## 3 Setup  
The following setup is used for the numerical simulations, whose results are described in Sections 4 and 5. The angular range is discretized into \(M=360\) linearly-spaced angles \(\{\phi_{0}=0^{\circ}\), \(\phi_{1}=1^{\circ},\ldots,\phi_{359}=360^{\circ}\}\). The frequency range covers the range of \(f_{p=0}=0\,\mathrm{Hz}\) to \(f_{p=256}=24\,\mathrm{kHz}\) generated at a sampling rate of \(f_{s}=48\,\mathrm{kHz}\) using a filter length of 512 samples. This results in \(P=257\) linear spaced frequency bins. This frequency range covers the spectral content of music [12] that is to be recorded by these microphone arrays. To obtain impulse responses of the filters, the complex spectrum was mirrored, conjugated and transformed towards the time domain via an iff. The microphone array consists of \(N=9\) omnidirectional microphones and has a total length of \(Length=1\,\mathrm{m}\). The array design is done with \(l_{spread}\approx 35\), so that the smallest microphone spacing \((s)\) in the center of the array is \(s=0.01\,\mathrm{m}\). Following that, the spatial aliasing frequency can be maximized to a frequency of \(f_{al}\approx 17,000\,\mathrm{Hz}\). For practical reasons, the limitation is set to \(s=0.01\,\mathrm{m}\) to ensure enough space for the microphones. The absolute microphone positions are set as follows (displayed in millimeter precision): \(x_{n=-4}=-0.500\,\mathrm{m}\), \(x_{n=-3}=-0.150\,\mathrm{m}\), \(x_{n=-2}=-0.043\,\mathrm{m}\), \(x_{n=-1}=-0.010\,\mathrm{m}\), \(x_{n=0}=0\,\mathrm{m}\), \(x_{n=1}=0.010\,\mathrm{m}\), \(x_{n=2}=0.043\,\mathrm{m}\), \(x_{n=3}=0.150\,\mathrm{m}\), \(x_{n=4}=0.500\,\mathrm{m}\). After having specified the microphone positions, the convex functions of the CLSB, shown in Section 2.5, are solved utilizing CVX, a package for specifying and solving convex programs [13; 14]. Parts of these convex functions are the WNG constraint and the loop design. For the WNG constraint, the lower bound \(\gamma\) for the WNG \(A(f_{p})\) is set up as follows:  
\[\gamma(f_{p})= \begin{cases}5&\text{for }f_{p}=0\,\mathrm{Hz}\\ \text{CSI}&\text{for }0\,\mathrm{Hz}<f_{p}<187.5\,\mathrm{Hz}\\ 1&\text{for }187.5\,\mathrm{Hz}\leq f_{p}\leq f_{s}/2\,\mathrm{Hz}\end{cases}  \tag{25}\]  
The lower bound starts with \(\gamma(f_{p}=0\,\mathrm{Hz})=5\) and ends with \(\gamma(f_{p}\geq 187.5\,\mathrm{Hz})=1\). In the intermediate zone, a Cubic Spline Interpolation (CSI) connects both points. The CSI in the intermediate zone avoids rapid changes of the directivity pattern across frequency below \((f_{p}<187.5\,\mathrm{Hz})\). In the high frequency range \((f_{p}\geq 187.5\,\mathrm{Hz})\), a lower bound of \(\gamma=1\) ensures a robust beamforming design. For the loop design, the constants are set up as follows:  
\[K=100,\,K_{temp}=K_{step}=10,\,\alpha=0.01\,\mathrm{cf. }\,\text{Equation \eqref{An increase of the value of the constant \(K\) leads to an improvement in the beam area (lower \(error_{beam}\) values), because the step size \(\alpha\) is smaller. The validity (cf. Section 2.5) of more possible directivity patterns with small \(error_{beam}\) values is checked by the loop design. In fact, to find a valid solution, Loop 2 has to increase \(e_{stop}\) further than before, which leads to a worsening in the stop area (higher \(error_{stop}\) values). A decrease of the value of the constant \(K\) leads consequently to the opposite effect. An increase of the values of the constants \(K_{temp}\) and \(K_{step}\) leads to a worsening in the beam area (higher \(error_{beam}\) values), because the first end point of Loop 1 \(\epsilon_{beam}^{k=K_{temp}}\), as well as all of the other ones \(\epsilon_{beam}^{k=K_{temp}+K_{step}+K_{step}+\cdots}\) is now higher. More possible directivity patterns with high \(error_{beam}\) values are checked by the loop design: Loop 2 does not have to increase \(e_{stop}\) so much than before, because these directivity patterns are in general more likely to be valid. This leads then to an improvement in the stop area (lower \(error_{stop}\) values).