# Ear Centering for Accurate Synthesis of Near-Field Head-Related Transfer Functions +
## 4 Effectiveness of SW Ear Centering in the Encoding Stage of HRTF Synthesis  
In the far field, spherical waves are similar to plane waves, thus the reason why many models consider any far-field source as a plane wave. This similarity may tempt us to change the SW direct translation operator by the PW one, while maintaining the inverse SW translation operator because in the near-field the similarities between plane waves and spherical waves disappear. However, we investigate the possible reasons why this approach is not a viable option for our ear centering scheme. Plane waves and spherical waves are similar in the far field. However, translation operators are neither plane nor spherical waves, but ratios of them. This subtle difference greatly impacts the synthesis performance. Magnitudes and phases of direct translation operators are identified in (3) and (4) as  
\[|\mathcal{T}_{\text{PW}}(\mathbf{a},\mathbf{r}_{\text{ref}}\mapsto\mathbf{r}_{ \text{ear}})|=1, \tag{22}\]  
\[\angle\mathcal{T}_{\text{PW}}(\mathbf{a},\mathbf{r}_{\text{ref}}\mapsto\mathbf{ r}_{\text{ear}})=k(r_{\text{ref}}\cos\Theta_{\mathbf{a},\mathbf{r}_{\text{ ref}}}-r_{\text{ear}}\cos\Theta_{\mathbf{a},\mathbf{r}_{\text{ear}}}), \tag{23}\]  
\[|\mathcal{T}_{\text{SW}}(\mathbf{a},\mathbf{r}_{\text{ref}}\mapsto\mathbf{r}_{ \text{ear}})|=\frac{\|\mathbf{a}-\mathbf{r}_{\text{ear}}\|}{\|\mathbf{a}- \mathbf{r}_{\text{ref}}\|}, \tag{24}\]  
\[\angle\mathcal{T}_{\text{SW}}(\mathbf{a},\mathbf{r}_{\text{ref}}\mapsto \mathbf{r}_{\text{ear}})=-k(\|\mathbf{a}-\mathbf{r}_{\text{ref}}\|-\|\mathbf{ a}-\mathbf{r}_{\text{ear}}\|). \tag{25}\]  
In Figure 6, the top panel is a color legend for panels (a), (b), (c), and (d). The color dots in the top panel indicate the positions \(\mathbf{a}\) of twelve point sources placed at the vertices of the icosahedron, at a distance of 100 cm from the center of the head \(\mathbf{r}_{\text{ref}}\) marked with a circle. The coordinate system used in the top panel is the same described in Figure 1. The positive \(x\)-axis indicates the front, and the \(y\)-axis is the interaural axis. The left ear \(\mathbf{r}_{\text{ear}}\) lies on the positive \(y\)-axis and is marked with an asterisk. Point sources with the same color  
Figure 5: Synthesis error calculated with (21) in dB. Black-dashed lines indicate \(f_{\text{max}}\) in (16). indicate that they are at the same distance from \(\mathbf{r}_{\text{ear}}\). The point sources lie in three regions of space:  
* Blue and yellow dots: \(\|\mathbf{a}-\mathbf{r}_{\text{ear}}\|>\|\mathbf{a}-\mathbf{r}_{\text{ref}}\|\),
* Green dots: \(\|\mathbf{a}-\mathbf{r}_{\text{ear}}\|\approx\|\mathbf{a}-\mathbf{r}_{\text{ ref}}\|\),
* Purple and red dots: \(\|\mathbf{a}-\mathbf{r}_{\text{ear}}\|<\|\mathbf{a}-\mathbf{r}_{\text{ref}}\|\). In Figure 6, panels (a) and (b) show the magnitude of the PW and SW translation operators identified in (22) and (24) respectively, expressed in logarithmic scale from \(-1\) dB to \(1\) dB; and panels (c) and (d) show the phase of the PW and SW translation operators identified in (23) and (25), respectively, expressed in degrees from \(-1080\) to \(1080\) degrees. When comparing panels (a) and (b) in Figure 6, we observe that panel (b) shows changes in amplitude of around \(\pm 0.5\) dB. There are no changes in panel (a) from point to point because the magnitude of the PW direct translation operator, identified in (22), remains the same for every direction. In panel (b), we observe that blue and yellow  
Figure 6: PW and SW direct translation operators for twelve different directions along the vertices of the icosahedron (top panel), at a distance of \(100\) cm from the center of the head. (**a**) Magnitude in (22); (**b**) Magnitude in (24); (**c**) Phase in (23); (**d**) Phase in (25). point sources yield positive magnitudes in dB, green point sources are in the median plane and yield approximately 0 dB in magnitude, and purple and red point sources yield negative magnitudes in dB.