## 2 Three Traditional Sound Field Control Methods
### PM Method [3]  
As shown in Figure 1, it is assumed that \(M\) loudspeakers \(\overrightarrow{l_{1}}\), \(\overrightarrow{l_{2}},\ldots,\overrightarrow{l_{M}}\) (they also indicate the location of loudspeakers) are placed on the same ring, with \(s\) sampling points \(\overrightarrow{b_{1}},\overrightarrow{b_{2}},\ldots,\overrightarrow{b_{s}}\) (they also indicate the location of sampling points) in the bright zone and \(t\) sampling points \(\overrightarrow{d_{1}},\overrightarrow{d_{2}},\ldots,\overrightarrow{d_{t}}\) in the dark zone. We suppose that the sound pressure produced by the original source in the bright and the dark zone, respectively, are  
\[ \begin{array}{rl}\mathbf{p_{bo}}&=\Big{(}p(\overrightarrow{b_{1}}),p( \overrightarrow{b_{2}}),\ldots,p(\overrightarrow{b_{s}})\Big{)}^{T}\\ \mathbf{p_{do}}&=\delta\Big{(}p(\overrightarrow{d_{1}}),p(\overrightarrow{d_{2 }}),\ldots,p(\overrightarrow{d_{t}})\Big{)}^{T}\end{array}  \tag{3}\]  
where \(\delta\) is the amplitude modulation factor. Suppose that the sound pressure generated by \(M\) loudspeakers at point \(\overrightarrow{b}\) (\(\overrightarrow{b}\in\{\overrightarrow{b_{1}},\overrightarrow{b_{2}}, \ldots,\overrightarrow{b_{s}}\}\)) in the bright zone and at point \(\overrightarrow{d}\) (\(\overrightarrow{d}\in\{\overrightarrow{d_{1}},\overrightarrow{d_{2}}, \ldots,\overrightarrow{d_{t}}\}\)) in the dark zone are, respectively,  
\[ \begin{array}{rl}p_{br}&=\sum\limits_{n=1}^{M}h(\overrightarrow{l_{n}}, \overrightarrow{b})q_{n}\\ p_{dr}&=\sum\limits_{n=1}^{M}h(\overrightarrow{l_{n}},\overrightarrow{d})q _{n}\end{array}  \tag{4}\]  
The sound pressure transfer function is labeled \(h(\overrightarrow{l_{n}},\overrightarrow{b})\)[32] or \(h(\overrightarrow{l_{n}},\overrightarrow{d})\), where  
\[h(\overrightarrow{l_{n}},\overrightarrow{b})=\frac{e^{-ik|\overrightarrow{b }-\overrightarrow{l_{n}}|}}{4\pi|\overrightarrow{b}-\overrightarrow{l_{n}}|} \tag{5}\]  
Figure 1: Loudspeaker array and sound zone diagram. The strength of the \(n\)-th loudspeaker is labeled \(q_{n}\). Formula (4) can be expressed as matrices:  
\[\mathbf{p_{br}} =\mathbf{H_{b}}\mathbf{q} \tag{6}\] \[\mathbf{p_{dr}} =\mathbf{H_{d}}\mathbf{q}\]  
where  
\[\mathbf{H_{b}}=\left( \begin{array}{cccc}h(\overrightarrow{l_{1}},\overrightarrow {b_{1}})&h(\overrightarrow{l_{2}},\overrightarrow{b_{1}})&\ldots&h( \overrightarrow{l_{M}},\overrightarrow{b_{1}})\\ h(\overrightarrow{l_{1}},\overrightarrow{b_{2}})&h(\overrightarrow{l_{2} },\overrightarrow{b_{2}})&\ldots&h(\overrightarrow{l_{M}},\overrightarrow {b_{2}})\\ \cdots&\cdots&\cdots&\cdots\\ h(\overrightarrow{l_{1}},\overrightarrow{b_{s}})&h(\overrightarrow{l_{2} },\overrightarrow{b_{s}})&\ldots&h(\overrightarrow{l_{M}},\overrightarrow {b_{s}})\end{array} \right) \tag{7}\]  
\[\mathbf{H_{d}}=\left( \begin{array}{cccc}h(\overrightarrow{l_{1}},\overrightarrow {d_{1}})&h(\overrightarrow{l_{2}},\overrightarrow{d_{1}})&\ldots&h( \overrightarrow{l_{M}},\overrightarrow{d_{1}})\\ h(\overrightarrow{l_{1}},\overrightarrow{d_{2}})&h(\overrightarrow{l_{2} },\overrightarrow{d_{2}})&\ldots&h(\overrightarrow{l_{M}},\overrightarrow {d_{2}})\\ \cdots&\cdots&\cdots&\cdots\\ h(\overrightarrow{l_{1}},\overrightarrow{d_{t}})&h(\overrightarrow{l_{2} },\overrightarrow{d_{t}})&\ldots&h(\overrightarrow{l_{M}},\overrightarrow {d_{t}})\end{array} \right) \tag{8}\]  
\[\mathbf{q}=(q_{1},q_{2},\ldots,q_{M})^{T} \tag{9}\]  
If we want to recover the sound pressure produced by the original source in the bright zone, the equation constructed by the PM method is  
\[\mathbf{p_{bo}}=\mathbf{p_{br}} \tag{10}\]  
Formula (10) can also be expressed as  
\[\mathbf{p_{bo}}=\mathbf{H_{b}}\mathbf{q} \tag{11}\]  
The solution of Equation (11) is \(\mathbf{q}=\mathbf{H_{b}}^{-1}\mathbf{p_{bo}}\); the inverse of a matrix is labeled \(-1\). We call this method the PM method in the bright zone.