## 6 Conclusions  
In this paper, the stability bound of feedback ANC was obtained as a form of a closed-form equation including physical domain design parameters such as noise frequency, length of the adaptive filter, and the mismatch of the secondary path estimation. The performed analysis and the obtained equation not only provide the direct calculation of the stability bound among the design parameters, but they also provide a clear understanding of the behavior of the adaptive filter response and the secondary path modeling error term on the feedback ANC system. Since the stability depends on how much the product of the adaptive filter response term and secondary path error term is close to Nyquist point (\(-1\), \(0\)), the wide band adaptive filter response due to the short filter length and the wide bandwidth of the target noise or spectral distortion of the filter response can cause the stability bound of the feedback ANC to become narrower than the bound of the case of pure delay secondary  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Stability Criterions/Simulations**} & \multicolumn{5}{c}{**Number of Adaptive Filter Taps N**} \\ \cline{2-5}  & **50** & **100** & **200** & **400** & **600** \\ \hline Nyquist Stability Criterion & 8 (1.00 ms) & 8 (1.00 ms) & 8 (1.00 ms) & 9 (1.125 ms) & 9 (1.125 ms) \\ Proposed Equation & 6 (0.750 ms) & 7 (0.875 ms) & 8 (1.00 ms) & 9 (1.125 ms) & 9 (1.125 ms) \\ Simulation w/Pure Delay Model & 7 (0.875 ms) & 8 (1.00 ms) & 8 (1.00 ms) & 9 (1.125 ms) & 9 (1.125 ms) \\ Simulation w/Measured Response & 7 (0.875 ms) & 8 (1.00 ms) & 8 (1.00 ms) & 9 (1.125 ms) & 9 (1.125 ms) \\ \hline \hline \end{tabular}  
* Frequency of primary noise \(f_{0}=150\) Hz.  
\end{table}
Table 1: The Stability Bounds of Delay error obtained for different number of adaptive filter taps.  
Figure 8: Comparison of stability bounds obtained with the analytic methods and simulations. Nyquist criterion (dotted line), proposed equation (solid line), simulation with the secondary path model measured from a commercial headphones (star), simulation with pure delay secondary path model (diamond).  
path model, pure tone primary noise. In the ideal condition of the pure delay secondary path model, the pure tone primary noise, and the infinite adaptive filter length, the stability error bound converges to \(\pi/3\) phase difference of the secondary path model. Although some approximations of the formula were used in the derivation process, the analysis result does not violate the original Nyquist stability condition. The performed analysis was verified by conducting simulations for both a pure delay secondary path model and a practical secondary path model. From the simulation results, when tested above 100 Hz primary noise frequency with the filter length \(N=200\), it is shown that the stability bound of the proposed equation agrees well with both the original Nyquist condition and the simulation results showing less than 0.250 ms difference. Such a closed-form, equation-based stability analysis can provide direct and fast computation of the parameter bounds for stability, so that it is expected to be very useful for the design and real-time operation of the feedback ANC system.  
This research was supported by the Korea National Research Foundation under Grant 20100786.  
[PERSON] conceived and developed the ideas behind the research. [PERSON] carried out the performance analysis and simulations, and wrote the paper under supervision of [PERSON]. [PERSON] supervised the research and finalized the paper.  
The authors declare no conflict of interest.