## 4 Derivation of the Stability Bound of the Feedback ANC System
\begin{array}{l}\left|W(e^{j\omega})\right|\cdot\left|e^{-j\omega\Delta}-e^ {-j\omega\hat{\Delta}}\right|\omega=\omega_{180}\\ \cong\left|\sin c\left(\frac{N\lambda}{2}\right)+\delta\right|\cdot 2\left|\sin \frac{\omega_{180}(\Delta-\hat{\Delta})}{2}\right|=\left|\sin c\left(\frac{N \lambda}{2}\right)+\delta\right|\cdot 2\left|\sin\frac{\lambda+\omega_{0}}{2} \frac{\pi-\lambda(2\hat{\Delta}+N-1)}{\lambda+4\omega_{0}}\right|\\ =\left|\sin c\left(\frac{N\lambda}{2}\right)+\delta\right|\cdot 2\left|\sin \left(\frac{\pi}{2}-\frac{\lambda(2\hat{\Delta}+N-1)}{2}\right)\right|=2 \left|\sin c\left(\frac{N\lambda}{2}\right)+\delta\right|\cdot\left|\cos \left(\frac{\lambda(2\hat{\Delta}+N-1)}{2}\right)\right|.\end{array}  \tag{18}\]  
In Figure 5, examples of the sinc function and the cosine function terms of (18) are illustrated on \(\lambda\)-axis. Each first zero crossing points of the sinc function and the cosine function are located at \(\lambda=2\pi/N\) and \(\lambda=\pi/(2\,\Delta^{\ast}+N-1)\), respectively. From (18), the stability bound is expected to be around \(\lambda=2\pi/3(2\,\Delta^{\ast}+N-1)\), in which the cosine function becomes 0.5 and the sinc function is slightly smaller than 1.0. We simplify the sinc function and the cosine function into 1 st order linear equations by performing the Taylor expansion at the point \(\lambda_{0}=2\pi/3(2\,\Delta^{\ast}+N-1)\) as  
\[\begin{array}{l}\sin c\left(\frac{N\lambda}{2}\right)+\delta\cong\\ \sin c\left(\frac{N\lambda_{0}}{2}\right)+\frac{N}{2}\left\{\frac{\cos(\frac{N _{0}}{2})}{\frac{N_{0}}{2}}-\frac{\sin(\frac{N_{0}}{2})}{\left(\frac{N_{0}}{2 }\right)^{2}}\right\}(\lambda-\lambda_{0})+\delta\endThe proposed stability bound can be simplified into a quadratic equation by substituting (19) and (20) into (18) as  
\[\frac{2}{N\lambda_{0}}\bigg{[}\sin\frac{N\lambda_{0}}{2}+\frac{N\lambda_{0}\delta }{2}-\frac{N\rho}{2}(\lambda-\lambda_{0})\bigg{]}\cdot\bigg{[}1-\frac{\pi}{ \sqrt{3}\lambda_{0}}(\lambda-\lambda_{0})\bigg{]}<1.0, \tag{21}\]  
in which  
\[\rho=\frac{2}{N\lambda_{0}}\sin\frac{N\lambda_{0}}{2}-\cos\frac{N\lambda_{0}}{ 2}.\]  
This quadratic equation is satisfied when  
\[\lambda>\lambda_{0}\Bigg{[}1+\frac{\gamma}{\lambda_{0}}+\frac{\sqrt{3}}{2\pi} -\sqrt{\left(\frac{\gamma}{\lambda_{0}}-\frac{\sqrt{3}}{2\pi}\right)^{2}+ \frac{\sqrt{3}}{\pi\rho}}\Bigg{]}, \tag{22}\]  
in which  
\[\gamma=\frac{1}{N\rho}\bigg{\{}\sin\biggl{(}\frac{N\lambda_{0}}{2}\biggr{)}+ \delta\biggl{(}\frac{N\lambda_{0}}{2}\biggr{)}\bigg{\}}.\]  
From (16) and (22), the delay bound to guarantee the stability of the FBANC system is derived as  
\[\Delta<\hat{\Delta}+\frac{\pi-\lambda(2\hat{\Delta}+N-1)}{\lambda+\omega_{0}}, \tag{23}\]  
Similarly, for the case of \(\Delta<\hat{\Delta}\), the stability bound is  
\[\Delta>\hat{\Delta}-\frac{\pi+\lambda^{\prime}(2\hat{\Delta}+N-1)}{\lambda^{ \prime}+\omega_{0}}, \tag{24}\]  
in which  
\[\lambda^{\prime}<\lambda_{0}\Bigg{[}-1-\left(\frac{\gamma}{\lambda_{0}}+\frac {\sqrt{3}}{2\pi}\right)+\sqrt{\left(\frac{\gamma}{\lambda_{0}}-\frac{\sqrt{3} }{2\pi}\right)^{2}+\frac{\sqrt{3}}{\pi\rho}}\Bigg{]}.\]  
Figure 5: An example illustration of the sinc function and the cosine function terms of (18) on \(\lambda\)-axis. The obtained equation enables the direct computation of stability bound among delay error in secondary path model, frequency of primary noise, and the adaptive filter length. The interpretation of the result is that the stability bounds increase as primary noise's frequency \(\omega_{0}\) decreases and the adaptive filter length \(N\) increases.