## 2 Virtual Artificial Head: Review of Methods and Configurations Chosen for the Present Study
### Calculation of Spectral Weights Using Constrained Optimization  
The virtual artificial head (VAH) consists of \(N\) spatially distributed microphones and is designed as a filter-and-sum beamformer. The aim of the VAH is to synthesize the desired directivity pattern \(\mathrm{D}(f,\Theta_{k})\) of the left or right HRTFs, where \(f\) denotes the frequency and \(\Theta_{k}=(\theta_{k},\phi_{k}),k=1,2,\ldots,P\), with azimuth \(\theta_{k}\) and elevation \(\phi_{k}\), denotes \(P\) discrete directions. Considering the \(N\)\(\times\)1 steering vector \(\mathbf{d}(f,\Theta_{k})\), defined as the free-field acoustic transfer function between a source at direction \(\Theta_{k}\) and \(N\) microphones in the array, the resulting directivity pattern \(\mathrm{H}(f,\Theta_{k})\) of the VAH is given by  
\[\mathrm{H}(f,\Theta_{k})=\mathbf{w}^{H}(f)\mathbf{d}(f,\Theta_{k}), \tag{1}\]  
where \(\{.\}^{H}\) denotes the Hermitian transpose operator and the \(N\)\(\times\)1 vector \(\mathbf{w}(f)\) contains the complex-valued spectral weights for \(N\) microphones. These spectral weights were calculated by minimizing the narrow-band least-squares cost function:  
\[\mathrm{J}_{\mathrm{LS}}(\mathbf{w}(f))=\sum_{k=1}^{P}|\mathrm{H}(f,\Theta_{ k})-\mathrm{D}(f,\Theta_{k})|^{2}=\sum_{k=1}^{P}|\mathbf{w}^{H}(f)\mathbf{d}(f, \Theta_{k})-\mathrm{D}(f,\Theta_{k})|^{2}, \tag{2}\]  
where both the amplitudes and the phases of \(\mathrm{H}(f,\Theta_{k})\) and \(\mathrm{D}(f,\Theta_{k})\) were considered. In this study, the cost function in Equation (2) was minimized, subject to two sets of constraints [11]. One set of constraints was imposed onto the spectral distortion (SD), defined as the spectral difference in dB between the desired and synthesized HRTFs in each direction \(\Theta_{k}\), i.e.,  
\[\mathrm{L}_{\mathrm{Low}}\leq\mathrm{SD}(f,\Theta_{k})=10\log_{10}\frac{| \mathbf{w}^{H}(f)\mathbf{d}(f,\Theta_{k})|^{2}}{|\mathrm{D}(f,\Theta_{k})|^{2} }\mathrm{dB}\leq\mathrm{L}_{\mathrm{Up}},\quad k=1,2,\ldots,P, \tag{3}\]  
where \(\mathrm{L}_{\mathrm{Up}}\) and \(\mathrm{L}_{\mathrm{Low}}\) denote the upper and lower boundaries, respectively. The constraints on SD were imposed to achieve good synthesis accuracy in a large number of directions. An additional constraint was imposed onto the _mean_ white noise gain (\(\mathrm{WNG}_{\mathrm{m}}\)), defined as the ratio between the mean output power of the microphone array in the \(P\) considered directions and the output power for spatially uncorrelated noise [9], i.e.,  
\[\mathrm{WNG}_{\mathrm{m}}(\mathbf{w}(f))=10\log_{10}(\frac{1}{P}\sum_{k=1}^{ P}\frac{|\mathbf{w}^{H}(f)\mathbf{d}(f,\Theta_{k})|^{2}}{\mathbf{w}^{H}(f) \mathbf{w}(f)})\mathrm{dB}\geq\beta, \tag{4}\]  
where \(\beta\) denotes the minimum desired \(\mathrm{WNG}_{\mathrm{m}}\). The constraint on \(\mathrm{WNG}_{\mathrm{m}}\) was applied to increase the robustness of the VAH synthesis against microphone self-noise or deviations in the microphone characteristics and positions [9]. An interior-point algorithm was used to minimize the cost function in Equation (2) subject to the constraints defined in Equations (3) and (4).