## 2 Dual-Zone Active Noise Control
\begin{array}{c}\mathbf{G}_{se}\\ \mathbf{G}_{sc}\end{array} \right]q_{s}+\left[ \begin{array}{c}\mathbf{G}_{pe} \\ 0\end{array} \right]q_{p}\right)^{\mathrm{H}}\left(\left[ \begin{array}{c} \mathbf{G}_{se}\\ \mathbf{G}_{sc}\end{array} \right]q_{s}+\left[ \begin{array}{c}\mathbf{G}_{pc} \\ 0\end{array} \right]q_{p}\right) \tag{4}\]  
Then, the least square solution to Equation (4) can be written as,  
\[\mathbf{q}_{s}^{\mathrm{\ }CIS}=-\left(\left[ \begin{array}{c}\mathbf{G}_{se} \\ \mathbf{G}_{sc}\end{array} \right]^{H}\left[ \begin{array}{c}\mathbf{G}_{se}\\ \mathbf{G}_{sc}\end{array} \right]+\lambda\mathbf{I}\right)^{-1}\left[ \begin{array} []{c}\mathbf{G}_{se}\\ \mathbf{G}_{sc}\end{array} \right]^{H}\left[ \begin{array}{c}\mathbf{G}_{pe} \\ 0\end{array} \right]q_{p} \tag{5}\]  
where \(\mathbf{I}\) is an identity matrix and \(\lambda\) is the regularization parameter. Equation (5) is the secondary source strength when the CLS algorithm is applied to dual-zone ANC. The cost function of the ACC algorithm does not contribute to noise reduction when it is applied to dual-zone ANC unless it is equalized [14; 15]. The optimal strength of the secondary source when it is applied in dual-zone ANC is given as,  
\[\mathbf{q}_{s}^{\mathrm{\ }ACC}=\frac{\mathrm{abs}\big{[}\mathbf{w}_{opt}{}^{H} \mathbf{G}_{se}{}^{H}\mathbf{P}_{cp}\big{]}}{\mathbf{w}_{opt}{}^{H}(\mathbf{ G}_{se}{}^{H}\mathbf{G}_{se})\mathbf{w}_{opt}}\exp\left(j\angle\big{[}\mathbf{w}_{opt}{}^{H} \mathbf{G}_{se}{}^{H}\mathbf{P}_{cp}\big{]}\right)\mathbf{w}_{opt} \tag{6}\]  
Figure 2: Block diagram of the dual-zone ANC system. where \(\mathbf{w}_{opt}\) equals the unit eigenvector, which corresponds to the largest eigenvalue of the matrix \(\left(\mathbf{G}_{sc}^{H}\mathbf{G}_{sc}+\lambda\mathbf{I}\right)^{-1}\!\left( \mathbf{G}_{sc}^{H}\mathbf{G}_{sc}\right)\). Additionally, \(abs\{\cdot\}\) and \(\angle\{\cdot\}\) represent the modulus, and argument of the elements, respectively. When SFR-ACC is applied to dual-zone ANC, the cost function can be rewritten as,  
\[ \begin{split}\min\lVert\mathbf{G}_{sc}\mathbf{q}_{s}+\mathbf{P} _{opt}\rVert_{2}^{2}\\ s.t.10\log C\geq 10\log C_{st}\end{split}  \tag{7}\]  
where \(C\) denotes the acoustic contrast, which can be defined as the ratio of acoustic potential energy between the control zone and the noncontrol zone. \(C_{st}\) is a constant that is smaller than \(C_{\text{max}}\), and \(C_{\text{max}}\) is the largest eigenvalue of the matrix \(\left(\mathbf{G}_{sc}^{H}\mathbf{G}_{sc}+\lambda\mathbf{I}\right)^{-1}\! \left(\mathbf{G}_{sc}^{H}\mathbf{G}_{sc}\right)\). Because this function is a non-convex problem, it has to be transferred into a convex problem [13] and then, can be solved by the CVX package [16]. Once the strength of the secondary source is obtained by Equations (5)-(7), the filter coefficient \(\mathbf{q}_{s}\), can be written as,  
\[\mathbf{q}_{s}\mathbf{{}^{\prime}}=\frac{\mathbf{q}_{s}}{\mathbf{X}} \tag{8}\]  
where \(\mathbf{X}\) is the reference signal of \(L\) frequencies. Therefore, the wideband control filter coefficients of the dual-zone ANC systems are obtained in Equation (8), based on different algorithms. Unlike sound field reproduction, evaluating the performance of the three algorithms in dual-zone ANC requires redefining new criteria. The first criterion refers to the noise reduction in the control zone, which can be expressed as,  
\[\mathbf{Re}=10\log\!\left(\frac{\lVert\mathbf{P}_{et}\rVert_{2}^{2}}{\lVert \mathbf{P}_{ep}\rVert_{2}^{2}}\right) \tag{9}\]  
and stands for the noise reduction performance of the secondary loudspeaker array in the control zone.