# Particle-Velocity-Based Mixed-Source Sound Field Translation for Binaural Reproduction
## 3 Particle-Velocity-Based Mixed-Source Expansion
### Particle-Velocity-Based Expansion  
In this subsection, we develop the theory of particle-velocity-based mixed-source expansion. The particle velocity is related to the sound pressure by the [PERSON]'s equation [55],  
\[\mathbf{V}(\mathbf{x},k)=\frac{i}{k\rho_{0}c}\vec{\
abla}P(\mathbf{x},k), \tag{18}\]  
where \(\vec{\
abla}P(\mathbf{x},k)\) denotes the gradient of \(P(\mathbf{x},k)\) in terms of \(r\), \(\theta\), and \(\phi\). Similarly, for the mixed source, we have  
\[\mathbf{V}(\mathbf{x},k;R_{X}\hat{y})=\frac{i}{k\rho_{0}c}\vec{\
abla}P(\mathbf{x},k;R_{X} \hat{y}), \tag{19}\]  
where \(R_{X}\in\{R_{N},R_{F}\}\); therefore, taking the gradient operation on both sides of (17), we can obtain the particle-velocity-based mixed-source expansion  
\[V_{\Psi}(\mathbf{x},k)=\int\psi(k,R_{N}\hat{y};\mathbf{o})V_{\Psi}(\mathbf{x},k;R_{N}\hat {y})d\hat{y}+\int\psi(k,R_{F}\hat{y};\mathbf{o})V_{\Psi}(\mathbf{x},k;R_{F}\hat{y})d \hat{y}, \tag{20}\]  
where \(V_{\Psi}(\mathbf{x},k;R_{N}\hat{y})\) and \(V_{\Psi}(\mathbf{x},k;R_{F}\hat{y})\) represent the particle velocity of the virtual sources in the near field and the far field, respectively, in the \(\Psi\) direction. In practice, the particle-velocity-based mixed-source expansion (20) can be approximated by a finite set of \(L\) known virtual sources for each virtual sphere; therefore, (20) can be expressed as  
\[V_{\Psi}(\mathbf{x},k)=\sum_{\ell=1}^{L}w_{\ell}\psi(k,R_{N}\hat{y};\mathbf{o})V_{ \Psi}(\mathbf{x},k;R_{N}\hat{y}_{\ell})+\sum_{\ell=1}^{L}w_{\ell}\psi(k,R_{F}\hat{y }_{\ell};\mathbf{o})V_{\Psi}(\mathbf{x},k;R_{F}\hat{y}_{\ell}), \tag{21}\]  
where \(w_{\ell}\) are the the sampling weights for the source distribution, and \(\hat{y}_{\ell}\) denotes the position of the \(\ell\)th virtual source on each sphere. Note that the parameters of the mixed-source expansion need to be carefully selected in the implementations. The near-field radius of the virtual sphere can be selected based on the desired maximum translation distance, because the listener cannot move beyond the near-field sphere. As for the far-field sphere, it can be placed anywhere so long as it is in the far-field. Moreover, the selection of the number of the virtual sources per sphere is a trade-off between angular resolution and computation cost. The more virtual sources there are, the more opportunity they have to match the true sound direction of arrival no matter the higher computation cost the system has. By replacing \(\alpha_{nm}(k)\) with \(\beta_{nm}^{(\ell)}\) (i.e., the pressure coefficients of the \(\ell\)th virtual source) in (6)-(8), and then substituting them into (5), the particle velocity due to the \(\ell\)th virtual source can be written as  
\[V_{\Psi}(\mathbf{x},k;R_{X}\hat{\mathbf{y}}_{\ell})=\sum_{p=0}^{Q_{\Psi}}\sum_{q=-p}^{ p}X_{pq}^{(\Psi,\ell)}(k_{r},|\mathbf{x}|;R_{X}\hat{\mathbf{y}}_{\ell})Y_{pq}(\hat{\mathbf{x}}), \tag{22}\]  
where \(X_{pq}^{(\Psi,\ell)}(k,|\mathbf{x}|)\) are the velocity coefficients of the \(\ell\)th virtual source. Substituting (5) and (22) into (21), (21) can be simplified as  
\[X_{pq}^{(\Psi)}(k,|\mathbf{x}|)=\sum_{\ell=1}^{2L}\omega_{\ell}\psi(k,R_{\ell}\hat {\mathbf{y}}_{\ell};\mathbf{o})X_{pq}^{(\Psi,\ell)}(k,|\mathbf{x}|;R_{\ell}\hat{\mathbf{y}}_{ \ell}), \tag{23}\]  
where \(R_{\ell}=R_{N}\) for \(\ell\in[1,L]\), and for \(\ell\in[L+1,2L]\) we have \(R_{\ell}=R_{F}\), and \(w_{\ell}=w_{\ell-L}\).