# Time-Domain Sound Field Reproduction with Pressure and Particle Velocity Jointly Controlled
## 2 Review: Frequency-Domain Velocity-Assisted Sound Field Reproduction
### System Formulation  
Now, we discuss the problem of sound field reproduction in the time domain. Assuming the room impulse responses (RIRs) are pre-calibrated through measurements, the reproduced sound pressure and particle velocity at the \(m\)th (\(1\leqslant m\leqslant M\)) control point \(\mathbf{x}_{m}\), generated by \(L\) loudspeakers located at \(\mathbf{y}_{1},\ldots,\mathbf{y}_{L}\), can be expressed as  
\[p_{n}(\mathbf{x}_{m})=\sum_{\ell=1}^{L}s_{n}*q_{n}^{\ell}*h_{p,n}(\mathbf{x}_ {m}|\mathbf{y}_{\ell}) \tag{8}\]  
\[\mathbf{v}_{n}(\mathbf{x}_{m})=\sum_{\ell=1}^{L}s_{n}*q_{n}^{\ell}*\mathbf{h}_ {0,n}(\mathbf{x}_{m}|\mathbf{y}_{\ell}), \tag{9}\]  
where \(*\) denotes the linear convolution operator and \(n\) denotes the sampling index. \(s_{n}\) denotes the input sound signal, \(q_{n}^{\ell}\) denotes the control filter for the \(\ell\)th loudspeaker, \(h_{p,n}(\mathbf{x}_{m}|\mathbf{y}_{\ell})\) and \(\mathbf{h}_{v,n}(\mathbf{x}_{m}|\mathbf{y}_{\ell})\) denote the RIRs of pressure and velocity components, respectively, from the \(\ell\)-th loudspeaker to the \(m\)-th control point. Note that for particle velocity vector, we follow the convention to define three components along x, y and z axes, that is,  
\[\mathbf{v}_{n}(\mathbf{x}_{m})=[v_{n,x}(\mathbf{x}_{m}),v_{n,y}(\mathbf{x}_{m }),v_{n,z}(\mathbf{x}_{m})]^{T}\]  
and  
\[\mathbf{h}_{v,n}(\mathbf{x}_{m}|\mathbf{y}_{\ell})=[h_{v,n}^{x}(\mathbf{x}_{m }|\mathbf{y}_{\ell}),h_{v,n}^{y}(\mathbf{x}_{m}|\mathbf{y}_{\ell}),h_{v,n}^{z }(\mathbf{x}_{m}|\mathbf{y}_{\ell})]^{T}.\]  
Represent (8) and (9) in matrix form, we have  
\[p_{n}(\mathbf{x}_{m})=\sum_{\ell=1}^{L}\mathbf{q}_{\ell}^{T}\mathbf{H}_{p}( \mathbf{x}_{m}|\mathbf{y}_{\ell})\mathbf{s}_{n}=\mathbf{q}^{T}\mathbf{H}_{p}( \mathbf{x}_{m})\mathbf{s}_{n} \tag{10}\]  
\[\mathbf{v}_{n,c}(\mathbf{x}_{m})=\sum_{\ell=1}^{L}\mathbf{q}_{\ell}^{T} \mathbf{H}_{v,c}(\mathbf{x}_{m}|\mathbf{y}_{\ell})\mathbf{s}_{n}=\mathbf{q}^{ T}\mathbf{H}_{v,c}(\mathbf{x}_{m})\mathbf{s}_{n},c\in x,y,z, \tag{11}\]  
where given the \(K\)-tap long RIR and \(J\)-tap long control filter,  
\[\mathbf{s}_{n}=\Big{[}s_{n},s_{n-1},\ldots,s_{n-(K+J-2)}\Big{]}^{T},\]  
\[\mathbf{q}_{\ell}=\Big{[}q_{1}^{\ell},q_{2}^{\ell},\ldots,q_{J}^{\ell}\Big{]}^ {T},\]  
and the RIR matrices \(\mathbf{H}_{p}(\mathbf{x}_{m}|\mathbf{y}_{\ell})\) and \(\mathbf{H}_{v,c}(\mathbf{x}_{m}|\mathbf{y}_{\ell})\) are Toeplitz matrices of size \(J\times(K+J-1)\). The first row vector and the first column vector of \(\mathbf{H}_{p}(\mathbf{x}_{m}|\mathbf{y}_{\ell})\) are defined as \(\left[h_{p,1}(\mathbf{x}_{m}|\mathbf{y}_{\ell}),\ldots,h_{p,K}(\mathbf{x}_{m }|\mathbf{y}_{\ell}),\underbrace{0,\ldots,0}_{J-1}\right]\) and \(\left[h_{p,1}(\mathbf{x}_{m}|\mathbf{y}_{\ell}),\underbrace{0,\ldots,0}_{J- 1}\right]^{T},\) respectively. The same formulation is adopted for each particle velocity component. Then, we have  
\[\mathbf{q}=\Big{[}\mathbf{q}_{1}^{T},\mathbf{q}_{2}^{T},\ldots,\mathbf{q}_{ L}^{T}\Big{]}^{T},\]  
\[\mathbf{H}_{p}(\mathbf{x}_{m})=\Big{[}\mathbf{H}_{p}^{T}(\mathbf{x}_{m}| \mathbf{y}_{1}),\mathbf{H}_{p}^{T}(\mathbf{x}_{m}|\mathbf{y}_{2}),\ldots, \mathbf{H}_{p}^{T}(\mathbf{x}_{m}|\mathbf{y}_{L})\Big{]}^{T},\]  
\[\mathbf{v}_{n}(\mathbf{x}_{m})=\sum_{\ell=1}^{L}s_{n}*q_{n}^{\ell}*\mathbf{h}_ {0,n}(\mathbf{x}_{m}|\mathbf{y}_{\ell}), \tag{9}\]0  
which are of size \(LJ\times 1\), \(LJ\times(K+J-1)\), and \(LJ\times(K+JIn a similar way, the desired sound pressure and particle velocity at position \(\mathbf{x}_{m}\) can be expressed as  
\[\mathbf{v}_{n}(\mathbf{x}_{m})=\sum_{\ell=1}^{L}s_{n}*q_{n}^{\ell}*\mathbf{h}_ {0,n}(\mathbf{x}_{m}|\mathbf{y}_{\ell}), \tag{9}\]1 \[\mathbf{v}_{n}(\mathbf{x}_{m})=\sum_{\ell=1}^{L}s_{n}*q_{n}^{\ell}*\mathbf{h}_ {0,n}(\mathbf{x}_{m}|\mathbf{y}_{\ell}), \tag{9}\]2  
where \(\mathbf{g}_{p}(\mathbf{x}_{m})\) and \(\mathbf{g}_{v}(\mathbf{x}_{m})\) denote the RIR of pressure and particle velocity, respectively, from the virtual desired source to the \(m\)th control point.