}=1000\) Hz (dashed line), \(f_{p}=4000\) Hz (dashed-dotted line) and \(f_{p}=8000\) Hz (dotted line).