## 4 Derivation of the Stability Bound of the Feedback ANC System
As \(\omega\) approaches to \(\omega_{0}\), \(W_{o}(e^{i\omega})\) in (8) is approximated as  
\[W_{o}(e^{i\omega})\,\cong e^{i\omega_{0}\Delta}\frac{\sin\left[\frac{N}{2}( \omega-\omega_{0})\right]}{\frac{N}{2}(\omega-\omega_{0})}\,+\,e^{-j\left[ \frac{N-1}{2}(\omega+\omega_{0})+\omega_{0}\Delta\right]}\frac{\sin\left[ \frac{N}{2}(\omega+\omega_{0})\right]}{N\sin\left[\frac{1}{2}(\omega+\omega_{ 0})\right]} \tag{25}\]  
In the case of infinite filter length, \(N\rightarrow\infty\), (25) is further reduced to  
\[W_{o}(e^{i\omega})\,\cong e^{i\omega_{0}\Delta}\]  
Consequently, the overall frequency response of the optimum ANC filter is  
\[\rho=\frac{2}{N\lambda_{0}}\sin\frac{N\lambda_{0}}{2}-\cos\frac{N\lambda_{0}}{ 2}.\]0  
The stability bound is guaranteed when (26) is less than 1.0. Therefore, the stability bound is obtained as,  
\[\rho=\frac{2}{N\lambda_{0}}\sin\frac{N\lambda_{0}}{2}-\cos\frac{N\lambda_{0}}{ 2}.\]1  
which represents the theoretical stability bound in the ideal condition, i.e., pure delay secondary path model, pure tone primary noise, and infinite adaptive filter length. The theoretical limit of the stability bound is that phase difference of the secondary path model caused by the delay estimation error is less than \(\pi/3\). The stability bound in practical situation can be inferred from the proposed stability definition (4), which is composed of product of the adaptive filter response and the secondary path modeling error term. The filter response of feedback ANC reflects the noise's spectral characteristic and its frequency resolution depends on the filter length \(N\). Therefore, increase of the primary noise's bandwidth or decrease of number of the FIR filter taps results in decrease of the stability bound. For instance, a rotating machine noise has wide bandwidth from tens to hundreds of Hz that can be modeled with the 2 nd order autoregressive process [4]. The noise's bandwidth property results in wider frequency response than the sinc function-shaped response in (8). In practice, the secondary path includes a spectral distortion component as well as a pure delay component. The spectral distortion is mainly caused by the physical characteristics of the electro-acoustic coupling devices such as a microphone and a speaker. Within narrow frequency range of a single engine noise, the fluctuations of frequency response generally do not show more than several decibels as exemplified in [4; 14; 15; 16]. Critical distortion exists out of the operation range, in the sub bass frequency range lower than 50 Hz. Considering that the disturbance of the secondary path model can be controlled by the feedback mechanism and such moderate level of fluctuation can be accommodated in the adaptive filter update process, the utilization of the simple pure delay model is justifiable [2; 4]. When the spectral magnitude response of the secondary path is not correctly estimated, the error can cause convergence problem of the adaptive algorithm [7], in which the stability condition of the adaptive algorithm of the FBANC system is obtained as (28) by the averaging method combined with the frequency-domain technique. \[\rho=\frac{2}{N\lambda_{0}}\sin\frac{N\lambda_{0}}{2}-\cos\frac{N\lambda_{0}}{ 2}.\]2in which the spectral magnitude mismatch \(\varepsilon_{l}\) is expressed as \(\varepsilon_{l}=e^{-j(\frac{2\pi\lambda n}{L})}(S_{l}-\hat{S_{l}})\). The DFT elements \(S_{l}\), \(\hat{S_{l}^{*}}\), \(W_{l}\), and \(\varepsilon_{l}\) mean the \(l\)-th element of the L-point DFT of \(S(z)\), \(\hat{S}^{*}(z)\), \(W(z)\), and \(\varepsilon(z)\), respectively. The operation \(\left[\cdot\right]_{+}\) indicates that the causal part is taken from the inverse transform of the quantity in the square bracket [17]. In the next section, the proposed method and the result equation's validity was confirmed by comparing the results (23) and (24) with both the original Nyquist stability condition and simulation results.