# Time-Domain Sound Field Reproduction with Pressure and Particle Velocity Jointly Controlled
## 4 Simulation
### Performance Evaluation Metrics  
Two performance evaluation metrics adopted are as follows:  
* The normalized mean squared error (NMSE) of reproduced sound intensity, which is defined as \[\epsilon=\frac{\frac{1}{NM}\sum_{m=1}^{M}\sum_{n=1}^{N}\|\mathbf{I}_{n}(\mathbf{ x}_{m})-\mathbf{I}_{n}^{d}(\mathbf{x}_{m})\|^{2}}{\frac{1}{NM}\sum_{m=1}^{M}\sum_{n=1}^{ N}\|\mathbf{I}_{n}^{d}(\mathbf{x}_{m})\|^{2}}\] (20) with the sound intensity vector calculated as follows [23]: \[\mathbf{I}_{n}(\mathbf{x}_{m})=p_{n}(\mathbf{x}_{m})\mathbf{v}_{n}(\mathbf{x} _{m}).\] (21) \begin{table}
\begin{tabular}{l} \hline
**INITIALIZATION**: \\
1. Calculate the spatial autocorrelation matrix \(\mathbf{R}\) and the spatial cross-correlation vector \(\mathbf{r}\) using (16) and (17) \\
2. Set the initial value of the filter \(\hat{\mathbf{q}}_{1}=\mathbf{0}_{LJ\times 1}\), the initial search direction vector \(\mathbf{d}_{1}=\mathbf{r}\), and the initial residual error \\
**e\({}_{1}=\mathbf{r}\)** \\
3. Set the number of iterations \(I\) \\
**LOOP**: for \(i=1,2,\dots,I\) \\
1. Determine the step \(\alpha_{i}\) of the \(i\)th iteration according to \(\alpha_{i}=\frac{\mathbf{e}_{i}^{T}\mathbf{e}_{i}}{\hat{\mathbf{d}}_{i}^{T} \mathbf{R}\mathbf{d}_{i}}\) \\
2. Update the estimates of the control filter \(\hat{\mathbf{q}}_{i+1}=\hat{\mathbf{q}}_{i}+\alpha_{i}\mathbf{d}_{i}\) and the residual error \(\mathbf{e}_{i+1}=\mathbf{e}_{i}-\alpha_{i}\mathbf{R}\mathbf{d}_{i}\) \\
3. Calculate the factor \(\beta_{i+1}\) that satisfies the conjugation condition, that is, \(\beta_{i+1}=\frac{\mathbf{e}_{i+1}^{T}\mathbf{e}_{i+1}}{\mathbf{e}_{i}^{T} \mathbf{e}_{i}}\) \\
4. Calculate the \(i+1\)th search direction vector \(\mathbf{d}_{i+1}=\mathbf{e}_{i+1}+\beta_{i+1}\mathbf{d}_{i}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Conjugate gradient algorithm for implementing the proposed time-domain sound field reproduction system. In Figure 2, we plot the NMSE of the reproduced intensity \(\epsilon\) varying with the tuning parameter \(\tau\). The case of \(\tau=0\) or \(\tau=1\) corresponds to controlling only the pressure or particle velocity, for which the reproduced sound intensity has a large error. The minimum intensity reproduction error occurs at \(\tau=0.5\), which is about 5 dB and 3 dB lower compared to the case of controlling only the pressure and the particle velocity, respectively. These results demonstrate that when the pressure and velocity are controlled with equal weights, which approximates sound-intensity control, the best reproduction performance can be obtained. As stated in literature, the sound intensity is closely related to source location perception [15], the proposed method also achieves the optimal reproduction results with sound intensity control under the regular loudspeaker array layout. We further investigate the validity of the proposed method in different reverberation conditions with varying control filter length. Figure 3 plots the NMSE of the reproduced sound intensity with the reverberation time \(RT60\) increasing from 0.2 s to 0.7 s while the control filter length is set as a constant value of \(J=400\). As the reverberation time \(RT60\) increase, the reproduction error also increases. In Figure 4, we change the control filter length, i.e., \(J=100,200,400,800,1000,1600\), to examine the reproduction performance. When \(J<800\), the error decreases rapidly with the increasing control filter length; however, when \(J\) is more than 800, the trend of the error decreases tends to be stable. Figure 2: The NMSE of the reproduced intensity at control points with different tuning parameter \(\tau\). The reverberation time \(RT60=200\) ms and the control filter length \(J=400\). We then investigate the reproduction performance within the entire reproduction region, especially the reproduction results at uncontrolled points.