# Time-Domain Sound Field Reproduction with Pressure and Particle Velocity Jointly Controlled
## 2 Review: Frequency-Domain Velocity-Assisted Sound Field Reproduction  
As a starting point, we briefly review the concept of frequency-domain velocity assisted sound field reproduction. At an arbitrary observation position \(\mathbf{x}\), the particle velocity \(\mathbf{v}(\mathbf{x},\omega)\) and the complex-valued sound pressure \(p(\mathbf{x},\omega)\) with time-dependency \(e^{i\omega t}\) have a relationship established by [PERSON]'s equation,  
\[-\
abla p(\mathbf{x},\omega)=i\omega\rho\mathbf{v}(\mathbf{x},\omega), \tag{1}\]  
where \(i\) is the imaginary unit, \(\omega=2\pi f\) denotes the angular frequency, \(\rho\) is the density of the propagation medium, and \(\
abla\) represents the gradient operation along the direction of the particle velocity vector. The components of the particle velocity vector can be defined either in the Cartesian coordinate, i.e., \(\mathbf{v}\equiv\{\mathbf{v}_{x},\mathbf{v}_{y},\mathbf{v}_{z}\}\), or the polar coordinate, such as \(\mathbf{v}\equiv\{\mathbf{v}_{\mathrm{rad}},\mathbf{v}_{\theta},\mathbf{v}_{ \phi}\}\) along the radial direction, the elevation and azimuth angular direction, respectively. In sound field reconstruction, we consider the reproduced sound generated by an array of \(L\) loudspeakers located positioned at \(\mathbf{y}_{l}\) with \(\ell=1,\dots,L\) surrounding the listening area. We define the acoustic transfer function (ATF) for the sound pressure component from the \(\ell\)th loudspeaker to the control point \(\mathbf{x}\) as \(T_{p}(\mathbf{x}|\mathbf{y}_{\ell},\omega)\). A special case is when the loudspeakers are modeled as point sources, and by assuming free-field propagation, the ATF is represented by the Green's function, that is  
\[T_{p}^{\text{free-field}}(\mathbf{x}|\mathbf{y}_{\ell},\omega)=\frac{1}{4\pi} \frac{e^{k\|\mathbf{y}_{\ell}-\mathbf{x}\|_{2}}}{\|\mathbf{y}_{\ell}-\mathbf{x }\|_{2}}, \tag{2}\]  
where \(k=\omega/c_{0}\) is the wave number, \(c_{0}\) denotes the sound speed, and \(\|\cdot\|_{2}\) denotes the \(L2\)-norm. Then, the reproduced sound pressure at position \(\mathbf{x}\) can be expressed as  
\[ \begin{split} p(\mathbf{x},\omega)&=\sum_{\ell=1} ^{L}T_{p}(\mathbf{x}|\mathbf{y}_{\ell},\omega)w_{\ell}(\omega)S(\omega)\\ &=\mathbf{t}_{p}^{T}(\mathbf{x},\omega)\mathbf{w}(\omega)S( \omega),\end{split}  \tag{3}\]  
where \(\mathbf{t}_{p}(\mathbf{x},\omega)=\left[T_{p}(\mathbf{x}|\mathbf{y}_{1}, \omega),\dots,T_{p}(\mathbf{x}|\mathbf{y}_{L},\omega)\right]^{T}\) is a column vector containing the ATFs for all the loudspeakers to the position \(\mathbf{x}\), \(\mathbf{w}(\omega)=\left[w_{1}(\omega),\dots,w_{L}(\omega)\right]^{T}\) is the vector consisting of the frequency-domain loudspeaker weights, and \(S(\omega)\) is the source audio signal. \((\cdot)^{T}\) denotes the transpose operator. Similarly, we can define the ATF for the particle velocity, i.e., \(\mathbf{t}_{v}(\mathbf{x}|\mathbf{y}_{\ell},\omega)\), which is a column vector of length 3 for each component of \(\mathbf{v}\), and has the following representation for the reproduced particle velocity:  
\[ \begin{split}\mathbf{v}(\mathbf{x},\omega)&=\sum_{ \ell=1}^{L}\mathbf{t}_{v}(\mathbf{x}|\mathbf{y}_{\ell},\omega)w_{\ell}(\omega )S(\omega)\\ &=\mathbf{T}_{v}^{T}(\mathbf{x},\omega)\mathbf{w}(\omega)S( \omega),\end{split}  \tag{4}\]  
where \(\mathbf{T}_{v}(\mathbf{x},\omega)=\left[\mathbf{t}_{v}(\mathbf{x}|\mathbf{y}_ {1},\omega),\dots,\mathbf{t}(\mathbf{x}|\mathbf{y}_{L},\omega)\right]^{T}\) is a matrix of size \(L\times 3\). Given that in sound field reproduction applications, we aim to reproduce the desired sound within a certain region of interest, by matching the sound pressure and particle velocity at multiple control points.