# Particle-Velocity-Based Mixed-Source Sound Field Translation for Binaural Reproduction
## 5 Simulation Study
### Discussion  
Based on the above simulation results, we give the following comments:  
* The particle-velocity-based expansion is derived directly from the sound-pressure-based expansion by the gradient calculation. Therefore, the least squares solution of the particle-velocity-based expansion is mathematically parallel to the closed-form solution of the sound-pressure-based expansion, both of which aim to reconstruct the original truncated recording by distributing energy throughout all virtual sources and  
Figure 11: The directional error \(\zeta_{E}\) and the magnitude of \(r_{E}\) as a function of translation distance along the positive \(y\)-axis. The frequency of the source is 1500 Hz. (**a**) Directional error \(\zeta_{E}\); (**b**) magnitude of \(r_{E}\).  
Figure 10: The directional error \(\zeta_{V}\) and the magnitude of \(r_{V}\) as a function of translation distance along the positive \(y\)-axis. The frequency of the source is 300 Hz. (**a**) Directional error \(\zeta_{V}\); (**b**) magnitude of \(r_{V}\).  
inherit the spatial artifacts caused by the truncated measurement. This leads to poor reproduction outside the receiver region.
* Sparse solutions can improve the performance of reproduction outside the receiver region due to the fact that most sound fields can be reproduced accurately by a single or a few virtual sources, and thus exhibit sparsity in space. Therefore, provided that the desired sound is sparse in space, the sparse solutions can be used to relax the restriction we mentioned in Section 2. We should note that the region of accurate translation in the multiple sources scenario becomes smaller than that in the single source scenario for the sparse solution. In addition, the sparse solution is less applicable to highly reverberant fields where the sparsity does not hold.
* Particle velocity contains the direction information of a sound field. For the sparse solutions, we can achieve more accurate sound field reproduction by controlling particle velocity than sound pressure. In other words, the particle-velocity-based solution outperforms the sound-pressure-based solution for sound field reproduction in the cases where there are a limited number of sources. Furthermore, particle velocity reflects the interaural time difference (ITD); therefore, the velocity vector is directly related to the localization predictor for human perception at low frequencies. Particle velocity is also one of the quantities that determine the sound intensity vector (the localization predictor at high frequencies), which reflects another human localization cue of interaural level difference (ILD). Therefore, the sparse solution of particle-velocity-based expansion is expected to provide an enhanced perceptual immersion for the listener.
* Sparse solutions extrapolate the sound field at the translated positions beyond the receiver region according to the sound field within the receiver region by the mixed-source expansion where we use multiple virtual sources to estimate the original source. Therefore, the more the virtual sources are used, the less error the mixed-source expansion has; however, the higher computation cost the system has. In addition, the particle-velocity-based solutions would be more computationally expensive than the sound-pressure-based solutions due to the the extra spherical harmonic decomposition of particle velocity.  
We examine the perceptual advantages of the proposed sparse solution of particle-velocity-based expansion by experiments in the next section.