# Particle-Velocity-Based Mixed-Source Sound Field Translation for Binaural Reproduction
## 2 Problem Formulation  
Consider a listener within an acoustic environment, where the listening space of the environment is centered by the origin \(\mathbf{o}\). The listener is positioned at \(\mathbf{d}=(r,\theta,\phi)\), where \(\theta\in[0,\pi]\) and \(\phi\in[0,2\pi)\) in spherical coordinates, with respect to the origin. Let there be a total of \(K\) sound sources outside the listening space, with the \(\kappa\)th source located at \(\mathbf{b}_{\kappa}\). The free-field binaural sound perceived by the listener can be synthesized by filtering the source signals with the listener's head-related transfer functions (HRTFs), expressed as  
\[P_{\text{L,R}}(\mathbf{d},k)=\sum_{\kappa=1}^{K}H_{\text{L,R}}(k,\mathbf{b}_{\kappa}; \mathbf{d})s_{\kappa}(k), \tag{1}\]  
where \(P_{\text{L,R}}(\mathbf{d},k)\) is the sound pressure at the listener's left and right ear, \(H_{\text{L,R}}(k,\mathbf{b}_{\kappa};\mathbf{d})\) are the HRTFs between the \(\kappa\)th source and the listener's ears, \(s_{\kappa}(k)\) is the sound signal of the \(\kappa\)th source, \(k=2\pi f/c\) is the wave number, \(f\) is frequency, and \(c\) is the speed of sound propagation.  
We aim to record and reproduce the auditory experience of (1) for every possible listening position. The acoustic environment can be characterized using a spherical harmonic decomposition of the sound pressure, expressed as  
\[P(\mathbf{x},k)=\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\alpha_{nm}(k)j_{n}(k|\mathbf{x}|)Y_ {nm}(\hat{\mathbf{x}}), \tag{2}\]  
where \(\mathbf{x}\) is an arbitrary position in the space, \(|\cdot|\equiv r_{r}\,\hat{\cdot}\equiv(\theta,\phi)\), \(\alpha_{nm}(k)\) are the pressure coefficients which completely describe the source-free acoustic environment centered about \(\omega_{i},j_{n}(\cdot)\) is the \(n\)th order spherical Bessel function of the first kind, and \(Y_{nm}(\hat{\mathbf{x}})\) is the spherical harmonic of order \(n\) and degree \(m\), defined by  
\[Y_{nm}(\hat{\mathbf{x}})\triangleq\sqrt{\frac{(2n+1)}{4\pi}\frac{(n-m)!}{(n+m)!}}p_{ nm}(\cos\theta)e^{im\phi}, \tag{3}\]  
where \(P_{nm}(\cos\theta)\) is the associated Legendre function. In practice, the pressure coefficients describing the acoustic environment \(\{\alpha_{nm}(k)\}\) can be estimated from recordings of a higher order microphone or a microphone array [52]. We now consider an \(N\)th order receiver centered at \(\omega\), such as a spherical array [53] or planar array [54], that we use to estimate the acoustic environment for \(\{\alpha_{nm}(k)\}\) up to order \(N\). However, the limited order \(N\) introduces a spatial reproduction constraint that the sound field can only be accurately reproduced within the receiver region (i.e., \(|\mathbf{x}|\leq R\)), where \(R\) is the radius of the receiver region that is related by \(N=\lceil kR\rceil\). This constraint restricts the listener to move within the receiver region (i.e., \(|\mathbf{d}|\leq R\)). If the listener attempts to move beyond the receiver region, they would experience spectral distortions and a loss in perceptual immersion.  
The objective of this work is to perceptually enhance the performance of translation to \(|\mathbf{d}|>R\) for binaural reproduction from a mode-limited measurement so that a listener can move over a large space in a virtual reconstruction with a sustained perceptual immersion, through the proposed particle-velocity-based mixed-source expansion.