# Ear Centering for Accurate Synthesis of Near-Field Head-Related Transfer Functions +
## 2 Ear Centering for Near-Field HRTF Synthesis
To include the distance information, we propose to use the following SW translation operator:  
\[\mathcal{T}_{\text{SW}}(\mathbf{a},\mathbf{r}_{\text{ref}}\mapsto\mathbf{r}_ {\text{ear}})=\frac{\|\mathbf{a}-\mathbf{r}_{\text{ear}}\|}{\|\mathbf{a}- \mathbf{r}_{\text{ref}}\|}e^{-jk(\|\mathbf{a}-\mathbf{r}_{\text{ref}}\|-\| \mathbf{a}-\mathbf{r}_{\text{ear}}\|)}, \tag{4}\]  
where \(\|\cdot\|\) denotes Euclidean norm. Considering an SW emanating from \(\mathbf{a}\), (4) stems from the ratio of SW observations at \(\mathbf{r}_{\text{ref}}\) and \(\mathbf{r}_{\text{ear}}\). Figure 2: Geometry for near-field HRTF synthesis. The SFT of the ear-centered \(\mathcal{H}(\mathbf{a},\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ear}})\) is defined by  
\[\mathcal{H}_{mm}(a,\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ear}})=\int_{-\pi} ^{\pi}\int_{-\frac{\pi}{2}}^{\pi}\mathcal{H}(\mathbf{a},\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ear}})Y_{n}^{m}(\theta_{a},\phi_{a})\cos(\phi_{a})d\phi_{a}d \theta_{a}. \tag{5}\]  
Here, the SFT basis functions are complex-valued spherical harmonic functions \(Y_{n}^{m}\) of order \(n\) and degree \(m\), defined as  
\[Y_{n}^{m}(\theta,\phi)=N_{nm}P_{n}^{|m|}(\sin\phi)e^{jm\theta}, \tag{6}\]  
where \(P_{n}^{m}\) is the non-normalized associated Legendre polynomial [25], the symbol \(|\cdot|\) denotes absolute value, and \(N_{nm}\) is the following normalization factor:  
\[N_{nm}=(-1)^{|m|}\sqrt{\frac{2n+1}{4\pi}\frac{(n-|m|)!}{(n+|m|)!}}. \tag{7}\]  
Distance variation from \(a\) to \(b\) is performed in the SFT domain according to the following expression:  
\[\mathcal{H}_{mm}(b,\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ear}})=\mathcal{ D}_{n}(a,b)\mathcal{H}_{mm}(a,\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ear}}). \tag{8}\]  
Here, \(\mathcal{D}_{n}\) denotes the spherical DVF of order \(n\) defined as  
\[\mathcal{H}(\mathbf{r},\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ref}})= \frac{\Psi(\mathbf{r},\mathbf{r}_{\text{ear}})}{\Psi_{FF}(\mathbf{r},\mathbf{r }_{\text{ref}})}, \tag{1}\]0  
where \(h_{n}^{(1)}\) is the spherical Hankel function of the first kind and order \(n\)[26]. The inverse spherical Fourier transform (ISFT) extracts HRTFs for arbitrary directions using the following expression:  
\[\mathcal{H}(\mathbf{r},\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ref}})= \frac{\Psi(\mathbf{r},\mathbf{r}_{\text{ear}})}{\Psi_{FF}(\mathbf{r},\mathbf{r }_{\text{ref}})}, \tag{1}\]1  
\[\mathcal{H}(\mathbf{r},\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ref}})= \frac{\Psi(\mathbf{r},\mathbf{r}_{\text{ear}})}{\Psi_{FF}(\mathbf{r},\mathbf{r }_{\text{ref}})}, \tag{1}\]2 Near-field HRTF synthesis with ear centering based on translation operators. Finally, inverse ear centering is performed with the inverse operator \(\mathcal{T}^{-1}\) that translates the reference from \(\mathbf{r}_{\text{ear}}\) to \(\mathbf{r}_{\text{ref}}\):  
\[\mathcal{H}(\mathbf{r},\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ref}})= \frac{\Psi(\mathbf{r},\mathbf{r}_{\text{ear}})}{\Psi_{FF}(\mathbf{r},\mathbf{r }_{\text{ref}})}, \tag{1}\]3  
The inverse PW translation operator [21] can be expressed as  
\[\mathcal{H}(\mathbf{r},\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ref}})= \frac{\Psi(\mathbf{r},\mathbf{r}_{\text{ear}})}{\Psi_{FF}(\mathbf{r},\mathbf{r }_{\text{ref}})}, \tag{1}\]4  
Considering a PW emanating from \(\mathbf{b}\), (12) stems from the ratio of PW observations at \(\mathbf{r}_{\text{ear}}\) and \(\mathbf{r}_{\text{ref}}\). The proposed inverse SW translation operator takes the form  
\[\mathcal{H}(\mathbf{r},\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ref}})= \frac{\Psi(\mathbf{r},\mathbf{r}_{\text{ear}})}{\Psi_{FF}(\mathbf{r},\mathbf{r }_{\text{ref}})}, \tag{1}\]5  
Considering a SW emanating from \(\mathbf{b}\), (13) stems from the ratio of SW observations at \(\mathbf{r}_{\text{ear}}\) and \(\mathbf{r}_{\text{ref}}\). In summary, in Figure 3, blocks 1 and 2 compose the encoding stage, block 3 performs distance variation, and blocks 4 and 5 compose the decoding stage.