## 4 Derivation of the Stability Bound of the Feedback ANC System  
In this section, the closed-form stability bound equation is derived by utilizing the magnitude response and phase response of the two terms of the open-loop frequency response, i.e., \(W(e^{i\omega})\) and \([e^{-j\omega\Delta}-e^{-j\omega\Delta}]\). The frequency response of the adaptive filter \(W(e^{i\omega})\) can be specified as a simple formation in assumption that the primary noise \(d(n)\) is a single-tone sinusoid \(d(n)=\cos(\omega n+\phi)\), and the coefficients of the ANC filter \(W(e^{i\omega})\) converge to the optimum state [8]. Where, \(\omega\) is frequency of the single-tone primary noise \(d(n)\) and \(\phi\) is phase. The optimum filter response for the single tone noise \(d(n)\) can be obtained from (3) as  
\[W_{o}(e^{i\omega})=e^{i\omega\hat{\Delta}}, \tag{5}\]  
which represents a pure time-advance system obtained by making the overall frequency response become zero at the frequency of the primary noise as \(H(e^{i\omega})=0\). For the single-tone signal that has a fixed frequency \(\omega_{0}\), i.e., \(d(n)=\cos(\omega_{0}n+\phi)\), the frequency response at \(\pm\omega_{0}\) is \(W_{o}(e^{\pm j\omega_{0}})=e^{\pm j\omega_{0}\hat{\Delta}}\). Then, the frequency response to the single-tone \(d(n)\), \(W_{o}(e^{j\omega})\), can be considered as a summation of the two complex-valued phase delay terms at \(\pm\omega_{0}\) as  
\[W_{o}(e^{j\omega})=\frac{1}{2}\Big{[}\delta(\omega-\omega_{0})e^{j\omega_{0} \hat{\Delta}}+\delta(\omega+\omega_{0})e^{-j\omega_{0}\hat{\Delta}}\Big{]} \tag{6}\]  
Therefore, the time-domain filter coefficients \(w_{0}(n)\) are obtained as inverse Fourier transform of (6) as,  
\[ \begin{array}{ll}w_{o}(n)&=\frac{1}{2}\Big{[}e^{i\omega_{0}n}\cdot e^{j\omega _{0}\hat{\Delta}}+e^{-j\omega_{0}n}\cdot e^{-j\omega_{0}\hat{\Delta}}\Big{]}= \frac{1}{2}\Big{[}e^{i\omega_{0}(n+\hat{\Delta})}+e^{-j\omega_{0}(n+\hat{ \Delta})}\Big{]}\\ &=\cos(\omega_{0}n+\hat{\Delta}).\end{array} \]  
The coefficients of the N-tap FIR ANC filter for the optimum response is obtained by applying N-tap rectangular window \(w_{R}(n)\) as  
\[ \begin{array}{ll}w_{o}(n)&=\left\{\begin{array}{ll}\frac{2}{N}\cos[\omega_ {0}(n+\hat{\Delta})]&for\;n=0,\;1,\;\cdots,\;N-1\\ 0&\mbox{elsewhere}\end{array}\right.,\\ &=\cos[\omega_{0}(n+\hat{\Delta})]\cdot w_{R}(n).\end{array}  \tag{7}\]  
in which,  
\[w_{R}(n)=\left\{ \begin{array}{ll}\frac{2}{N}&for\;n=0,\;1,\;\cdots,\;N-1\\ 0&\mbox{elsewhere}.\end{array} \right.\]  
The overall frequency response of the optimum ANC filter \(W_{o}(e^{j\omega})\) can be expressed as the convolution of the discrete-time Fourier transform of the cosine function and that of the rectangular window as  
\[W_{o}\Big{(}e^{j\omega}\Big{)}=\mathcal{F}\big{[}\cos[\omega_{0}(n+\hat{ \Delta})]\big{]}*\mathcal{F}[w_{R,i}]\,.\]  
in which \(\mathcal{F}[\cdot]\) is the discrete-time Fourier transform operation and \(*\) is the convolution operator. The discrete-time Fourier transform of the cosine term is obtained as  
\[\mathcal{F}\big{[}\cos[\omega_{0}(n+\hat{\Delta})]\big{]}=\frac{e^{j\omega_{0 }\hat{\Delta}}}{2}\delta(\omega-\omega_{0})+\frac{e^{-j\omega_{0}\hat{\Delta}} }{2}\delta(\omega+\omega_{0})\]  
The discrete-time Fourier transform of the rectangular window function is obtained as  
\[ \begin{array}{ll}\mathcal{F}[w_{R,i}]=e^{-j\frac{N-1}{2}\omega}\frac{\frac{ 2}{N}\sin\left(\frac{N}{2}\omega\right)}{\sin\left(\frac{1}{2}\omega\right)} \end{array} \]  
The overall frequency response of the optimum ANC filter \(W_{o}(e^{j\omega})\) can be expressed as  
\[