## 2 Virtual Artificial Head: Review of Methods and Configurations Chosen for the Present Study
### VAH Implementations and Constraint Parameters Used in This Study  
In this study, two different microphone arrays, referred to as VAH1 (a planar array with \(N\) = 24 microphones, 20 cm \(\times\) 20 cm [10]) and VAH2 (a 3D array with \(N\) = 31 microphones, 11 cm \(\times\) 11 cm \(\times\) 6 cm), as shown in Figure 1, were used. In both VAHs, the microphones were spatially distributed such that the inter-microphone distances were as different as possible in all possible directions. This was achieved by placing the microphones usinga Golomb ruler [14]. A Golomb ruler is a numerical series such that the distance between any two points in this series is different than the distance between any other two points. It was shown in [15] that the Golomb-based microphone distribution for a VAH leads to smaller synthesis errors over a wider frequency range compared to a randomly chosen microphone distribution. Due to the smaller extension of VAH2 compared to VAH1 and consequently smaller microphone spacing, grating lobes appeared for VAH2 at higher frequencies than for VAH1. However, for both VAHs, the grating lobes appeared at frequencies above 5 kHz, which was assumed to be above the frequency range of interest for the test signal (speech--see Section 3.1.5) used in this study. Throughout this study, the SD constraint parameters \(\mathrm{L_{Up}}\) and \(\mathrm{L_{Low}}\) in Equation (3) were chosen as 0.5 and \(-1.5\) dB, respectively, as in [11; 13]. This led to a maximum deviation of 2 dB in the interaural level difference (ILD) in all directions \(P\), which was assumed to be reasonable based on the reported just noticeable differences in ILD deviations [16]. The minimum desired \(\mathrm{WNG_{m}}\) in Equation (4) was chosen as \(\beta\) = 0 dB, based on the results in [13]. Although this value of \(\beta\) was evaluated in [13] for VAH1 only, the same value was chosen for VAH2 as well. Consistently with the previous study in [13], two sets of directions were considered in the optimization (Equations (2)-(4)): \(P\) = 72 directions, equally spaced along the horizontal plane (i.e., 5\({}^{\circ}\) resolution), and \(P\) = 3 \(\times\) 72 = 216 directions, i.e., the 72 directions from the horizontal plane and the same 72 azimuths at two elevations \(\pm\)15\({}^{\circ}\). A summary of the constraint parameter \(P\) and the VAHs used in this study is given in Table 1. The cases V11 and V13 were already evaluated in a previous study [13]. VAH2 was not evaluated previously. As discussed in Section 1, the VAH can be virtually rotated to a specific head orientation of the listener simply by adjusting the spectral weights. For a given head orientation \(\Theta_{h}=(\theta_{h},\phi_{h})\), spectral weights can be calculated by using the HRTF directivity pattern \(\mathrm{D}(f,\Theta_{k}),k=1,2,\ldots,P\) and the shifted steering vector \(\mathbf{d}(f,\Theta_{s})\) with \(\Theta_{s}=(\theta_{k}+\theta_{h},\phi_{k}+\phi_{h})\) in Equations (1)-(4). For both VAHs in this study, spectral weights labeled as V11, V13, V21 and V23 (see Table 1) were each calculated for 37 \(\times\) 5 = 185 head \begin{table}
\begin{tabular}{c c} \hline \hline
**Label** & **Constraint Parameter \(P\) and the Used VAH** \\ \hline V11 & VAH1 - \(P\) = 72 (Elevation: 0\({}^{\circ}\)) \\ \hline V13 & VAH1 - \(P\) = 3 \(\times\) 72 = 216 (Elevations: \(-15^{\circ}\), 0\({}^{\circ}\), 15\({}^{\circ}\)) \\ \hline V21 & VAH2 - \(P\) = 72 (Elevation: 0\({}^{\circ}\)) \\ \hline V23 & VAH2 - \(P\) = 3 \(\times\) 72 = 216 (Elevations: \(-15^{\circ}\), 0\({}^{\circ}\), 15\({}^{\circ}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: An overview of parameter \(P\) and the VAHs used to calculate the spectral weights. Figure 1: VAHs used in this study. VAH1: a planar microphone array with 24 microphones. VAH2: a 3D microphone array with 31 microphones.