# Ear Centering for Accurate Synthesis of Near-Field Head-Related Transfer Functions +
## 1 Introduction
[24] extended the work in [20,21] applying a free-field spherical-wave translation operator to incorporate distance information, as well as DVFs to synthesize HRTFs at close distances from the center of the head from both sparse and dense far-field HRTF datasets. In summary, translation operators can include distance information [19,24] or not [20, 21, 22, 23], can be applied in the unit sphere domain [20, 21, 22, 23] or in the SFT domain [19], and can consider acoustic propagation in the free field [19, 20, 21, 24] or include an acoustically rigid scatterer that mimics a simple head [22,23]. Hitherto, ear centering with free-field translation operators based on a plane-wave (PW) model, applied to HRTF datasets on the unit sphere, have yielded optimum use of SFT basis functions and accurate synthesis when distances between the sound source and the ears are large [21]. However, when PW translation operators are used to synthesize near-field HRTFs, the accuracy is affected because the PW model does not consider the distance information. Following this approach, it would be useful to have a translation operator that considers the distance between the sound source and the ears to synthesize near-field HRTFs. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Reference** & **Distance** & **Translation Point** & **Domain** & **Translation Model** \\ \hline [PERSON] et al., 2014 [19] & Yes & Optimized point around the ear & SFT domain & Ratio of hankel functions \\ [PERSON] et al., 2018 [20] & No & \(y\)-axis with 8.5 cm radius & Unit sphere & Plane-wave \\ [PERSON] et al., 2019 [21] & No & \(y\)-axis with 8.75 cm radius & Unit sphere & Plane-wave \\ [PERSON] et al., 2019 [22] & No & \(y\)-axis with 9.19 cm radius & Unit sphere & Plane-wave with rigid sphere \\ [PERSON] et al., 2021 [23] & No & \(y\)-axis with 9.19 cm radius & Unit sphere & Plane-wave with rigid sphere \\ [PERSON] et al., 2021 [24] & Yes & Ear position & Unit sphere & Spherical-wave \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Overview of previous research in translation operators for ear centering in HRTF synthesis. In this paper, we propose to use a free-field translation operator based on a spherical-wave (SW) model for ear centering in near-field HRTF synthesis. Although the basic idea of our SW-based method has been proposed in our previous conference paper [24], there are still some unsolved problems. The main contributions of this paper when contrasted with [24] are as follows:  
1. The new SW translation operators translate the reference position and maintain fixed ear positions, whereas the previous translation operators translate the ear positions. The new operators are therefore consistent with the definition of HRTFs. 2. The open question on whether PW and SW translation operators are equivalent when encoding HRTF datasets in the far field is addressed in this paper. The results are presented as a novel content in Section 4. 3. The use of complex-valued SFT basis functions in this paper generalizes previous formulations based on real-valued SFT basis functions. This generalization allows for more precise analyses in terms of magnitude and phase. 4. The review structured in Table 1 is new. 5. This paper generalizes the theory, whereas the conference paper was oriented to implementing a particular case. The reminder of this paper is organized as follows. Section 2 formulates ear centering for near-field HRTFs using translation operators. Section 3 compares PW and SW translation operators with calculated data. Section 4 dives into the effectiveness of using the proposed SW translation operator in the encoding stage of synthesis. Finally, Section 5 states the conclusions.