## 4 Synthesis of the Repository of Binaural Audio Recordings
### Convolution  
As mentioned above, the binaural excerpts were synthesized by the convolution of the monophonic multitrack recordings with the HRTFs. This procedure was undertaken using the following equation:  
\[y_{c}[n]=\sum_{i=1}^{N}\sum_{k=0}^{K-1}h_{c,\,\Theta_{i},\Phi_{i}}[n-k]\ x_{i}[k] \tag{1}\]  
where \(y_{c}[n]\) denotes an output binaural signal for an audio channel \(c\) (left or right) for a given music recording and sample \(n\); \(x_{i}[k]\) represents a \(k\)-th sample of an \(i\)-th monophonic track (individual music source); and \(h_{c,\Theta_{i},\Phi_{i}}[n]\) is a HRIR for an audio channel \(c\), azimuth \(\Theta_{i}\) and elevation \(\Phi_{i}\) for an \(i\)-th monophonic track. As mentioned earlier, the total number \(N\) of the monophonic tracks varied across the music recordings, with a minimum value of 5 and a maximum of 62 (a median value was 10). The upper summation limit \(K\) was determined by the duration of the binaural excerpts (7 s) and amounted to 336 \(\times\) 10\({}^{3}\), given the sample rate of 48 kHz. It is assumed that the signals are in discrete-time domain.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**HRTF Set** & **Acronym** & **Head** & **Type** & **Radius** & **Source** \\ \hline
1 & Subject pp2 & & & & Huawei Technologies, TU Berlin, \\
2 & HUTUBS & Subject pp3 & Human & 1.47 m & Munich Research Centre, Sennheiser \\
3 & Subject pp4 & & & Electronic [35] \\ \hline
4 & Subject H3 & & & & \\
5 & SADIE & Subject H4 & Human & 1.2 m & University of York [36] \\
6 & Subject H5 & & & & \\ \hline
7 & & & & & \\
8 & TH KöLN & Neumann KU 100 & Artificial & 1 m & TH Köln, TU Berlin, TU Ilmenau [37] \\
9 & & & & 1.5 m & \\ \hline
10 & & FABIAN HATO 0\({}^{\circ}\) & & & \\
11 & TU BERLIN & FABIAN HATO 10\({}^{\circ}\) & Artificial & 1.7 m & TU Berlin, Carl von Ossietzky University, \\
12 & & FABIAN HATO 350\({}^{\circ}\) & & & RWTH Aachen University [38] \\ \hline \hline \end{tabular}
\end{table}
Table 1: HRTF sets used to synthesize binaural excerpts.  
Due to a relatively high spatial resolution of the HRTF sets selected for this study, no interpolation of HRTFs was performed. This means that for each individual track, an intended angle was \"quantized\" to that of the nearest HRTF within the measurement grid (constrained by the ensemble boundary). Figure 3 illustrates examples of individual music audio sources (red dots) within the front-, back-, up-, and down-located ensembles, respectively, after the quantization of the intended positions to the nearest available ones. Black dots represent the gird of the measured HRTFs. The figure shows the quantization effects for the four selected HRTF sets differing in their spatial resolution. In these examples, the width of the ensembles was fixed to \(\pm 30^{\circ}\) as signified by blue areas.  
Figure 3: Examples of spatial scenes generated using four selected HRTFs. Black dots represent a grid of the measured HRTFs. Red dots signify music sound sources. Front, back, up, and down-located music ensembles of fixed width (\(\pm 30^{\circ}\)) are illustrated using blue areas.  
All individual multitrack signals were RMS normalized prior to the convolution. The duration of each synthesized binaural excerpt was limited to 7 s. The excerpts were stored in two-channel uncompressed audio files with a 48 kHz sample rate and 32-bit resolution. In total, 43,776 binaural excerpts were synthesized (152 music recordings \(\times\) 12 HRTF sets \(\times\) 4 ensemble positions \(\times\) 6 ensemble widths).