## 4 Derivation of the Stability Bound of the Feedback ANC System
\begin{array}{ll}W_{o}(e^{j\omega})&=\mathcal{F}\big{[}\cos[\omega_{0}(n+ \hat{\Delta})]\big{]}\times\mathcal{F}[w_{R,i}]\\ &=e^{-j[\frac{N-1}{2}(\omega-\omega_{0})-\omega_{0}\hat{\Delta}]}\frac{\sin \left[\frac{N}{2}(\omega-\omega_{0})\right]}{N\sin\left[\frac{1}{2}(\omega- \omega_{0})\right]}+e^{-j[\frac{N-1}{2}(\omega+\omega_{0})+\omega_{0}\hat{ \Delta}]}\frac{\sin\left[\frac{N}{2}(\omega+\omega_{0})\right]}{N\sin\left[ \frac{1}{2}(\omega+\omega_{0})\right]}\end{array} \]For the frequency range of the first term's main lobe, the magnitude response of the optimum ANC filter can be approximated as  
\[|W_{o}(e^{i\omega})|\ \cong\ \left|\sin\mathrm{c}\left[\frac{N}{2}(\omega-\omega_{0}) \right]+\delta\right|, \tag{9}\]  
in which \(\delta\) is a constant value to approximate the effect of second term of (8) on the magnitude response. The constant value is chosen as its potential maximum value existing in the right half of the main lobe as expressed as  
\[\delta=\frac{1}{N\sin[\omega_{0}]}. \tag{10}\]  
Since we assume that the filter length \(N\) is long enough to cover a single period of the primary noise \(N\cdot 2\pi/\omega_{0}\), the potential effect of the second term of (8) is negligibly small for the frequency range of the first term's main lobe. Therefore, the phase response \(\angle W(e^{i\omega})\) within the frequency range of the first term's main lobe, i.e., \(\omega_{0}-\frac{2\pi}{N}<\omega<\omega_{0}+\frac{2\pi}{N}\), can be approximated as  
\[\angle W_{o}(e^{i\omega})\ \cong\ -\frac{N-1}{2}\omega+\bigg{(}\hat{\Delta}+ \frac{N-1}{2}\bigg{)}\omega_{0},\quad\text{for}\ \omega_{0}-\frac{2\pi}{N}<\omega<\omega_{0}+\frac{2\pi}{N}. \tag{11}\]  
The second term of the open loop frequency response \([e^{-j\omega\Delta}-e^{-j\omega\hat{\Delta}}]\) can be derived as  
\[e^{-j\omega\Delta}-e^{-j\omega\hat{\Delta}}=2j\sin\frac{\omega(\Delta-\hat{ \Delta})}{2}\cdot e^{-j\frac{\omega(\Delta+\hat{\Delta})}{2}}. \tag{12}\]  
The second term's magnitude response \(|e^{-j\omega\Delta}-e^{-j\omega\hat{\Delta}}|\) can be viewed as a length of base of an isosceles triangle that has unit-length legs and a vertex angle \(\omega\), as shown in Figure 4, and can be expressed as  
\[|e^{-j\omega\Delta}-e^{-j\omega\hat{\Delta}}|\ =2\cdot\left|\sin\frac{\omega( \Delta-\hat{\Delta})}{2}\right|. \tag{13}\]  
Figure 3: An example of the magnitude response of the optimum ANC filter, \(\left|W_{o}(e^{i\omega})\right|\), for a single-tone primary noise. The phase response \(\angle[e^{-j\omega\Delta}-e^{-j\omega\hat{\Delta}}]\) is obtained as  
\[\angle[e^{-j\omega\Delta}-e^{-j\omega\hat{\Delta}}]=\left\{ \begin{array}{ll}- \frac{\omega(\Delta+\hat{\Delta})}{2}-\frac{\pi}{2}&\text{for}\ \Delta>\hat{\Delta},\\ -\frac{\omega(\Delta+\hat{\Delta})}{2}+\frac{\pi}{2}&\text{for}\ \Delta<\hat{\Delta}.\end{array} \right. \tag{14}\]  
For the case of \(\Delta>\hat{\Delta}\), the overall phase response can be obtained by summing the separate phase responses (11) and (14) as  
\[\angle W(e^{j\omega})[e^{-j\omega\Delta}-e^{-j\omega\hat{\Delta}}]=\angle W (e^{j\omega})+\angle[e^{-j\omega\Delta}-e^{-j\omega\hat{\Delta}}] \tag{15}\] \[=-\frac{\Delta+\hat{\Delta}+N-1}{2}\omega+\frac{2\hat{\Delta}+N- 1}{2}\omega_{0}-\frac{\pi}{2}\,.\]  
From (15), the frequency point \(\omega_{180}\), where the phase response \(\angle W(e^{j\omega})[e^{-j\omega\Delta}-e^{-j\omega\hat{\Delta}}]\) equals \(\pm\pi\), is obtained as  
\[\omega_{180}=\omega_{0}+\frac{\pi-(\Delta-\hat{\Delta})\omega_{0}}{\Delta+ \hat{\Delta}+N-1}=\omega_{0}+\lambda. \tag{16}\]  
The overall magnitude response can be obtained using (9) and (13) as following. \[\left|W(e^{j\omega})\right|\cdot\left|e^{-j\omega\Delta}-e^{-j \omega\hat{\Delta}}\right| \tag{17}\] \[\cong\left|\sin c\left(\frac{N}{2}[\omega-\omega_{0}]\right)+ \delta\right|\cdot 2\left|\sin\frac{\omega(\Delta-\hat{\Delta})}{2}\right|\,.\]  
By using (16) and (17), the proposed stability formula (4) is derived as  
\[