# Ear Centering for Accurate Synthesis of Near-Field Head-Related Transfer Functions +
## 2 Ear Centering for Near-Field HRTF Synthesis  
In the spherical coordinate system shown in Figure 1, a point in space \(\mathbf{r}=(r,\theta,\phi)\) is specified by its radial distance \(r\), azimuthal angle \(\theta\in[-\pi,\pi]\), and elevation angle \(\phi\in[-\frac{\pi}{2},\frac{\pi}{2}]\). The azimuthal angle \(\theta\) is measured from the \(x\)-axis, whereas the elevation angle \(\phi\) is measured from the \(xy\)-plane. Positions in front of the listener lie along the positive \(x\)-axis or the direction \((\theta=0,\phi=0)\). All of what follows considers acoustic waves satisfying the Helmholtz equation with time-harmonic dependence \(e^{jkct}\), where \(k\) denotes the wave number, \(c\) is the speed of sound in air, and \(j\) is the imaginary unit. The relation between frequency \(f\) and wave number \(k\) is \(2\pi f=kc\). Figure 2 shows the top-view geometry for theoretical near-field HRTF synthesis. The center of the head coincides with the origin \(\mathbf{0}=(0,0,0)\) and the ear position is denoted by \(\mathbf{r}_{\text{ear}}=(r_{\text{ear}},\theta_{\text{ear}},\phi_{\text{ear}})\). Let \(\mathbf{a}=(a,\theta_{a},\phi_{a})\) be a point in a continuous, spherical distribution at a far distance \(a\). Let \(\mathbf{b}=(b,\theta_{b},\phi_{b})\) be an arbitrary point at a near distance \(b\)  
Figure 1: Spherical coordinate system. The HRTF is defined as the sound pressure at the ear position due to a point source at the source position, divided by the sound pressure at the reference position when the head is not present [5]. The HRTF, denoted by \(\mathcal{H}\), depends on the source position \(\mathbf{r}\), the ear positions \(\mathbf{r}_{\text{ear}}\), and the reference position \(\mathbf{r}_{\text{ref}}\). The HRTF is defined as follows:  
\[\mathcal{H}(\mathbf{r},\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ref}})= \frac{\Psi(\mathbf{r},\mathbf{r}_{\text{ear}})}{\Psi_{FF}(\mathbf{r},\mathbf{r }_{\text{ref}})}, \tag{1}\]  
where \(\Psi(\mathbf{r}_{\text{s}},\mathbf{r}_{\text{r}})\) denotes the pressure emanated from a source position \(\mathbf{r}_{\text{s}}\) measured at a receiver position \(\mathbf{r}_{\text{r}}\) and the sub-index \(FF\) stands for \"free-field\", which indicates that the head is not present. Throughout this paper, the reference position is the head center \(\mathbf{r}_{\text{ref}}=\mathbf{0}\). Figure 3 overviews the synthesis process with ear centering. The input is a continuous, spherical distribution of HRTFs from \(\mathbf{a}\) to \(\mathbf{r}_{\text{ear}}\), denoted by \(\mathcal{H}(\mathbf{a},\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ref}})\), whereas the output is a synthesized HRTF from \(\mathbf{b}\) to \(\mathbf{r}_{\text{ear}}\), denoted by \(\mathcal{H}(\mathbf{b},\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ref}})\). For simplicity, only the left ear is considered; however, the formulations below that relate the output to the input hold for both ears. Direct ear centering is performed by an operator \(\mathcal{T}\) that translates the reference of \(\mathcal{H}\) from \(\mathbf{r}_{\text{ref}}\) to \(\mathbf{r}_{\text{ear}}\) as follows:  
\[\mathcal{H}(\mathbf{a},\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ear}})= \mathcal{T}(\mathbf{a},\mathbf{r}_{\text{ref}}\mapsto\mathbf{r}_{\text{ear}}) \mathcal{H}(\mathbf{a},\mathbf{r}_{\text{ear}},\mathbf{r}_{\text{ref}}). \tag{2}\]  
The existing PW translation operator proposed in [21] can be formulated as  
\[\mathcal{T}_{\text{PW}}(\mathbf{a},\mathbf{r}_{\text{ref}}\mapsto\mathbf{r}_ {\text{ear}})=e^{jk(r_{\text{ref}}\cos\Theta_{\mathbf{a}_{\text{ref}}}-r_{ \text{ear}}\cos\Theta_{\mathbf{a}_{\text{ref}}})}, \tag{3}\]  
where \(\Theta_{\mathbf{a},\mathbf{r}_{\text{ear}}}\) denotes the angle between \(\mathbf{a}\) and \(\mathbf{r}_{\text{ear}}\) and \(\Theta_{\mathbf{a},\mathbf{r}_{\text{ref}}}\) denotes the angle between \(\mathbf{a}\) and \(\mathbf{r}_{\text{ref}}\). Considering a PW emanating from \(\mathbf{a}\), (3) stems from the ratio of PW observations at \(\mathbf{r}_{\text{ref}}\) and \(\mathbf{r}_{\text{ear}}\).