# Particle-Velocity-Based Mixed-Source Sound Field Translation for Binaural Reproduction
## 1 Introduction
A similar approach was described in [34], which extends to support arbitrary listener movement; however, with multiple source directions and distances. In [35], the authors improve the spatial localization accuracy by additional spatial information obtained from multi-perspective Ambisonics recordings. Higher order Ambisonics signals are used in [36], which translates the sound field based on multi-directional decomposition that estimates source distances. Another parametric decomposition from a higher order Ambisonics signal has been proposed in [37], offering excellent localization performance even for strong spatial displacements. However, it also requires a prior knowledge of source distance information and an additional analysis stage of direction of arrival estimation. The sound field translation with very few measurements in a room is investigated in [38]. Recently, an alternative secondary source model for extrapolated virtual reproduction has been developed, where it sparsely expands the recording into the equivalent virtual environment using a mix of near-field and far-field virtual sources [39]. It has been shown that this method can relax the limitation and offer an improved perceptual experience through perceptual listening tests [40]. However, this method expands the sound field based on sound pressure, which is not an acoustic quantity directly linked with human perception of sound localization. Since [PERSON] first developed velocity theory of sound localization based on binaural phase cues for reproducing psychoacoustically optimum sound [41], particle velocity (vector), which is recently controlled in sound field reproduction systems [42, 43, 44], has been one of the objective metrics predicting sound direction. Thus, having particle velocity as the design criterion contributes to a more precise reconstruction of the interaural phase difference, which is crucial for human perception of sound directions and, therefore,contributes to improved localization of the reconstructed sound, especially for low frequencies (below 700 Hz) [45]. For example, a particle-velocity-based sound field reproduction method is proposed to improve the perception experience for listeners away from the center of the reproduction area [46]. Furthermore, the particle velocity-controlled sound field shows an accurate reproduction of sound intensity in the reproduction region [47; 48], where sound intensity is another important acoustic quantity for localization perception; however, it is most appropriate at high frequencies (above 500 Hz) [49; 50; 51]. In this paper, we propose a new mixed-source expansion based on particle velocity for sound field translation and synthesis. We aim to reproduce a spatial acoustic scene by matching particle velocity of the sound field over a region with a sparse distribution of virtual sound sources placed in both the near and far field. We will show that sparsely expanding the sound field in terms of particle velocity offers a more immersive perceptual experience at the translated positions, especially for source localization. The remainder of this paper is structured as follows. First, the sound field translation problem is formulated in Section 2. We have recently formulated continuous particle velocity expressions over space in the spherical harmonic domain [44], which contain the directivity information of a sound field over continuous spatial regions. In Section 3, we review the continuous particle velocity theory and develop the particle-velocity-based expansion by exploiting this theory in the mixed-source model. Section 4 translates the sound field and synthesizes binaural signals for translated listeners. In Section 5, we introduce two localization metrics, in addition to the relative pressure error, as evaluation criteria. With these criteria, we demonstrate the reproduction accuracy of the proposed method by comparing it with the state-of-the-art method, sound-pressure-based mixed-source expansion, through numerical simulations.