## 2 Dual-Zone Active Noise Control  
The geometry of the dual-zone ANC system is shown in Figure 1. One zone is defined as the control zone and the other zone is defined as the noncontrol zone. The control zone is the quiet zone and the noncontrol zone's sound power remains unchanged when ANC is on. Therefore, the dual-zone ANC system in this paper is not aimed at realizing ANC in both zones but at realizing a directional ANC. The primary source is located on the negative half of the \(y\)-axis. The secondary source array consisting of \(N_{s}\) loudspeakers is evenly distributed on the \(x\)-axis. The control zone and the noncontrol zone have \(N_{e}\) and \(N_{c}\) uniformly distributed microphones, respectively, and are symmetrically distributed in the \(XOY\) plane. The distance between the primary source and the origin is \(L_{ps}\). The length of the loudspeaker array is \(L_{s}\) with spacing \(d_{ss}\). The distance between the center of the control points and the origin is \(R_{cc}\), and the distance between the control points is \(d_{e}\). The noncontrol points are identical in configuration to the control points with symmetric coordinate points on the \(y\)-axis. The control zone and the noncontrol zone are located in the \(-\theta\), and \(\theta\) directions, respectively, relative to the center. The block diagram of the dual-zone ANC system is shown in Figure 2. When the primary source and the secondary source are working together, the broadband sound pressure at the control points can be represented as,  
\[\mathbf{P}_{el}=\mathbf{P}_{ep}+\mathbf{P}_{es}=\mathbf{G}_{pe}d_{p}+\mathbf{G }_{sc}d_{s} \tag{1}\]  
Figure 1: The geometry of the dual-zone active noise control (ANC) system. and the sound pressure at the noncontrol points can be written as,  
\[\mathbf{P}_{ct}=\mathbf{P}_{cp}+\mathbf{P}_{cs}=\mathbf{G}_{pc}q_{p}+\mathbf{G}_ {sc}q_{s} \tag{2}\]  
where \(\mathbf{P}_{cp}\) and \(\mathbf{P}_{cp}\) are the sound pressure of the control points and the noncontrol points produced by the primary source, respectively. \(\mathbf{P}_{cp}\) is an \(N_{e}\times L\) vector, and \(\mathbf{P}_{cp}\) is an \(N_{c}\times L\) vector, where \(L\) is the number of frequencies. \(\mathbf{P}_{cs}\) and \(\mathbf{P}_{cs}\) are the sound pressure of the control points and noncontrol points produced by the secondary source, respectively. \(\mathbf{P}_{cs}\) is an \(N_{e}\times L\) vector, and \(\mathbf{P}_{cs}\) is an \(N_{c}\times L\) vector. \(\mathbf{G}_{pc}\) and \(\mathbf{G}_{pc}\) are the transfer functions between the primary source and the control points and noncontrol points, points, respectively. \(\mathbf{G}_{pc}\) is an \((N_{e}\times 1)\times L\) matrix, and \(\mathbf{G}_{pc}\) is an \((N_{c}\times 1)\times L\) matrix. \(\mathbf{G}_{sc}\) and \(\mathbf{G}_{sc}\) are the transfer functions between the secondary source and the control points and noncontrol points, respectively. \(\mathbf{G}_{se}\) is an \((N_{e}\times N_{s})\times L\) matrix, and \(\mathbf{G}_{sc}\) is an \((N_{c}\times N_{s})\times L\) matrix. In addition, \(q_{p}\) and \(q_{s}\) are the strength of the primary source and the secondary source, respectively. The key concept of the dual-zone ANC is how to solve the strength of the secondary source. There are two targets: One is to minimize the total energy in the control zone, corresponding to noise reduction; the other is to minimize the input energy to the noncontrol zone produced by the secondary sources, corresponding to forming directional outputs. When the constraint LS algorithm is applied to dual-zone ANC, the total pressure of the control zone can be represented as in (1). However, the total pressure of the noncontrol zone is expressed as,  
\[\mathbf{P}_{ct}=\mathbf{P}_{cp}+\mathbf{P}_{cs}=\mathbf{G}_{pc}q_{p} \tag{3}\]  
where the response of the secondary loudspeaker array at the noncontrol zone is constrained to zero. Therefore, the cost function of the dual-zone ANC can be written as,  
\[\mathbf{J}_{\mathrm{CLS}}=\left(\left[