## 2 Methods
### Modification of the Measured BRIR
#### 2.2.1 Modification of the Direct Part in the BRIR  
A previous study in [17] investigated the role of spectral details in direct parts of BRIRs on perceived externalization. For that, different levels of smoothing were achieved by using gammatone filters with various bandwidth factors \(B\) (bandwidth factors are the values relative to one equivalent rectangular bandwidth (ERB) [25]) ranging from 0.316-63.1. The smoothed spectral magnitude of the direct part, \(|\mathrm{BRTF}_{\mathrm{direct},\,smooth}(r,\theta,\varphi,f_{c})|\), for each center frequency, \(f_{c}\), was calculated according to [PERSON] and [PERSON] [26]:  
\[|\mathrm{BRTF}_{\mathrm{direct},\,smooth}(r,\theta,\varphi,f_{c})|=\sqrt{ \frac{\int_{0}^{\infty}\left|\mathrm{BRTF}_{\mathrm{direct}}(r,\theta, \varphi,f)\right|^{2}\left|\mathrm{H}(f,f_{c})\right|^{2}df}{\int_{0}^{ \infty}\left|\mathrm{H}(f,f_{c})\right|^{2}df}}\, \tag{3}\]  
where \(|\mathrm{BRTF}_{\mathrm{direct}}(r,\theta,\varphi,f)|\) represents the spectral magnitude of the original direct part in the BRIR. \(|\mathrm{H}(f,f_{c})|\) denotes the spectral magnitude of a 4 th-order gammatone filter at the center frequency of \(f_{c}\) with a bandwidth of \(b(f_{c})\), which is described in [26; 27] and can be expressed as:  
\[|\mathrm{H}(f,f_{c})|=\left|\left(\frac{b(f_{c})}{\mathrm{j}(f-f_{c})+b(f_{c })}\right)^{4}\right|\, \tag{4}\]  
Figure 2: Illustration of the setup of BRIR measurements. with  
\[b(f_{c})=B\times\frac{24.7\ (0.00437\times f_{c}+1)}{2\sqrt{2^{1/4}-1}},\ \text{and j}=\sqrt{-1}. \tag{5}\]  
The results showed that the externalization ratings of sound sources synthesized by such modified BRIRs (smoothed magnitude spectra of direct parts) decreased noticeably by increasing \(B\) above 1 ERB. In the case of the largest bandwidth factor in their study (\(B=63.1\)), the synthesized binaural sound was perceived as internalized (\"in the head\" [28]). In the present study, 4 th-order gammatone filters with \(B\) of 63.1 were applied to smooth \"maximally\" the magnitude spectra of direct parts in BRIRs. The reason for using this smoothing method is to compare the results with the observations by [PERSON] et al. [17]. Other methods, e.g., truncating the FS of the magnitude spectra [8], 1/N-octave smoothing [29], etc., could also be used to smooth the magnitude spectra of direct sound components. To study the importance of spectral details in direct sounds of contralateral and ipsilateral ear signals on perceived externalization, the magnitude spectra of direct parts were smoothed (Equations (3)-(5)) in (i) BRIRs of the left ear (condition \"SL\"), (ii) BRIRs of the right ear (condition \"SR\"), and (iii) BRIRs of both ears (condition \"SB\"). The binaural sounds rendered by non-processed BRIRs were used as references (condition \"SN\"). It should be noted that in the case of \(-30^{\circ}\), \(-60^{\circ}\), and \(-90^{\circ}\) sound sources, the left ear is the ipsilateral (\"same-side\") ear, and the right ear is the contralateral (\"opposite-side\") ear. In contrast, for \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\) sound sources, the left and right ear are the contralateral and ipsilateral ear, respectively. For a frontal (\(0^{\circ}\)) sound source, both ears are facing the loudspeaker, and they are neither contralateral, nor ipsilateral ears. Figure 3 shows the processed spectral magnitude of the direct part in the pair of BRIRs at \(60^{\circ}\) for the \"SN\" (left top), \"SL\" (right top), \"SR\" (left bottom), and \"SB\" (right bottom) conditions. After the smoothing process, the spectral magnitude of the direct part was almost constant across frequencies, and the specific notches and peaks disappeared. The reverberant parts in BRIRs remained unchanged and were added to the processed direct parts to generate modified BRIRs. Figure 4 shows the resulting magnitude spectra of modified BRIRs at \(60^{\circ}\) for different conditions. The magnitude spectra showed in each panel of both figures were normalized to the maximum magnitude level of the right ear BRIR.