# A Review of Sound Field Control
## 3 Personal Audio Systems  
### Basic Theory  
Personal audio systems aim to focus sound to a listening zone without disturbing others, which can be realized by multi-zone sound field reproduction technologies. Multi-zone sound field reproduction creates two different kinds of zones: the bright zone where a desired sound field is reproduced, and the dark zone where the sound pressure level needs to be attenuated. Figure 1 is the schematic diagram of the multi-zone sound reproduction. Assuming that there are \(M\) discrete control points in the target area, the sound pressure generated by the loudspeaker array composed of \(L\) units in the target area can be expressed as:  
\[\mathbf{p}=\mathbf{G}\mathbf{q} \tag{2}\]  
where \(\mathbf{p}=[p(\tau_{1}),p(\tau_{2}),\cdots,p(\tau_{M})]^{\mathrm{T}}\) is the sound pressure vector at all control points, \(\mathbf{G}\) is the acoustic transfer function matrix, and \(\mathbf{q}=[q_{1},q_{2},\cdots,q_{L}]^{\mathrm{T}}\) is the weight vector. The matrix \(\mathbf{G}\) reads:  
\[\mathbf{G}=\begin{pmatrix}g(\tau_{1}|\tau_{s1})&\cdots&g(\tau_{1}|\tau_{sL})\\ \vdots&\ddots&\vdots\\ g(\tau_{M}|\tau_{s1})&\cdots&g(\tau_{M}|\tau_{sL})\end{pmatrix} \tag{3}\]  
where \(g(\tau_{m}|\tau_{sL})\) represents the transfer function from the \(l\)th loudspeaker to the \(m\)th control point.  
Figure 1: Schematic diagram of multi-zone sound reproduction.  
The primary evaluation metric for a personal audio system is the acoustic contrast, which is defined as the ratio of acoustic potential energy density between the bright and the dark zone:  
\[\max_{q}AC=\frac{\mathbf{q}^{H}\mathbf{R}_{b}\mathbf{q}}{\mathbf{q}^{H}\mathbf{R}_{d}\mathbf{q}} \tag{4}\]  
where \(\mathbf{R}_{b}=\mathbf{G}_{b}^{\mathrm{H}}\mathbf{G}_{b}\) and \(\mathbf{R}_{d}=\mathbf{G}_{d}^{\mathrm{H}}\mathbf{G}_{d}\) are the spatial correlation matrices of the bright and dark areas, respectively.  
The acoustic contrast control (ACC) method obtains the weight vector by maximizing the acoustic contrast. However, the main drawbacks of the ACC method are the lack of phase control and the uneven distribution of energy in the bright zone. To deal with these problems, pressure matching (PM) [38] and planarity control (PC) [39] were proposed. As mentioned above, the PM method minimizes the errors between the desired and reproduced sound pressure at the control points. In personal audio systems, the desired sound pressure in the bright zone is set according to the target sound field, and the desired sound pressure in the dark zone is equal to zero. The PC method uses superdirective beamforming for spatial filtering of the sound pressure at the control points and then maximizes the bright zone energy after spatial filtering with the dark zone energy constraints.  
Both the PM and PC methods can reproduce a directional sound field in the bright zone, but their acoustic contrasts are less than that of the ACC method. Considering that the main purpose of personal audio system is the higher acoustic contrast, the ACC method is more appropriate. For the lack of phase control and other issues, many studies have proposed improvements on robustness and sound quality based on the ACC method. These improvements are described in detail in the following subsection.