## 3 Optimization Methods for Feedback Controllers  
How to optimize the controller is one of the key issues for the design of feedback ANC headphones. The basic principle of feedback controller optimization is to enhance the NR performance at low frequencies as much as possible with the restrictions that the feedback loop should have enough stability margins and that the noise enhancement at high frequencies should be limited. In this study, cascade biquad filters [14; 20] are used as feedback controllers, which is in accordance with most commercial applications of feedback ANC headphones. Specifically, two prototypes of parametric biquad filters with minimum phase frequency responses are used, whose effectiveness has been confirmed in a previous work [20]. The transfer functions are listed as follows:  
\[\text{H}_{0}(\text{z})=\frac{(1+\alpha\text{A})-2\cos\omega\text{z}^{-1}+(1- \alpha\text{A})\text{z}^{-2}}{(1+\alpha/\text{A})-2\cos\omega\text{z}^{-1}+(1- \alpha/\text{A})\text{z}^{-2}} \tag{6}\]\[\mathrm{H_{1}(z)}=\mathrm{A}\frac{\left((\mathrm{A}+1)+(\mathrm{A}-1)\cos\omega+2 \alpha\sqrt{\mathrm{A}}\right)-2((\mathrm{A}-1)+(\mathrm{A}+1)\cos\omega)z^{-1} +\left((\mathrm{A}+1)+(\mathrm{A}-1)\cos\omega-2\alpha\sqrt{\mathrm{A}}\right)z^ {-2}}{\left((\mathrm{A}+1)-(\mathrm{A}-1)\cos\omega+2\alpha\sqrt{\mathrm{A}} \right)+2((\mathrm{A}-1)-(\mathrm{A}+1)\cos\omega)z^{-1}+\left((\mathrm{A}+1)- (\mathrm{A}-1)\cos\omega-2\alpha\sqrt{\mathrm{A}}\right)z^{-2}} \tag{7}\]  
where  
\[\mathrm{A}=10^{6/\omega},\quad\omega=\frac{2\pi\mathrm{f}}{\mathrm{f}_{\mathrm{ s}}},\quad\alpha=\frac{\sin\omega}{2\mathrm{Q}} \tag{8}\]  
and \(\mathrm{f_{s}}\) is the sampling frequency. It could be seen that both (6) and (7) are depending on a parameter set g, f and Q. H\({}_{0}\)(z) in (6) is a peak/notch filter since the FR is enhanced by g dB at frequency f. The FR of H\({}_{1}\)(z) is enhanced by g dB within low frequency band and f can be seen as the cutoff frequency. Hence, H\({}_{1}\)(z) in (7) is called the low/high shelf filter. For both H\({}_{0}\)(z) and H\({}_{1}\)(z), Q plays the role of the quality factor. Some typical FRs of H\({}_{0}\)(z) and H\({}_{1}\)(z) with different combinations of g, f and Q can be found in [20]. The feedback controller is constructed with N biquad filters. Since the controller has a cascade structure, its FR should be the product of all individual FRs of each biquad filter. Thus, the FR of the feedback controller is depending on the following parameter vector:  
\[\mathbf{X}=\left[\mathrm{g}_{1},\mathrm{f}_{1},\mathrm{Q}_{1},\ldots,\mathrm{ g}_{\mathrm{N}},\mathrm{f}_{\mathrm{N}},\mathrm{Q}_{\mathrm{N}},\mathrm{Gain}\right] \tag{9}\]  
where \(\mathrm{g_{i}}\), \(\mathrm{f_{i}}\), \(\mathrm{Q}_{i}\) correspond to the i-th biquad filter and Gain is an extra gain of the controller with dB as its unit. The dimension of \(\mathbf{X}\) is \(3\mathrm{N}+1\). In order to find the optimal feedback controller according to the basic principle described above, an objective function is proposed in this work as follows:  
\[\mathrm{Obj}=\sum_{\mathrm{k}}^{\mathrm{f}_{\mathrm{start}}\leq \mathrm{f}_{\mathrm{k}}\leq\mathrm{f}_{\mathrm{stop}}}\mathrm{w_{k}}|\mathrm{ G}(\mathrm{f}_{\mathrm{k}})|^{2}+\mathrm{\xi}_{1}(\mathrm{u}(\mathrm{L}_{\mathrm{ min}}-\mathrm{L}(\mathbf{X}))+\mathrm{u}(\mathrm{\gamma}_{\mathrm{min}}- \mathrm{\gamma}(\mathbf{X}))) \tag{10}\] \[+\mathrm{\xi}_{2}\mathrm{u}\left(\max_{\mathrm{f}_{\mathrm{k}}> \mathrm{f}_{\mathrm{stop}}}(|\mathrm{G}(\mathrm{f}_{\mathrm{k}})|/\mathrm{ \delta}_{\mathrm{k}})-1\right)+\mathrm{\xi}_{3}[\mathrm{sum}(\mathrm{u}( \mathrm{l}\mathbf{b}-\mathbf{X}))+\mathrm{sum}(\mathrm{u}(\mathbf{X}-\mathrm{u }\mathbf{b}))]\]  
where \(\mathrm{f_{k}}(\mathrm{k}=1,\,\ldots,\mathrm{K})\) are discrete frequencies over the whole frequency band and \(\mathrm{u}(\mathrm{x})\) is the unit step function. \[\mathrm{u}(\mathrm{x})=\left\{ \begin{array}{cl}1&\mathrm{x}\geq 0\\ 0&\mathrm{x}<0\end{array} \right.