## 2 Methods
### Desired Directivity Pattern  
The playback of the recorded signals should be in a stereophonic configuration, as mentioned in Section 1 and illustrated in Figure 3a. The playback approach proposed by [PERSON] and [PERSON] [3] uses two loudspeakers for the direct sound reproduction with a typical base angle of \(\phi_{base}=60^{\circ}\) relative to the listener's position [9]. There are several approaches to shift a phantom source from one loudspeaker to the other, utilizing phase differences \(\Delta phase=phase_{1}-phase_{2}\) and/or level differences (amplitude panning) \(\Delta Level=Level_{1}-Level_{2}\) applied on the two loudspeaker signals. Based on this playback configuration, the recording configuration presented in this paper consists of two crossed end-fire microphone arrays with a \(60^{\circ}\) opening angle, sharing one center microphone and using omnidirectional microphones, illustrated in Figure 3b. The microphone positions in this figure can only be considered as a sketch, the absolute positions can be found in Section 3. The phantom-source shifting approaches of the playback configuration can be used to formulate either the correct phase and/or level differences between the two arrays. In this way, the perceived location of the sound source in the playback situation is identical to the one of the  
Figure 3: The stereophonic recording configuration is based on the playback one. Recorded level and phase differences with the two end-fire microphone arrays generate a phantom source between the two loudspeakers in the playback configuration. The signal emitted from Loudspeaker 1 has the level \(Level_{1}\) and the phase \(phase_{1}\). The signal emitted from Loudspeaker 2 has the level \(Level_{2}\) and the phase \(phase_{2}\). (**a**) Typical stereophonic playback configuration [9]; (**b**) proposed stereophonic recording configuration with sketched microphone positions. The absolute microphone positions are shown in Section 3.  
recording provided that the distribution of recorded sound sources does not span more than 60\({}^{\circ}\) of angle. Although not evaluated here, in principle, a different opening angle could be used for the microphone arrays, thus effectively compressing or expanding the reproduced sound stage. We restrict our proposed method to have only level differences, and for this reason, the desired directivity pattern \(\hat{b}\) is purely real valued. With this desired directivity pattern, the phase of the directivity pattern is mainly controlled by the array design, which will be explained in Section 2.4. In this paper, the phantom source shifting approach of amplitude panning is used for formulating the desired directivity pattern of Array 1 \(\hat{b}_{array1}\) and Array 2 \(\hat{b}_{array2}\)[9]:  
\[\hat{b}_{array1}(\phi_{\delta}) =\sqrt{\left(1+\left(\frac{\tan(\phi_{\delta})-\tan(\phi_{b}/2)} {\tan(\phi_{\delta})+\tan(\phi_{b}/2)}\right)^{2}\right)^{-1}} \tag{8}\] \[\hat{b}_{array2}(\phi_{\delta}) =\sqrt{\left(1+\left(\frac{\tan(\phi_{\delta})+\tan(\phi_{b}/2)} {\tan(\phi_{\delta})-\tan(\phi_{b}/2)}\right)^{2}\right)^{-1}}\]  
The angle area \(\phi_{\delta}\) between both arrays is defined by:  
\[\phi_{\delta}=\{\phi_{m}|-\phi_{b}/2\leq\phi_{m}\leq\phi_{b}/2\} \tag{9}\]  
with the constant \(\phi_{b}=\phi_{base}=60^{\circ}\). The derivation of the desired directivity patterns according to [9] gives two possible recording room assumptions: an anechoic chamber or a real room. The latter one is chosen for Equation (8) since the microphone array configuration will be used in real rooms, such as concert halls. The desired directivity pattern of the one array is the mirror-flipped version of the other array. This symmetry of the recording configuration makes it possible to formulate one desired directivity pattern, which is the same for both arrays.