# Particle-Velocity-Based Mixed-Source Sound Field Translation for Binaural Reproduction
## 5 Simulation Study
### Simulation Setup and Criteria  
In this simulation, we firstly consider an acoustic environment that contains a single free-field point source with a location of \(\mathbf{b}_{1}=(1,\pi/3,\pi/4)\). We then consider another scenario where multiple sources exist, which is constructed by adding two more point sources with randomly selected locations of \(\mathbf{b}_{2}=(0.5,3\pi/4,4\pi/3)\) and \(\mathbf{b}_{3}=(1.8,\pi/2,\pi/2)\), respectively, in the former acoustic environment. The true sound pressure at an arbitrary position \(\mathbf{x}\) due to the \(j\)th point source is  
\[P_{j}(\mathbf{x},k)=\frac{e^{ik||\mathbf{b}_{j}-\mathbf{x}||}}{4\pi||\mathbf{b}_{j}-\mathbf{x}||}, \quad j=\{1,2,3\}. \tag{33}\]  
The total sound pressure due to multiple sources follows the principle of linear superposition. We assume that the receiver can record the sound field up to the 4 th order (i.e., \(N=4\)), and the radius of the receiver region is \(R=0.042\) m (We simulate the radius of the EigenMike, which is a popular commercial microphone array [53]). The pressure coefficients describing the acoustic environment \(\{\alpha_{nm}(k)\}\) can be extracted from the receiver's recording. Here, for simplicity, we use the theoretical pressure coefficients up to the \(N\)th order, expressed as  
\[\alpha_{nm}(k)= \begin{cases}ikh_{n}(k|\mathbf{b}_{1}|)Y_{nm}^{*}(\hat{\mathbf{b}}_{1}),\ n\in[0,N],\ \text{for S1},\\ \\ \sum_{j=1}^{3}ikh_{n}(k|\mathbf{b}_{j}|)Y_{nm}^{*}(\hat{\mathbf{b}}_{j}),\ n\in[0,N],\ \text{for S2},\end{cases}  \tag{34}\]  
where S1 denotes the single source scenario and S2 denotes the multiple sources scenario. The mixed-source model consists of two sets of \(L=36\) virtual sources distributed on the two spheres, both with the positions arranged on Fliege nodes [64]. The first set is placed in the near field at \(R_{N}=2\) m, and the second set is placed in far field at \(R_{F}=20\) m.  
Throughout the simulations, the frequency of the sources is \(f=1500\) Hz, except the evaluation in terms of the frequency. The sound speed is \(c=343\) m/s and the air density is \(\rho_{0}=1.29\) kg/m\({}^{3}\). The truncation orders of the velocity coefficients in the \(r\), \(\theta\), \(\phi\) direction are \(Q_{r}=N\), \(Q_{\theta}=2N\), and \(Q_{\phi}=2N\), respectively, according to the truncation theorem of the velocity expressions [44]. For the sparse solution, we use a 500 iterations LASSO. As the first objective performance measure for the reproduction method, we define the mismatch between the reproduced sound field and the true sound field at any point \(\mathbf{x}\) as  
\[\epsilon(\mathbf{x},k)=\frac{|P(\mathbf{x},k)-\widetilde{P}(\mathbf{x},k)|^{2}}{|P(\mathbf{x},k)|^{2}}\times 100(\%). \tag{35}\]A value of 0% means the reproduced sound field is exactly the same as the true sound field at the point. The larger the percentage, the greater the difference between the reproduced sound field and the true sound field at the point. A good perception of sound localization requires accurate reproduction of sound direction. To evaluate the performance of the reproduction of sound direction at the translated position \(\mathbf{d}\), two localization metrics are introduced [22]. The first one is related to the velocity vector at the translated position, which is given by  
\[\mathbf{r}_{V}(k;\mathbf{d})=\text{Re}\bigg{(}\frac{\sum_{\ell=1}^{\mathcal{C}}\mathcal{ U}_{\ell}(k)\mathbf{\hat{z}}_{\ell}}{\sum_{\ell=1}^{\mathcal{C}}\mathcal{U}_{\ell}(k)} \bigg{)}, \tag{36}\]  
where the subscript \(V\) denotes velocity, \(\text{Re}(\cdot)\)denotes the real part, and \(\mathcal{U}_{\ell}(k)\) are the effective source gains (accounting for point-source radiation), expressed as  
\[\mathcal{U}_{\ell}(k)=\frac{e^{\mu(\mathbf{k}|\mathbf{z}_{\ell}|-R_{\ell})}}{|\mathbf{z}_{ \ell}|/R_{\ell}}\psi(k,R_{\ell}\mathbf{\hat{y}}_{\ell};\mathbf{o}). \tag{37}\]  
\(\mathbf{r}_{V}(k)\) is used to predict localization at low frequencies (below 700 Hz).