# Particle-Velocity-Based Mixed-Source Sound Field Translation for Binaural Reproduction
## 1 Introduction  
Sound field translation plays an important role in binaural reproduction systems [1], which enables a listener to move about an acoustic space with a sustained perceptual immersion. A typical example of its application is in virtual-reality reproductions of real-world experiences, where the listener is allowed to virtually explore by moving their body/head and they can experience a spatially accurate perception that is the same as in the real-world space [2]. In this paper, we propose a particle-velocity-based mixed-source expansion that can provide a wide range of sound field translation with immersive human perception. There have been various studies of sound field translation that allow for a listener to interact and move within a sound field reproduction, and they can be divided into two main categories: interpolation-based techniques [3] and extrapolation-based techniques [4]. The interpolation-based techniques aim to interpolate the sound field to listeners during reproduction using a grid of higher-order microphones (inpractice, higher-order microphones are designed from several pressure microphones equally spaced on a solid sphere) over space [5]. For example, a simple interpolation-based technique is shown in [6], wherethe room impulse response is interpolated to enable real-time auralizations with head rotation. The interpolation-based techniques usually suffer from a significant localization error [5,7] and comb-filtering spectral distortions [8], in particular when the sound source is nearer to one microphone than to another. Although the comb-filtering spectral distortions can be avoided using a parametric Ambisonics room impulse response interpolation system [9], it requires extra effort to compensate for varying interference. In addition, the interpolation assumes multiple distributed microphones or even arrays of microphones, which is a significant hardware limitation in many Augmented Reality/Virtual Reality (AR/VR) applications [3,10]. To enhance the performance of perceptual source localization for displaced listeners, a modified method is proposed, where the directional components of the microphone nearest to the listener are emphasized [11]. This method also interpolates the sound field to listeners and is subject to the fundamental limitation. Methods that alleviate the fundamental limitation have been investigated; however, they usually require additional source localization and separation of direct sound field components [12, 13, 14]. On the other hand, the extrapolation-based techniques spherically extend the translation from the recording microphone to the nearest source, which can be achieved by Ambisonics [15,16], discrete virtual sources [17], plane-wave expansion [18, 19, 20], and the equivalent source method [21]. A comparison between different extrapolation-based techniques is given in [22]. To improve the performance of translation and source localization, especially for off-center positions, optimized decoding methods such as max-\(r_{\text{E}}\) (meaning max energy vector magnitude) decoding [23] and near-field-compensated decoding [24] have been proposed. However, most of the extrapolated-based techniques are limited by the order of the recordings, which is determined by both the upper frequency band and the microphone radius [25,26]. Consequently, for a low-order recording, the range of translation is significantly limited. Attempts to move beyond this limited range, even after extrapolation, result in spectral distortions [27, 28, 29], degraded source localization [22,30], and a poor perceptual listening experience. Parametric models have been shown to provide an efficient way to describe sound scenes [14,31,32], which can be used in binaural reproduction. To overcome the limitation of translation range discussed above, a parametric model is built to describe first-order Ambisonics recordings for translation with known information of a listener's position and source distances [33].